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Revision 3616 - (download) (as text) (annotate)
Tue Sep 13 16:24:21 2005 UTC (14 years, 3 months ago) by dpvc
File size: 175489 byte(s)
Fixed problem with recognizing MINF in NUM_CMP.

    1 # This file is PGanswermacros.pl
    2 # This includes the subroutines for the ANS macros, that
    3 # is, macros allowing a more flexible answer checking
    4 ####################################################################
    5 # Copyright @ 1995-2000 University of Rochester
    6 # All Rights Reserved
    7 ####################################################################
    8 #$Id$
    9 
   10 =head1 NAME
   11 
   12   PGanswermacros.pl -- located in the courseScripts directory
   13 
   14 =head1 SYNPOSIS
   15 
   16   Number Answer Evaluators:
   17     num_cmp() --  uses an input hash to determine parameters
   18 
   19     std_num_cmp(), std_num_cmp_list(), std_num_cmp_abs, std_num_cmp_abs_list()
   20     frac_num_cmp(), frac_num_cmp_list(), frac_num_cmp_abs, frac_num_cmp_abs_list()
   21     arith_num_cmp(), arith_num_cmp_list(), arith_num_cmp_abs, arith_num_cmp_abs_list()
   22     strict_num_cmp(), strict_num_cmp_list(), strict_num_cmp_abs, strict_num_cmp_abs_list()
   23     numerical_compare_with_units()  --  requires units as part of the answer
   24     std_num_str_cmp() --  also accepts a set of strings as possible answers
   25 
   26   Function Answer Evaluators:
   27     fun_cmp() --  uses an input hash to determine parameters
   28 
   29     function_cmp(), function_cmp_abs()
   30     function_cmp_up_to_constant(), function_cmp_up_to_constant_abs()
   31     multivar_function_cmp()
   32 
   33   String Answer Evaluators:
   34     str_cmp() --  uses an input hash to determine parameters
   35 
   36     std_str_cmp(), std_str_cmp_list(), std_cs_str_cmp(), std_cs_str_cmp_list()
   37     strict_str_cmp(), strict_str_cmp_list()
   38     ordered_str_cmp(), ordered_str_cmp_list(), ordered_cs_str_cmp(), ordered_cs_str_cmp_list()
   39     unordered_str_cmp(), unordered_str_cmp_list(), unordered_cs_str_cmp(), unordered_cs_str_cmp_list()
   40 
   41   Miscellaneous Answer Evaluators:
   42     checkbox_cmp()
   43     radio_cmp()
   44 
   45 =cut
   46 
   47 =head1 DESCRIPTION
   48 
   49 This file adds subroutines which create "answer evaluators" for checking
   50 answers. Each answer evaluator accepts a single input from a student answer,
   51 checks it and creates an output hash %ans_hash with seven or eight entries
   52 (the preview_latex_string is optional). The output hash is now being created
   53 with the AnswerHash package "class", which is located at the end of this file.
   54 This class is currently just a wrapper for the hash, but this might change in
   55 the future as new capabilities are added.
   56 
   57           score     =>  $correctQ,
   58           correct_ans   =>  $originalCorrEqn,
   59           student_ans   =>  $modified_student_ans
   60           original_student_ans  =>  $original_student_answer,
   61           ans_message   =>  $PGanswerMessage,
   62           type      =>  'typeString',
   63           preview_text_string =>  $preview_text_string,
   64           preview_latex_string  =>  $preview_latex_string
   65 
   66 
   67   $ans_hash{score}      --  a number between 0 and 1 indicating
   68                     whether the answer is correct. Fractions
   69                     allow the implementation of partial
   70                     credit for incorrect answers.
   71   $ans_hash{correct_ans}      --  The correct answer, as supplied by the
   72                     instructor and then formatted. This can
   73                     be viewed by the student after the answer date.
   74   $ans_hash{student_ans}      --  This is the student answer, after reformatting;
   75                     for example the answer might be forced
   76                     to capital letters for comparison with
   77                     the instructors answer. For a numerical
   78                     answer, it gives the evaluated answer.
   79                     This is displayed in the section reporting
   80                     the results of checking the student answers.
   81   $ans_hash{original_student_ans}   --  This is the original student answer. This is displayed
   82                     on the preview page and may be used for sticky answers.
   83   $ans_hash{ans_message}      --  Any error message, or hint provided by the answer evaluator.
   84                     This is also displayed in the section reporting
   85                     the results of checking the student answers.
   86   $ans_hash{type}       --  A string indicating the type of answer evaluator. This
   87                     helps in preprocessing the student answer for errors.
   88                     Some examples:
   89                       'number_with_units'
   90                       'function'
   91                       'frac_number'
   92                       'arith_number'
   93   $ans_hash{preview_text_string}    --  This typically shows how the student answer was parsed. It is
   94                     displayed on the preview page. For a student answer of 2sin(3x)
   95                     this would be 2*sin(3*x). For string answers it is typically the
   96                     same as $ans_hash{student_ans}.
   97   $ans_hash{preview_latex_string}   --  THIS IS OPTIONAL. This is latex version of the student answer
   98                     which is used to show a typeset view on the answer on the preview
   99                     page. For a student answer of 2/3, this would be \frac{2}{3}.
  100 
  101 Technical note: the routines in this file are not actually answer evaluators. Instead, they create
  102 answer evaluators. An answer evaluator is an anonymous subroutine, referenced by a named scalar. The
  103 routines in this file build the subroutine and return a reference to it. Later, when the student
  104 actually enters an answer, the problem processor feeds that answer to the referenced subroutine, which
  105 evaluates it and returns a score (usually 0 or 1). For most users, this distinction is unimportant, but
  106 if you plan on writing your own answer evaluators, you should understand this point.
  107 
  108 =cut
  109 
  110 BEGIN {
  111   be_strict(); # an alias for use strict.  This means that all global variable must contain main:: as a prefix.
  112 }
  113 
  114 
  115 my ($BR                   ,   # convenient localizations.
  116   $PAR                  ,
  117   $numRelPercentTolDefault    ,
  118   $numZeroLevelDefault      ,
  119   $numZeroLevelTolDefault     ,
  120   $numAbsTolDefault         ,
  121   $numFormatDefault         ,
  122   $functRelPercentTolDefault      ,
  123   $functZeroLevelDefault      ,
  124   $functZeroLevelTolDefault   ,
  125   $functAbsTolDefault         ,
  126   $functNumOfPoints         ,
  127   $functVarDefault          ,
  128   $functLLimitDefault         ,
  129   $functULimitDefault         ,
  130   $functMaxConstantOfIntegration  ,
  131   $CA                             ,
  132   $rh_envir                       ,
  133   $useBaseTenLog                  ,
  134   $inputs_ref                     ,
  135   $QUESTIONNAIRE_ANSWERS          ,
  136   $user_context,
  137   $Context,
  138 );
  139 
  140 
  141 
  142 
  143 sub _PGanswermacros_init {
  144 
  145      $BR                              =   main::PG_restricted_eval(q!$main::BR!);
  146      $PAR                             =   main::PG_restricted_eval(q!$main::PAR!);
  147 
  148     # import defaults
  149     # these are now imported from the %envir variable
  150      $numRelPercentTolDefault     = main::PG_restricted_eval(q!$main::numRelPercentTolDefault!);
  151      $numZeroLevelDefault       = main::PG_restricted_eval(q!$main::numZeroLevelDefault!);
  152      $numZeroLevelTolDefault      = main::PG_restricted_eval(q!$main::numZeroLevelTolDefault!);
  153      $numAbsTolDefault            = main::PG_restricted_eval(q!$main::numAbsTolDefault!);
  154      $numFormatDefault            = main::PG_restricted_eval(q!$main::numFormatDefault!);
  155      $functRelPercentTolDefault     = main::PG_restricted_eval(q!$main::functRelPercentTolDefault!);
  156      $functZeroLevelDefault       = main::PG_restricted_eval(q!$main::functZeroLevelDefault!);
  157      $functZeroLevelTolDefault      = main::PG_restricted_eval(q!$main::functZeroLevelTolDefault!);
  158      $functAbsTolDefault        = main::PG_restricted_eval(q!$main::functAbsTolDefault!);
  159      $functNumOfPoints            = main::PG_restricted_eval(q!$main::functNumOfPoints!);
  160      $functVarDefault           = main::PG_restricted_eval(q!$main::functVarDefault!);
  161      $functLLimitDefault        = main::PG_restricted_eval(q!$main::functLLimitDefault!);
  162      $functULimitDefault        = main::PG_restricted_eval(q!$main::functULimitDefault!);
  163      $functMaxConstantOfIntegration   = main::PG_restricted_eval(q!$main::functMaxConstantOfIntegration!);
  164      $rh_envir                          =   main::PG_restricted_eval(q!\%main::envir!);
  165      $useBaseTenLog                     =   main::PG_restricted_eval(q!$main::useBaseTenLog!);
  166      $inputs_ref                        =   main::PG_restricted_eval(q!$main::inputs_ref!);
  167      $QUESTIONNAIRE_ANSWERS       =   '';
  168 
  169      if (!main::PG_restricted_eval(q!$main::useOldAnswerMacros!)) {
  170        $user_context = main::PG_restricted_eval(q!\%context!);
  171        $Context = sub {Parser::Context->current($user_context,@_)};
  172      }
  173 }
  174 
  175 
  176 
  177 ##########################################################################
  178 
  179 #Note   use $rh_envir to read environment variables
  180 
  181 ##########################################################################
  182 ## Number answer evaluators
  183 
  184 =head2 Number Answer Evaluators
  185 
  186 Number answer evaluators take in a numerical answer, compare it to the correct answer,
  187 and return a score. In addition, they can choose to accept or reject an answer based on
  188 its format, closeness to the correct answer, and other criteria. There are two types
  189 of numerical answer evaluators: num_cmp(), which takes a hash of named options as parameters,
  190 and the "mode"_num_cmp() variety, which use different functions to access different sets of
  191 options. In addition, there is the special case of std_num_str_cmp(), which can evaluate
  192 both numbers and strings.
  193 
  194 Numerical Comparison Options
  195 
  196   correctAnswer   --  This is the correct answer that the student answer will
  197             be compared to. However, this does not mean that the
  198             student answer must match this exactly. How close the
  199             student answer must be is determined by the other
  200             options, especially tolerance and format.
  201 
  202   tolerance   --  These options determine how close the student answer
  203             must be to the correct answer to qualify. There are two
  204             types of tolerance: relative and absolute. Relative
  205             tolerances are given in percentages. A relative
  206             tolerance of 1 indicates that the student answer must
  207             be within 1% of the correct answer to qualify as correct.
  208             In other words, a student answer is correct when
  209               abs(studentAnswer - correctAnswer) <= abs(.01*relpercentTol*correctAnswer)
  210             Using absolute tolerance, the student answer must be a
  211             fixed distance from the correct answer to qualify.
  212             For example, an absolute tolerance of 5 means that any
  213             number which is +-5 of the correct answer qualifies as correct.
  214               Final (rarely used) tolerance options are zeroLevel
  215             and zeroLevelTol, used in conjunction with relative
  216             tolerance. if correctAnswer has absolute value less than
  217             or equal to zeroLevel, then the student answer must be,
  218             in absolute terms, within zeroLevelTol of correctAnswer, i.e.,
  219               abs(studentAnswer - correctAnswer) <= zeroLevelTol.
  220             In other words, if the correct answer is very near zero,
  221             an absolute tolerance will be used. One must do this to
  222             handle floating point answers very near zero, because of
  223             the inaccuracy of floating point arithmetic. However, the
  224             default values are almost always adequate.
  225 
  226   mode      --  This determines the allowable methods for entering an
  227             answer. Answers which do not meet this requirement will
  228             be graded as incorrect, regardless of their numerical
  229             value. The recognized modes are:
  230               'std' (default) --  allows any expression which evaluates
  231                         to a number, including those using
  232                         elementary functions like sin() and
  233                         exp(), as well as the operations of
  234                         arithmetic (+, -, *, /, ^)
  235               'strict'  --  only decimal numbers are allowed
  236               'frac'    --  whole numbers and fractions are allowed
  237               'arith'   --  arithmetic expressions are allowed, but
  238                         no functions
  239             Note that all modes allow the use of "pi" and "e" as
  240             constants, and also the use of "E" to represent scientific
  241             notation.
  242 
  243   format      --  The format to use when displaying the correct and
  244             submitted answers. This has no effect on how answers are
  245             evaluated; it is only for cosmetic purposes. The
  246             formatting syntax is the same as Perl uses for the sprintf()
  247             function. Format strings are of the form '%m.nx' or '%m.nx#',
  248             where m and n are described below, and x is a formatter.
  249               Esentially, m is the minimum length of the field
  250             (make this negative to left-justify). Note that the decimal
  251             point counts as a character when determining the field width.
  252             If m begins with a zero, the number will be padded with zeros
  253             instead of spaces to fit the field.
  254               The precision specifier (n) works differently, depending
  255             on which formatter you are using. For d, i, o, u, x and X
  256             formatters (non-floating point formatters), n is the minimum
  257             number of digits to display. For e and f, it is the number of
  258             digits that appear after the decimal point (extra digits will
  259             be rounded; insufficient digits will be padded with spaces--see
  260             '#' below). For g, it is the number of significant digits to
  261             display.
  262               The full list of formatters can be found in the manpage
  263             for printf(3), or by typing "perldoc -f sprintf" at a
  264             terminal prompt. The following is a brief summary of the
  265             most frequent formatters:
  266               d --  decimal number
  267               ld  --  long decimal number
  268               u --  unsigned decimal number
  269               lu  --  long unsigned decimal number
  270               x --  hexadecimal number
  271               o --  octal number
  272               e --  floating point number in scientific notation
  273               f --  floating point number
  274               g --  either e or f, whichever takes less space
  275             Technically, g will use e if the exponent is less than -4 or
  276             greater than or equal to the precision. Trailing zeros are
  277             removed in this mode.
  278               If the format string ends in '#', trailing zeros will be
  279             removed in the decimal part. Note that this is not a standard
  280             syntax; it is handled internally by WeBWorK and not by Perl
  281             (although this should not be a concern to end users).
  282             The default format is '%0.5f#', which displays as a floating
  283             point number with 5 digits of precision and no trailing zeros.
  284             Other useful format strings might be '%0.2f' for displaying
  285             dollar amounts, or '%010d' to display an integer with leading
  286             zeros. Setting format to an empty string ( '' ) means no
  287             formatting will be used; this will show 'arbitrary' precision
  288             floating points.
  289 
  290 Default Values (As of 7/24/2000) (Option -- Variable Name -- Value)
  291 
  292   Format          --  $numFormatDefault   --  "%0.5f#"
  293   Relative Tolerance    --  $numRelPercentTolDefault  --  .1
  294   Absolute Tolerance    --  $numAbsTolDefault   --  .001
  295   Zero Level        --  $numZeroLevelDefault    --  1E-14
  296   Zero Level Tolerance  --  $numZeroLevelTolDefault   --  1E-12
  297 
  298 =cut
  299 
  300 
  301 =head3 num_cmp()
  302 
  303 Compares a number or a list of numbers, using a named hash of options to set
  304 parameters. This can make for more readable code than using the "mode"_num_cmp()
  305 style, but some people find one or the other easier to remember.
  306 
  307 ANS( num_cmp( answer or answer_array_ref, options_hash ) );
  308 
  309   1. the correct answer, or a reference to an array of correct answers
  310   2. a hash with the following keys (all optional):
  311     mode      --  'std' (default) (allows any expression evaluating to
  312                 a number)
  313               'strict' (only numbers are allowed)
  314               'frac' (fractions are allowed)
  315               'arith' (arithmetic expressions allowed)
  316     format      --  '%0.5f#' (default); defines formatting for the
  317                 correct answer
  318     tol       --  an absolute tolerance, or
  319     relTol      --  a relative tolerance
  320     units     --  the units to use for the answer(s)
  321     strings     --  a reference to an array of strings which are valid
  322                 answers (works like std_num_str_cmp() )
  323     zeroLevel   --  if the correct answer is this close to zero,
  324                  then zeroLevelTol applies
  325     zeroLevelTol  --  absolute tolerance to allow when answer is close
  326                  to zero
  327 
  328     debug     --  if set to 1, provides verbose listing of
  329                 hash entries throughout fliters.
  330 
  331   Returns an answer evaluator, or (if given a reference to an array of
  332   answers), a list of answer evaluators. Note that a reference to an array of
  333   answers results is just a shortcut for writing a separate <code>num_cmp()</code> for each
  334   answer.
  335 
  336 EXAMPLES:
  337 
  338   num_cmp( 5 )          --  correct answer is 5, using defaults
  339                   for all options
  340   num_cmp( [5,6,7] )        --  correct answers are 5, 6, and 7,
  341                   using defaults for all options
  342   num_cmp( 5, mode => 'strict' )  --  correct answer is 5, mode is strict
  343   num_cmp( [5,6], relTol => 5 ) --  correct answers are 5 and 6,
  344                     both with 5% relative tolerance
  345   num_cmp( 6, strings => ["Inf", "Minf", "NaN"] )
  346                   --  correct answer is 6, "Inf", "Minf",
  347                    and "NaN" recognized as valid, but
  348                    incorrect answers.
  349   num_cmp( "-INF", strings => ["INF", "-INF"] )
  350                   --  correct answer is "-INF", "INF" and
  351                    numerical expressions recognized as valid,
  352                    but incorrect answers.
  353 
  354 
  355 =cut
  356 
  357 sub num_cmp {
  358   my $correctAnswer = shift @_;
  359   $CA = $correctAnswer;
  360   my @opt = @_;
  361   my %out_options;
  362 
  363 #########################################################################
  364 # Retain this first check for backword compatibility.  Allows input of the form
  365 # num_cmp($ans, 1, '%0.5f') but warns against it
  366 #########################################################################
  367   my %known_options = (
  368           'mode'      =>  'std',
  369           'format'    =>  $numFormatDefault,
  370           'tol'     =>  $numAbsTolDefault,
  371           'relTol'    =>  $numRelPercentTolDefault,
  372           'units'     =>  undef,
  373           'strings'   =>  undef,
  374           'zeroLevel'   =>  $numZeroLevelDefault,
  375           'zeroLevelTol'  =>  $numZeroLevelTolDefault,
  376           'tolType'       =>  'relative',
  377           'tolerance'     =>  1,
  378           'reltol'    =>  undef,      #alternate spelling
  379           'unit'      =>  undef,      #alternate spelling
  380           'debug'     =>  0
  381         );
  382 
  383   my @output_list;
  384   my( $relPercentTol, $format, $zeroLevel, $zeroLevelTol) = @opt;
  385 
  386   unless( ref($correctAnswer) eq 'ARRAY' || scalar( @opt ) == 0 ||
  387         ( defined($opt[0]) and exists $known_options{$opt[0]} ) ) {
  388     # unless the first parameter is a list of arrays
  389     # or the second parameter is a known option or
  390     # no options were used,
  391     # use the old num_cmp which does not use options, but has inputs
  392     # $relPercentTol,$format,$zeroLevel,$zeroLevelTol
  393     warn "This method of using num_cmp() is deprecated. Please rewrite this" .
  394           " problem using the options style of parameter passing (or" .
  395           " check that your first option is spelled correctly).";
  396 
  397     %out_options = (  'relTol'    => $relPercentTol,
  398           'format'    => $format,
  399           'zeroLevel'   => $zeroLevel,
  400           'zeroLevelTol'  => $zeroLevelTol,
  401           'mode'      => 'std'
  402     );
  403   }
  404 
  405 #########################################################################
  406 # Now handle the options assuming they are entered in the form
  407 # num_cmp($ans, relTol=>1, format=>'%0.5f')
  408 #########################################################################
  409   %out_options = @opt;
  410   assign_option_aliases( \%out_options,
  411         'reltol'    =>      'relTol',
  412         'unit'      =>      'units',
  413         'abstol'  =>    'tol',
  414         );
  415 
  416   set_default_options( \%out_options,
  417            'tolType'    =>  (defined($out_options{'tol'}) ) ? 'absolute' : 'relative',  # the existence of "tol" means that we use absolute tolerance mode
  418            'tolerance'    =>  (defined($out_options{'tolType'}) && $out_options{'tolType'} eq 'absolute' ) ? $numAbsTolDefault : $numRelPercentTolDefault,  # relative tolerance is the default
  419            'mode'       =>  'std',
  420            'format'   =>  $numFormatDefault,
  421            'tol'        =>  undef,
  422            'relTol'   =>  undef,
  423            'units'    =>  undef,
  424            'strings'    =>  undef,
  425            'zeroLevel'  =>  $numZeroLevelDefault,
  426            'zeroLevelTol' =>  $numZeroLevelTolDefault,
  427            'debug'    =>  0,
  428   );
  429 
  430   # can't use both units and strings
  431   if( defined( $out_options{'units'} ) && defined( $out_options{'strings'} ) ) {
  432     warn "Can't use both 'units' and 'strings' in the same problem " .
  433     "(check your parameters to num_cmp() )";
  434   }
  435 
  436   # absolute tolType and relTol are incompatible. So are relative tolType and tol
  437   if( defined( $out_options{'relTol'} ) &&  $out_options{'tolType'} eq 'absolute' )  {
  438     warn "The 'tolType' 'absolute' is not compatible with 'relTol' " .
  439     "(check your parameters to num_cmp() )";
  440   }
  441   if( defined( $out_options{'tol'} ) &&  $out_options{'tolType'} eq 'relative' )  {
  442     warn "The 'tolType' 'relative' is not compatible with 'tol' " .
  443     "(check your parameters to num_cmp() )";
  444   }
  445 
  446 
  447   # Handle legacy options
  448     if ($out_options{tolType} eq 'absolute')   {
  449     $out_options{'tolerance'}=$out_options{'tol'} if defined($out_options{'tol'});
  450     delete($out_options{'relTol'}) if exists( $out_options{'relTol'} );
  451   } else {
  452     $out_options{'tolerance'}=$out_options{'relTol'} if defined($out_options{'relTol'});
  453     # delete($out_options{'tol'}) if exists( $out_options{'tol'} );
  454   }
  455   # end legacy options
  456 
  457   # thread over lists
  458   my @ans_list = ();
  459 
  460   if ( ref($correctAnswer) eq 'ARRAY' ) {
  461     @ans_list = @{$correctAnswer};
  462   }
  463   else { push( @ans_list, $correctAnswer );
  464   }
  465 
  466   # produce answer evaluators
  467   foreach my $ans (@ans_list) {
  468     if( defined( $out_options{'units'} ) ) {
  469       $ans = "$ans $out_options{'units'}";
  470 
  471       push( @output_list, NUM_CMP(  'correctAnswer'       =>  $ans,
  472               'tolerance'   =>  $out_options{'tolerance'},
  473               'tolType'   =>  $out_options{'tolType'},
  474               'format'    =>  $out_options{'format'},
  475               'mode'      =>  $out_options{'mode'},
  476               'zeroLevel'   =>  $out_options{'zeroLevel'},
  477               'zeroLevelTol'  =>  $out_options{'zeroLevelTol'},
  478               'debug'     =>  $out_options{'debug'},
  479               'units'     =>  $out_options{'units'},
  480             )
  481       );
  482     } elsif( defined( $out_options{'strings'} ) ) {
  483 
  484 
  485       push( @output_list, NUM_CMP(  'correctAnswer' =>  $ans,
  486               'tolerance' =>  $out_options{tolerance},
  487               'tolType' =>  $out_options{tolType},
  488               'format'  =>  $out_options{'format'},
  489               'mode'    =>  $out_options{'mode'},
  490               'zeroLevel' =>  $out_options{'zeroLevel'},
  491               'zeroLevelTol'  =>  $out_options{'zeroLevelTol'},
  492               'debug'   =>  $out_options{'debug'},
  493               'strings' =>  $out_options{'strings'},
  494          )
  495          );
  496     } else {
  497       push(@output_list,
  498         NUM_CMP(  'correctAnswer'       =>  $ans,
  499           'tolerance'   =>  $out_options{tolerance},
  500           'tolType'   =>  $out_options{tolType},
  501           'format'    =>  $out_options{'format'},
  502           'mode'      =>  $out_options{'mode'},
  503           'zeroLevel'   =>  $out_options{'zeroLevel'},
  504           'zeroLevelTol'        =>  $out_options{'zeroLevelTol'},
  505           'debug'     =>  $out_options{'debug'},
  506         ),
  507       );
  508       }
  509   }
  510 
  511   return (wantarray) ? @output_list : $output_list[0];
  512 }
  513 
  514 #legacy code for compatability purposes
  515 sub num_rel_cmp {   # compare numbers
  516     std_num_cmp( @_ );
  517 }
  518 
  519 
  520 =head3 "mode"_num_cmp() functions
  521 
  522 There are 16 functions total, 4 for each mode (std, frac, strict, arith). Each mode has
  523 one "normal" function, one which accepts a list of answers, one which uses absolute
  524 rather than relative tolerance, and one which uses absolute tolerance and accepts a list.
  525 The "std" family is documented below; all others work precisely the same.
  526 
  527  std_num_cmp($correctAnswer) OR
  528  std_num_cmp($correctAnswer, $relPercentTol) OR
  529  std_num_cmp($correctAnswer, $relPercentTol, $format) OR
  530  std_num_cmp($correctAnswer, $relPercentTol, $format, $zeroLevel) OR
  531  std_num_cmp($correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol)
  532 
  533   $correctAnswer  --  the correct answer
  534   $relPercentTol  --  the tolerance, as a percentage (optional)
  535   $format   --  the format of the displayed answer (optional)
  536   $zeroLevel  --  if the correct answer is this close to zero, then zeroLevelTol applies (optional)
  537   $zeroLevelTol --  absolute tolerance to allow when correct answer is close to zero (optional)
  538 
  539   std_num_cmp() uses standard mode (arithmetic operations and elementary
  540   functions allowed) and relative tolerance. Options are specified by
  541   one or more parameters. Note that if you wish to set an option which
  542   is later in the parameter list, you must set all previous options.
  543 
  544  std_num_cmp_abs($correctAnswer) OR
  545  std_num_cmp_abs($correctAnswer, $absTol) OR
  546  std_num_cmp_abs($correctAnswer, $absTol, $format)
  547 
  548   $correctAnswer    --  the correct answer
  549   $absTol     --  an absolute tolerance (optional)
  550   $format     --  the format of the displayed answer (optional)
  551 
  552   std_num_cmp_abs() uses standard mode and absolute tolerance. Options
  553   are set as with std_num_cmp(). Note that $zeroLevel and $zeroLevelTol
  554   do not apply with absolute tolerance.
  555 
  556  std_num_cmp_list($relPercentTol, $format, @answerList)
  557 
  558   $relPercentTol    --  the tolerance, as a percentage
  559   $format     --  the format of the displayed answer(s)
  560   @answerList   --  a list of one or more correct answers
  561 
  562   std_num_cmp_list() uses standard mode and relative tolerance. There
  563   is no way to set $zeroLevel or $zeroLevelTol. Note that no
  564   parameters are optional. All answers in the list will be
  565   evaluated with the same set of parameters.
  566 
  567  std_num_cmp_abs_list($absTol, $format, @answerList)
  568 
  569   $absTol   --  an absolute tolerance
  570   $format   --  the format of the displayed answer(s)
  571   @answerList --  a list of one or more correct answers
  572 
  573   std_num_cmp_abs_list() uses standard mode and absolute tolerance.
  574   Note that no parameters are optional. All answers in the list will be
  575   evaluated with the same set of parameters.
  576 
  577  arith_num_cmp(), arith_num_cmp_list(), arith_num_cmp_abs(), arith_num_cmp_abs_list()
  578  strict_num_cmp(), strict_num_cmp_list(), strict_num_cmp_abs(), strict_num_cmp_abs_list()
  579  frac_num_cmp(), frac_num_cmp_list(), frac_num_cmp_abs(), frac_num_cmp_abs_list()
  580 
  581 Examples:
  582 
  583   ANS( strict_num_cmp( 3.14159 ) )  --  The student answer must be a number
  584     in decimal or scientific notation which is within .1 percent of 3.14159.
  585     This assumes $numRelPercentTolDefault has been set to .1.
  586   ANS( strict_num_cmp( $answer, .01 ) ) --  The student answer must be a
  587     number within .01 percent of $answer (e.g. 3.14159 if $answer is 3.14159
  588     or $answer is "pi" or $answer is 4*atan(1)).
  589   ANS( frac_num_cmp( $answer) ) or ANS( frac_num_cmp( $answer,.01 ))  --
  590     The student answer can be a number or fraction, e.g. 2/3.
  591   ANS( arith_num_cmp( $answer) ) or ANS( arith_num_cmp( $answer,.01 ))  --
  592     The student answer can be an arithmetic expression, e.g. (2+3)/7-2^.5 .
  593   ANS( std_num_cmp( $answer) ) or ANS( std_num_cmp( $answer,.01 ))  --
  594     The student answer can contain elementary functions, e.g. sin(.3+pi/2)
  595 
  596 =cut
  597 
  598 sub std_num_cmp {           # compare numbers allowing use of elementary functions
  599     my ( $correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
  600 
  601   my %options = ( 'relTol'        =>  $relPercentTol,
  602             'format'    =>  $format,
  603             'zeroLevel'   =>  $zeroLevel,
  604             'zeroLevelTol'  =>  $zeroLevelTol
  605     );
  606 
  607     set_default_options( \%options,
  608        'tolType'      =>  'relative',
  609        'tolerance'    =>  $numRelPercentTolDefault,
  610        'mode'       =>  'std',
  611        'format'     =>  $numFormatDefault,
  612        'relTol'     =>  $numRelPercentTolDefault,
  613        'zeroLevel'    =>  $numZeroLevelDefault,
  614        'zeroLevelTol' =>  $numZeroLevelTolDefault,
  615        'debug'        =>  0,
  616     );
  617 
  618     num_cmp([$correctAnswer], %options);
  619 }
  620 
  621 ##  Similar to std_num_cmp but accepts a list of numbers in the form
  622 ##  std_num_cmp_list(relpercentTol,format,ans1,ans2,ans3,...)
  623 ##  format is of the form "%10.3g" or "", i.e., a format suitable for sprintf(). Use "" for default
  624 ##  You must enter a format and tolerance
  625 
  626 sub std_num_cmp_list {
  627   my ( $relPercentTol, $format, @answerList) = @_;
  628 
  629   my %options = ( 'relTol'  =>      $relPercentTol,
  630       'format'        =>      $format,
  631   );
  632 
  633   set_default_options( \%options,
  634            'tolType'      =>      'relative',
  635            'tolerance'    =>      $numRelPercentTolDefault,
  636            'mode'         =>      'std',
  637            'format'       =>      $numFormatDefault,
  638            'relTol'       =>      $numRelPercentTolDefault,
  639            'zeroLevel'    =>      $numZeroLevelDefault,
  640            'zeroLevelTol' =>      $numZeroLevelTolDefault,
  641            'debug'        =>      0,
  642   );
  643 
  644   num_cmp(\@answerList, %options);
  645 
  646 }
  647 
  648 sub std_num_cmp_abs {     # compare numbers allowing use of elementary functions with absolute tolerance
  649   my ( $correctAnswer, $absTol, $format) = @_;
  650   my %options = ( 'tolerance'  => $absTol,
  651             'format'     => $format
  652   );
  653 
  654   set_default_options (\%options,
  655            'tolType'      =>      'absolute',
  656            'tolerance'    =>      $absTol,
  657            'mode'         =>      'std',
  658            'format'       =>      $numFormatDefault,
  659            'zeroLevel'    =>      0,
  660            'zeroLevelTol' =>      0,
  661            'debug'        =>      0,
  662   );
  663 
  664   num_cmp([$correctAnswer], %options);
  665 }
  666 
  667 ##  See std_num_cmp_list for usage
  668 
  669 sub std_num_cmp_abs_list {
  670   my ( $absTol, $format, @answerList ) = @_;
  671 
  672         my %options = ( 'tolerance'         =>      $absTol,
  673                         'format'            =>      $format,
  674   );
  675 
  676         set_default_options( \%options,
  677                              'tolType'      =>      'absolute',
  678                              'tolerance'    =>      $absTol,
  679                              'mode'         =>      'std',
  680                              'format'       =>      $numFormatDefault,
  681                              'zeroLevel'    =>      0,
  682                              'zeroLevelTol' =>      0,
  683                              'debug'        =>      0,
  684         );
  685 
  686         num_cmp(\@answerList, %options);
  687 }
  688 
  689 sub frac_num_cmp {            # only allow fractions and numbers as submitted answer
  690 
  691   my ( $correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
  692 
  693   my %options = ( 'relTol'   =>   $relPercentTol,
  694           'format'     =>   $format,
  695           'zeroLevel'  =>   $zeroLevel,
  696           'zeroLevelTol'   =>   $zeroLevelTol
  697   );
  698 
  699   set_default_options( \%options,
  700          'tolType'     =>   'relative',
  701          'tolerance'   =>   $relPercentTol,
  702          'mode'      =>   'frac',
  703          'format'    =>   $numFormatDefault,
  704          'zeroLevel'   =>   $numZeroLevelDefault,
  705          'zeroLevelTol'  =>   $numZeroLevelTolDefault,
  706          'relTol'    =>   $numRelPercentTolDefault,
  707          'debug'     =>   0,
  708    );
  709 
  710   num_cmp([$correctAnswer], %options);
  711 }
  712 
  713 ##  See std_num_cmp_list for usage
  714 sub frac_num_cmp_list {
  715   my ( $relPercentTol, $format, @answerList ) = @_;
  716 
  717   my %options = (      'relTol'  =>   $relPercentTol,
  718                'format'    =>   $format
  719   );
  720 
  721   set_default_options( \%options,
  722        'tolType'     =>   'relative',
  723        'tolerance'   =>   $relPercentTol,
  724        'mode'      =>   'frac',
  725        'format'    =>   $numFormatDefault,
  726        'zeroLevel'   =>   $numZeroLevelDefault,
  727        'zeroLevelTol'  =>   $numZeroLevelTolDefault,
  728        'relTol'    =>   $numRelPercentTolDefault,
  729        'debug'     =>   0,
  730   );
  731 
  732   num_cmp(\@answerList, %options);
  733 }
  734 
  735 sub frac_num_cmp_abs {      # only allow fraction expressions as submitted answer with absolute tolerance
  736     my ( $correctAnswer, $absTol, $format ) = @_;
  737 
  738     my %options = (             'tolerance'    =>     $absTol,
  739               'format'       =>     $format
  740     );
  741 
  742   set_default_options (\%options,
  743       'tolType'    =>   'absolute',
  744       'tolerance'    =>   $absTol,
  745       'mode'       =>   'frac',
  746       'format'     =>   $numFormatDefault,
  747       'zeroLevel'    =>   0,
  748       'zeroLevelTol' =>   0,
  749       'debug'      =>   0,
  750   );
  751 
  752     num_cmp([$correctAnswer], %options);
  753 }
  754 
  755 ##  See std_num_cmp_list for usage
  756 
  757 sub frac_num_cmp_abs_list {
  758     my ( $absTol, $format, @answerList ) = @_;
  759 
  760     my %options = (             'tolerance'    =>     $absTol,
  761               'format'       =>     $format
  762     );
  763 
  764     set_default_options (\%options,
  765        'tolType'      =>     'absolute',
  766        'tolerance'    =>     $absTol,
  767        'mode'         =>     'frac',
  768        'format'       =>     $numFormatDefault,
  769        'zeroLevel'    =>     0,
  770        'zeroLevelTol' =>     0,
  771        'debug'        =>     0,
  772     );
  773 
  774     num_cmp(\@answerList, %options);
  775 }
  776 
  777 
  778 sub arith_num_cmp {           # only allow arithmetic expressions as submitted answer
  779 
  780     my ( $correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
  781 
  782     my %options = (     'relTol'      =>     $relPercentTol,
  783       'format'         =>     $format,
  784       'zeroLevel'      =>     $zeroLevel,
  785       'zeroLevelTol'   =>     $zeroLevelTol
  786     );
  787 
  788     set_default_options( \%options,
  789                         'tolType'       =>     'relative',
  790                         'tolerance'     =>     $relPercentTol,
  791                         'mode'          =>     'arith',
  792                         'format'        =>     $numFormatDefault,
  793                         'zeroLevel'     =>     $numZeroLevelDefault,
  794                         'zeroLevelTol'  =>     $numZeroLevelTolDefault,
  795                         'relTol'        =>     $numRelPercentTolDefault,
  796                         'debug'         =>     0,
  797     );
  798 
  799     num_cmp([$correctAnswer], %options);
  800 }
  801 
  802 ##  See std_num_cmp_list for usage
  803 sub arith_num_cmp_list {
  804     my ( $relPercentTol, $format, @answerList ) = @_;
  805 
  806     my %options = (     'relTol'     =>     $relPercentTol,
  807                         'format'        =>     $format,
  808     );
  809 
  810     set_default_options( \%options,
  811                          'tolType'       =>     'relative',
  812                          'tolerance'     =>     $relPercentTol,
  813                          'mode'          =>     'arith',
  814                          'format'        =>     $numFormatDefault,
  815                          'zeroLevel'     =>     $numZeroLevelDefault,
  816                          'zeroLevelTol'  =>     $numZeroLevelTolDefault,
  817                          'relTol'        =>     $numRelPercentTolDefault,
  818                          'debug'         =>     0,
  819     );
  820 
  821     num_cmp(\@answerList, %options);
  822 }
  823 
  824 sub arith_num_cmp_abs {     # only allow arithmetic expressions as submitted answer with absolute tolerance
  825     my ( $correctAnswer, $absTol, $format ) = @_;
  826 
  827     my %options = (      'tolerance'    =>     $absTol,
  828                          'format'       =>     $format
  829     );
  830 
  831     set_default_options (\%options,
  832                          'tolType'      =>     'absolute',
  833                          'tolerance'    =>     $absTol,
  834                          'mode'         =>     'arith',
  835                          'format'       =>     $numFormatDefault,
  836                          'zeroLevel'    =>     0,
  837                          'zeroLevelTol' =>     0,
  838                          'debug'        =>     0,
  839     );
  840 
  841     num_cmp([$correctAnswer], %options);
  842 }
  843 
  844 ##  See std_num_cmp_list for usage
  845 sub arith_num_cmp_abs_list {
  846     my ( $absTol, $format, @answerList ) = @_;
  847 
  848     my %options = (      'tolerance'    =>     $absTol,
  849                          'format'       =>     $format
  850     );
  851 
  852     set_default_options (\%options,
  853                          'tolType'      =>     'absolute',
  854                          'tolerance'    =>     $absTol,
  855                          'mode'         =>     'arith',
  856                          'format'       =>     $numFormatDefault,
  857                          'zeroLevel'    =>     0,
  858                          'zeroLevelTol' =>     0,
  859                          'debug'        =>     0,
  860     );
  861 
  862     num_cmp(\@answerList, %options);
  863 }
  864 
  865 sub strict_num_cmp {          # only allow numbers as submitted answer
  866     my ( $correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
  867 
  868     my %options = (      'relTol'     =>     $relPercentTol,
  869                          'format'        =>     $format,
  870                          'zeroLevel'     =>     $zeroLevel,
  871                          'zeroLevelTol'  =>     $zeroLevelTol
  872     );
  873 
  874     set_default_options( \%options,
  875                          'tolType'       =>     'relative',
  876                          'tolerance'     =>     $relPercentTol,
  877                          'mode'          =>     'strict',
  878                          'format'        =>     $numFormatDefault,
  879                          'zeroLevel'     =>     $numZeroLevelDefault,
  880                          'zeroLevelTol'  =>     $numZeroLevelTolDefault,
  881                          'relTol'        =>     $numRelPercentTolDefault,
  882                          'debug'         =>     0,
  883     );
  884     num_cmp([$correctAnswer], %options);
  885 
  886 }
  887 
  888 ##  See std_num_cmp_list for usage
  889 sub strict_num_cmp_list {       # compare numbers
  890     my ( $relPercentTol, $format, @answerList ) = @_;
  891 
  892     my %options = (    'relTol'     =>     $relPercentTol,
  893        'format'        =>     $format,
  894     );
  895 
  896     set_default_options( \%options,
  897                          'tolType'       =>     'relative',
  898                          'tolerance'     =>     $relPercentTol,
  899                          'mode'          =>     'strict',
  900                          'format'        =>     $numFormatDefault,
  901                          'zeroLevel'     =>     $numZeroLevelDefault,
  902                          'zeroLevelTol'  =>     $numZeroLevelTolDefault,
  903                          'relTol'        =>     $numRelPercentTolDefault,
  904                          'debug'         =>     0,
  905     );
  906 
  907     num_cmp(\@answerList, %options);
  908 }
  909 
  910 
  911 sub strict_num_cmp_abs {        # only allow numbers as submitted answer with absolute tolerance
  912     my ( $correctAnswer, $absTol, $format ) = @_;
  913 
  914     my %options = (       'tolerance'    =>     $absTol,
  915                     'format'       =>     $format
  916     );
  917 
  918     set_default_options (\%options,
  919                          'tolType'      =>     'absolute',
  920                          'tolerance'    =>     $absTol,
  921                          'mode'         =>     'strict',
  922                          'format'       =>     $numFormatDefault,
  923                          'zeroLevel'    =>     0,
  924                          'zeroLevelTol' =>     0,
  925                          'debug'        =>     0,
  926     );
  927     num_cmp([$correctAnswer], %options);
  928 
  929 }
  930 
  931 ##  See std_num_cmp_list for usage
  932 sub strict_num_cmp_abs_list {     # compare numbers
  933     my ( $absTol, $format, @answerList ) = @_;
  934 
  935     my %options = (      'tolerance'    =>     $absTol,
  936                          'format'       =>     $format
  937     );
  938 
  939     set_default_options (\%options,
  940                          'tolType'      =>     'absolute',
  941                          'tolerance'    =>     $absTol,
  942                          'mode'         =>     'strict',
  943                          'format'       =>     $numFormatDefault,
  944                          'zeroLevel'    =>     0,
  945                          'zeroLevelTol' =>     0,
  946                          'debug'        =>     0,
  947     );
  948 
  949     num_cmp(\@answerList, %options);
  950 }
  951 
  952 ## sub numerical_compare_with_units
  953 ## Compares a number with units
  954 ## Deprecated; use num_cmp()
  955 ##
  956 ## IN:  a string which includes the numerical answer and the units
  957 ##    a hash with the following keys (all optional):
  958 ##      mode    --  'std', 'frac', 'arith', or 'strict'
  959 ##      format    --  the format to use when displaying the answer
  960 ##      tol   --  an absolute tolerance, or
  961 ##      relTol    --  a relative tolerance
  962 ##      zeroLevel --  if the correct answer is this close to zero, then zeroLevelTol applies
  963 ##      zeroLevelTol  --  absolute tolerance to allow when correct answer is close to zero
  964 
  965 # This mode is depricated.  send input through num_cmp -- it can handle units.
  966 
  967 sub numerical_compare_with_units {
  968   my $correct_answer = shift;  # the answer is a string which includes both the numerical answer and the units.
  969   my %options = @_;    # all of the other inputs are (key value) pairs
  970 
  971   # Prepare the correct answer
  972   $correct_answer = str_filters( $correct_answer, 'trim_whitespace' );
  973 
  974   # it surprises me that the match below works since the first .* is greedy.
  975   my ($correct_num_answer, $correct_units) = $correct_answer =~ /^(.*)\s+([^\s]*)$/;
  976   $options{units} = $correct_units;
  977 
  978   num_cmp($correct_num_answer, %options);
  979 }
  980 
  981 
  982 =head3 std_num_str_cmp()
  983 
  984 NOTE: This function is maintained for compatibility. num_cmp() with the
  985     'strings' parameter is slightly preferred.
  986 
  987 std_num_str_cmp() is used when the correct answer could be either a number or a
  988 string. For example, if you wanted the student to evaluate a function at number
  989 of points, but write "Inf" or "Minf" if the function is unbounded. This routine
  990 will provide error messages that do not give a hint as to whether the correct
  991 answer is a string or a number. For numerical comparisons, std_num_cmp() is
  992 used internally; for string comparisons, std_str_cmp() is used.  String answers
  993 must consist entirely of letters except that an initial minus sign is allowed.
  994 E.g. "inf" and "-inf" are valid strings where as "too-big" is not.
  995 
  996  std_num_str_cmp( $correctAnswer ) OR
  997  std_num_str_cmp( $correctAnswer, $ra_legalStrings ) OR
  998  std_num_str_cmp( $correctAnswer, $ra_legalStrings, $relPercentTol ) OR
  999  std_num_str_cmp( $correctAnswer, $ra_legalStrings, $relPercentTol, $format ) OR
 1000  std_num_str_cmp( $correctAnswer, $ra_legalStrings, $relPercentTol, $format, $zeroLevel ) OR
 1001  std_num_str_cmp( $correctAnswer, $ra_legalStrings, $relPercentTol, $format,
 1002           $zeroLevel, $zeroLevelTol )
 1003 
 1004   $correctAnswer    --  the correct answer
 1005   $ra_legalStrings  --  a reference to an array of legal strings, e.g. ["str1", "str2"]
 1006   $relPercentTol    --  the error tolerance as a percentage
 1007   $format     --  the display format
 1008   $zeroLevel    --  if the correct answer is this close to zero, then zeroLevelTol applies
 1009   $zeroLevelTol   --  absolute tolerance to allow when correct answer is close to zero
 1010 
 1011 Examples:
 1012   ANS( std_num_str_cmp( $ans, ["Inf", "Minf", "NaN"] ) );
 1013   ANS( std_num_str_cmp( $ans, ["INF", "-INF"] ) );
 1014 
 1015 =cut
 1016 
 1017 sub std_num_str_cmp {
 1018   my ( $correctAnswer, $ra_legalStrings, $relpercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
 1019   # warn ('This method is depreciated.  Use num_cmp instead.');
 1020   return num_cmp ($correctAnswer, strings=>$ra_legalStrings, relTol=>$relpercentTol, format=>$format,
 1021     zeroLevel=>$zeroLevel, zeroLevelTol=>$zeroLevelTol);
 1022 }
 1023 
 1024 sub NUM_CMP {                              # low level numeric compare (now uses Parser)
 1025   return ORIGINAL_NUM_CMP(@_)
 1026     if main::PG_restricted_eval(q!$main::useOldAnswerMacros!);
 1027 
 1028   my %num_params = @_;
 1029 
 1030   #
 1031   #  check for required parameters
 1032   #
 1033   my @keys = qw(correctAnswer tolerance tolType format mode zeroLevel zeroLevelTol debug);
 1034   foreach my $key (@keys) {
 1035       warn "$key must be defined in options when calling NUM_CMP"
 1036         unless defined($num_params{$key});
 1037   }
 1038 
 1039   my $correctAnswer = $num_params{correctAnswer};
 1040   my $mode          = $num_params{mode};
 1041   my %options       = (debug => $num_params{debug});
 1042 
 1043   #
 1044   #  Hack to fix up exponential notation in correct answer
 1045   #  (e.g., perl will pass .0000001 as 1e-07).
 1046   #
 1047   $correctAnswer = uc($correctAnswer)
 1048     if $correctAnswer =~ m/e/ && Value::isNumber($correctAnswer);
 1049 
 1050   #
 1051   #  Get an apppropriate context based on the mode
 1052   #
 1053   my $context;
 1054   for ($mode) {
 1055     /^strict$/i    and do {
 1056       $context = $Parser::Context::Default::context{LimitedNumeric}->copy;
 1057       last;
 1058     };
 1059     /^arith$/i     and do {
 1060       $context = $Parser::Context::Default::context{LegacyNumeric}->copy;
 1061       $context->functions->disable('All');
 1062       last;
 1063     };
 1064     /^frac$/i  and do {
 1065       $context = $Parser::Context::Default::context{'LimitedNumeric-Fraction'}->copy;
 1066       last;
 1067     };
 1068 
 1069     # default
 1070     $context = $Parser::Context::Default::context{LegacyNumeric}->copy;
 1071   }
 1072   $context->{format}{number} = $num_params{'format'};
 1073   $context->strings->clear;
 1074   #  FIXME:  should clear variables as well? Copy them from the current context?
 1075 
 1076   #
 1077   #  Add the strings to the context
 1078   #
 1079   if ($num_params{strings}) {
 1080     foreach my $string (@{$num_params{strings}}) {
 1081       my %tex = ($string =~ m/^(-?)inf(inity)?$/i)? (TeX => "$1\\infty"): ();
 1082       %tex = (TeX => "-\\infty") if uc($string) eq "MINF";
 1083       $context->strings->add(uc($string) => {%tex});
 1084     }
 1085   }
 1086 
 1087   #
 1088   #  Set the tolerances
 1089   #
 1090   if ($num_params{tolType} eq 'absolute') {
 1091     $context->flags->set(
 1092       tolerance => $num_params{tolerance},
 1093       tolType => 'absolute',
 1094     );
 1095   } else {
 1096     $context->flags->set(
 1097       tolerance => .01*$num_params{tolerance},
 1098       tolType => 'relative',
 1099     );
 1100   }
 1101   $context->flags->set(
 1102     zeroLevel => $num_params{zeroLevel},
 1103     zeroLevelTol => $num_params{zeroLevelTol},
 1104   );
 1105 
 1106   #
 1107   #  Get the proper Parser object for the professor's answer
 1108   #  using the initialized context
 1109   #
 1110   my $oldContext = &$Context($context); my $r;
 1111   if ($num_params{units}) {
 1112     $r = new Parser::Legacy::NumberWithUnits($correctAnswer);
 1113           $options{rh_correct_units} = $num_params{units};
 1114   } else {
 1115     $r = Value::Formula->new($correctAnswer);
 1116     die "The professor's answer can't be a formula" unless $r->isConstant;
 1117     $r = $r->eval; $r = new Value::Real($r) unless Value::class($r) eq 'String';
 1118     $r->{correct_ans} = $correctAnswer;
 1119     if ($mode eq 'phase_pi') {
 1120       my $pi = 4*atan2(1,1);
 1121       while ($r >  $pi/2) {$r -= $pi}
 1122       while ($r < -$pi/2) {$r += $pi}
 1123     }
 1124   }
 1125   #
 1126   #  Get the answer checker from the parser object
 1127   #
 1128   my $cmp = $r->cmp(%options);
 1129   $cmp->install_pre_filter(sub {
 1130     my $rh_ans = shift;
 1131     $rh_ans->{original_student_ans} = $rh_ans->{student_ans};
 1132     $rh_ans->{original_correct_ans} = $rh_ans->{correct_ans};
 1133     return $rh_ans;
 1134   });
 1135   $cmp->install_post_filter(sub {
 1136     my $rh_ans = shift;
 1137     $rh_ans->{student_ans} = $rh_ans->{student_value}->string
 1138       if ref($rh_ans->{student_value});
 1139     return $rh_ans;
 1140   });
 1141   $cmp->{debug} = $num_params{debug};
 1142   &$Context($oldContext);
 1143 
 1144   return $cmp;
 1145 }
 1146 
 1147 #
 1148 #  The original version, for backward compatibility
 1149 #  (can be removed when the Parser-based version is more fully tested.)
 1150 #
 1151 sub ORIGINAL_NUM_CMP {    # low level numeric compare
 1152   my %num_params = @_;
 1153 
 1154   my @keys = qw ( correctAnswer tolerance tolType format mode zeroLevel zeroLevelTol debug );
 1155   foreach my $key (@keys) {
 1156       warn "$key must be defined in options when calling NUM_CMP" unless defined ($num_params{$key});
 1157   }
 1158 
 1159   my $correctAnswer = $num_params{'correctAnswer'};
 1160   my $format        = $num_params{'format'};
 1161   my $mode        = $num_params{'mode'};
 1162 
 1163   if( $num_params{tolType} eq 'relative' ) {
 1164     $num_params{'tolerance'} = .01*$num_params{'tolerance'};
 1165   }
 1166 
 1167   my $formattedCorrectAnswer;
 1168   my $correct_units;
 1169   my $correct_num_answer;
 1170   my %correct_units;
 1171   my $corrAnswerIsString = 0;
 1172 
 1173 
 1174   if (defined($num_params{units}) && $num_params{units}) {
 1175     $correctAnswer  = str_filters( $correctAnswer, 'trim_whitespace' );
 1176             # units are in form stuff space units where units contains no spaces.
 1177 
 1178     ($correct_num_answer, $correct_units) = $correctAnswer =~ /^(.*)\s+([^\s]*)$/;
 1179     %correct_units = Units::evaluate_units($correct_units);
 1180     if ( defined( $correct_units{'ERROR'} ) ) {
 1181        warn ("ERROR: The answer \"$correctAnswer\" in the problem definition cannot be parsed:\n" .
 1182         "$correct_units{'ERROR'}\n");
 1183     }
 1184     # $formattedCorrectAnswer = spf($correct_num_answer,$num_params{'format'}) . " $correct_units";
 1185     $formattedCorrectAnswer = prfmt($correct_num_answer,$num_params{'format'}) . " $correct_units";
 1186 
 1187   } elsif (defined($num_params{strings}) && $num_params{strings}) {
 1188     my $legalString = '';
 1189     my @legalStrings = @{$num_params{strings}};
 1190     $correct_num_answer = $correctAnswer;
 1191     $formattedCorrectAnswer = $correctAnswer;
 1192     foreach $legalString (@legalStrings) {
 1193       if ( uc($correctAnswer) eq uc($legalString) ) {
 1194         $corrAnswerIsString = 1;
 1195 
 1196         last;
 1197       }
 1198     }     ## at this point $corrAnswerIsString = 0 iff correct answer is numeric
 1199   } else {
 1200     $correct_num_answer = $correctAnswer;
 1201     $formattedCorrectAnswer = prfmt( $correctAnswer, $num_params{'format'} );
 1202   }
 1203 
 1204   $correct_num_answer = math_constants($correct_num_answer);
 1205 
 1206   my $PGanswerMessage = '';
 1207 
 1208   my ($inVal,$correctVal,$PG_eval_errors,$PG_full_error_report);
 1209 
 1210   if (defined($correct_num_answer) && $correct_num_answer =~ /\S/ && $corrAnswerIsString == 0 ) {
 1211       ($correctVal, $PG_eval_errors,$PG_full_error_report) = PG_answer_eval($correct_num_answer);
 1212   } else { # case of a string answer
 1213     $PG_eval_errors = ' ';
 1214     $correctVal = $correctAnswer;
 1215   }
 1216 
 1217   if ( ($PG_eval_errors && $corrAnswerIsString == 0) or ((not is_a_number($correctVal)) && $corrAnswerIsString == 0)) {
 1218         ##error message from eval or above
 1219     warn "Error in 'correct' answer: $PG_eval_errors<br>
 1220           The answer $correctAnswer evaluates to $correctVal,
 1221           which cannot be interpreted as a number.  ";
 1222 
 1223   }
 1224   #########################################################################
 1225 
 1226   #construct the answer evaluator
 1227       my $answer_evaluator = new AnswerEvaluator;
 1228       $answer_evaluator->{debug} = $num_params{debug};
 1229       $answer_evaluator->ans_hash(
 1230                 correct_ans       =>  $correctVal,
 1231                 type          =>  "${mode}_number",
 1232                 tolerance       =>  $num_params{tolerance},
 1233               tolType         =>  $num_params{tolType},
 1234               units         =>  $correct_units,
 1235                 original_correct_ans  =>  $formattedCorrectAnswer,
 1236                 rh_correct_units    =>      \%correct_units,
 1237                 answerIsString      =>  $corrAnswerIsString,
 1238       );
 1239       my ($in, $formattedSubmittedAnswer);
 1240   $answer_evaluator->install_pre_filter(sub {my $rh_ans = shift;
 1241     $rh_ans->{original_student_ans} = $rh_ans->{student_ans}; $rh_ans;}
 1242   );
 1243 
 1244 
 1245 
 1246   if (defined($num_params{units}) && $num_params{units}) {
 1247       $answer_evaluator->install_pre_filter(\&check_units);
 1248   }
 1249   if (defined($num_params{strings}) && $num_params{strings}) {
 1250       $answer_evaluator->install_pre_filter(\&check_strings, %num_params);
 1251   }
 1252 
 1253   ## FIXME? - this pre filter was moved before check_units to allow
 1254   ##      for latex preview of answers with no units.
 1255   ##          seems to work but may have unintended side effects elsewhere.
 1256 
 1257   ##      Actually it caused trouble with the check strings package so it has been moved back
 1258   #       We'll try some other method  -- perhaps add code to fix_answer for display
 1259   $answer_evaluator->install_pre_filter(\&check_syntax);
 1260 
 1261   $answer_evaluator->install_pre_filter(\&math_constants);
 1262 
 1263   if ($mode eq 'std') {
 1264         # do nothing
 1265   } elsif ($mode eq 'strict') {
 1266     $answer_evaluator->install_pre_filter(\&is_a_number);
 1267   } elsif ($mode eq 'arith') {
 1268       $answer_evaluator->install_pre_filter(\&is_an_arithmetic_expression);
 1269     } elsif ($mode eq 'frac') {
 1270       $answer_evaluator->install_pre_filter(\&is_a_fraction);
 1271 
 1272     } elsif ($mode eq 'phase_pi') {
 1273       $answer_evaluator->install_pre_filter(\&phase_pi);
 1274 
 1275     } else {
 1276       $PGanswerMessage = 'Tell your professor that there is an error in his or her answer mechanism. No mode was specified.';
 1277       $formattedSubmittedAnswer = $in;
 1278     }
 1279 
 1280   if ($corrAnswerIsString == 0 ){   # avoiding running compare_numbers when correct answer is a string.
 1281     $answer_evaluator->install_evaluator(\&compare_numbers, %num_params);
 1282    }
 1283 
 1284 
 1285 ###############################################################################
 1286 # We'll leave these next lines out for now, so that the evaluated versions of the student's and professor's
 1287 # can be displayed in the answer message.  This may still cause a few anomolies when strings are used
 1288 #
 1289 ###############################################################################
 1290 
 1291   $answer_evaluator->install_post_filter(\&fix_answers_for_display);
 1292 
 1293       $answer_evaluator->install_post_filter(sub {my $rh_ans = shift;
 1294           return $rh_ans unless $rh_ans->catch_error('EVAL');
 1295           $rh_ans->{student_ans} = $rh_ans->{original_student_ans}. ' '. $rh_ans->{error_message};
 1296           $rh_ans->clear_error('EVAL'); } );
 1297       $answer_evaluator->install_post_filter(sub {my $rh_ans = shift; $rh_ans->clear_error('SYNTAX'); } );
 1298       $answer_evaluator->install_post_filter(sub {my $rh_ans = shift; $rh_ans->clear_error('UNITS'); } );
 1299       $answer_evaluator->install_post_filter(sub {my $rh_ans = shift; $rh_ans->clear_error('NUMBER'); } );
 1300       $answer_evaluator->install_post_filter(sub {my $rh_ans = shift; $rh_ans->clear_error('STRING'); } );
 1301       $answer_evaluator;
 1302 }
 1303 
 1304 
 1305 
 1306 ##########################################################################
 1307 ##########################################################################
 1308 ## Function answer evaluators
 1309 
 1310 =head2 Function Answer Evaluators
 1311 
 1312 Function answer evaluators take in a function, compare it numerically to a
 1313 correct function, and return a score. They can require an exactly equivalent
 1314 function, or one that is equal up to a constant. They can accept or reject an
 1315 answer based on specified tolerances for numerical deviation.
 1316 
 1317 Function Comparison Options
 1318 
 1319   correctEqn  --  The correct equation, specified as a string. It may include
 1320           all basic arithmetic operations, as well as elementary
 1321           functions. Variable usage is described below.
 1322 
 1323   Variables --  The independent variable(s). When comparing the correct
 1324           equation to the student equation, each variable will be
 1325           replaced by a certain number of numerical values. If
 1326           the student equation agrees numerically with the correct
 1327           equation, they are considered equal. Note that all
 1328           comparison is numeric; it is possible (although highly
 1329           unlikely and never a practical concern) for two unequal
 1330           functions to yield the same numerical results.
 1331 
 1332   Limits    --  The limits of evaluation for the independent variables.
 1333           Each variable is evaluated only in the half-open interval
 1334           [lower_limit, upper_limit). This is useful if the function
 1335           has a singularity or is not defined in a certain range.
 1336           For example, the function "sqrt(-1-x)" could be evaluated
 1337           in [-2,-1).
 1338 
 1339   Tolerance --  Tolerance in function comparisons works exactly as in
 1340           numerical comparisons; see the numerical comparison
 1341           documentation for a complete description. Note that the
 1342           tolerance does applies to the function as a whole, not
 1343           each point individually.
 1344 
 1345   Number of --  Specifies how many points to evaluate each variable at. This
 1346   Points      is typically 3, but can be set higher if it is felt that
 1347           there is a strong possibility of "false positives."
 1348 
 1349   Maximum   --  Sets the maximum size of the constant of integration. For
 1350   Constant of   technical reasons concerning floating point arithmetic, if
 1351   Integration   the additive constant, i.e., the constant of integration, is
 1352           greater (in absolute value) than maxConstantOfIntegration
 1353           AND is greater than maxConstantOfIntegration times the
 1354           correct value, WeBWorK will give an error message saying
 1355           that it can not handle such a large constant of integration.
 1356           This is to prevent e.g. cos(x) + 1E20 or even 1E20 as being
 1357           accepted as a correct antiderivatives of sin(x) since
 1358           floating point arithmetic cannot tell the difference
 1359           between cos(x) + 1E20, 1E20, and -cos(x) + 1E20.
 1360 
 1361 Technical note: if you examine the code for the function routines, you will see
 1362 that most subroutines are simply doing some basic error-checking and then
 1363 passing the parameters on to the low-level FUNCTION_CMP(). Because this routine
 1364 is set up to handle multivariable functions, with single-variable functions as
 1365 a special case, it is possible to pass multivariable parameters to single-
 1366 variable functions. This usage is strongly discouraged as unnecessarily
 1367 confusing. Avoid it.
 1368 
 1369 Default Values (As of 7/24/2000) (Option -- Variable Name -- Value)
 1370 
 1371   Variable      --  $functVarDefault      --  'x'
 1372   Relative Tolerance    --  $functRelPercentTolDefault    --  .1
 1373   Absolute Tolerance    --  $functAbsTolDefault     --  .001
 1374   Lower Limit     --  $functLLimitDefault     --  .0000001
 1375   Upper Limit     --  $functULimitDefault     --  1
 1376   Number of Points    --  $functNumOfPoints     --  3
 1377   Zero Level      --  $functZeroLevelDefault      --  1E-14
 1378   Zero Level Tolerance    --  $functZeroLevelTolDefault   --  1E-12
 1379   Maximum Constant    --  $functMaxConstantOfIntegration    --  1E8
 1380     of Integration
 1381 
 1382 =cut
 1383 
 1384 
 1385 
 1386 =head3 fun_cmp()
 1387 
 1388 Compares a function or a list of functions, using a named hash of options to set
 1389 parameters. This can make for more readable code than using the function_cmp()
 1390 style, but some people find one or the other easier to remember.
 1391 
 1392 ANS( fun_cmp( answer or answer_array_ref, options_hash ) );
 1393 
 1394   1. a string containing the correct function, or a reference to an
 1395     array of correct functions
 1396   2. a hash containing the following items (all optional):
 1397     var           --  either the number of variables or a reference to an
 1398                       array of variable names (see below)
 1399     limits            --  reference to an array of arrays of limits (see below), or:
 1400     mode            --  'std' (default) (function must match exactly), or:
 1401                     'antider' (function must match up to a constant)
 1402     relTol            --  (default) a relative tolerance (as a percentage), or:
 1403     tol           --  an absolute tolerance for error
 1404     numPoints         --  the number of points to evaluate the function at
 1405     maxConstantOfIntegration      --  maximum size of the constant of integration
 1406     zeroLevel         --  if the correct answer is this close to zero, then
 1407                       zeroLevelTol applies
 1408     zeroLevelTol          --  absolute tolerance to allow when answer is close to zero
 1409     test_points    -- a list of points to use in checking the function, or a list of lists when there is more than one variable.
 1410     params                an array of "free" parameters which can be used to adapt
 1411                     the correct answer to the submitted answer. (e.g. ['c'] for
 1412                     a constant of integration in the answer x^3/3 + c.
 1413     debug           --  when set to 1 this provides extra information while checking the
 1414                         the answer.
 1415 
 1416   Returns an answer evaluator, or (if given a reference to an array
 1417   of answers), a list of answer evaluators
 1418 
 1419 ANSWER:
 1420 
 1421   The answer must be in the form of a string. The answer can contain
 1422   functions, pi, e, and arithmetic operations. However, the correct answer
 1423   string follows a slightly stricter syntax than student answers; specifically,
 1424   there is no implicit multiplication. So the correct answer must be "3*x" rather
 1425   than "3 x". Students can still enter "3 x".
 1426 
 1427 VARIABLES:
 1428 
 1429   The var parameter can contain either a number or a reference to an array of
 1430   variable names. If it contains a number, the variables are named automatically
 1431   as follows: 1 variable  --  x
 1432       2 variables --  x, y
 1433       3 variables --  x, y, z
 1434       4 or more --  x_1, x_2, x_3, etc.
 1435   If the var parameter contains a reference to an array of variable names, then
 1436   the number of variables is determined by the number of items in the array. A
 1437   reference to an array is created with brackets, e.g. "var => ['r', 's', 't']".
 1438   If only one variable is being used, you can write either "var => ['t']" for
 1439   consistency or "var => 't'" as a shortcut. The default is one variable, x.
 1440 
 1441 LIMITS:
 1442 
 1443   Limits are specified with the limits parameter. You may NOT use llimit/ulimit.
 1444   If you specify limits for one variable, you must specify them for all variables.
 1445   The limit parameter must be a reference to an array of arrays of the form
 1446   [lower_limit. upper_limit], each array corresponding to the lower and upper
 1447   endpoints of the (half-open) domain of one variable. For example,
 1448   "vars => 2, limits => [[0,2], [-3,8]]" would cause x to be evaluated in [0,2) and
 1449   y to be evaluated in [-3,8). If only one variable is being used, you can write
 1450   either "limits => [[0,3]]" for consistency or "limits => [0,3]" as a shortcut.
 1451 
 1452 TEST POINTS:
 1453 
 1454   In some cases, the problem writer may want to specify the points
 1455   used to check a particular function.  For example, if you want to
 1456   use only integer values, they can be specified.  With one variable,
 1457   you can specify "test_points => [1,4,5,6]" or "test_points => [[1,4,5,6]]".
 1458   With more variables, specify the list for the first variable, then the
 1459   second, and so on: "vars=>['x','y'], test_points => [[1,4,5],[7,14,29]]".
 1460 
 1461   If the problem writer wants random values which need to meet some special
 1462   restrictions (such as being integers), they can be generated in the problem:
 1463   "test_points=>[random(1,50), random(1,50), random(1,50), random(1,50)]".
 1464 
 1465   Note that test_points should not be used for function checks which involve
 1466   parameters  (either explicitly given by "params", or as antiderivatives).
 1467 
 1468 EXAMPLES:
 1469 
 1470   fun_cmp( "3*x" )  --  standard compare, variable is x
 1471   fun_cmp( ["3*x", "4*x+3", "3*x**2"] ) --  standard compare, defaults used for all three functions
 1472   fun_cmp( "3*t", var => 't' )  --  standard compare, variable is t
 1473   fun_cmp( "5*x*y*z", var => 3 )  --  x, y and z are the variables
 1474   fun_cmp( "5*x", mode => 'antider' ) --  student answer must match up to constant (i.e., 5x+C)
 1475   fun_cmp( ["3*x*y", "4*x*y"], limits => [[0,2], [5,7]] ) --  x evaluated in [0,2)
 1476                                 y evaluated in [5,7)
 1477 
 1478 =cut
 1479 
 1480 sub fun_cmp {
 1481   my $correctAnswer = shift @_;
 1482   my %opt           = @_;
 1483 
 1484     assign_option_aliases( \%opt,
 1485         'vars'    =>  'var',    # set the standard option 'var' to the one specified as vars
 1486           'domain'  =>  'limits', # set the standard option 'limits' to the one specified as domain
 1487           'reltol'    =>  'relTol',
 1488           'param'   =>  'params',
 1489     );
 1490 
 1491     set_default_options(  \%opt,
 1492         'var'         =>  $functVarDefault,
 1493             'params'        =>  [],
 1494         'limits'        =>  [[$functLLimitDefault, $functULimitDefault]],
 1495         'test_points'   => undef,
 1496         'mode'          =>  'std',
 1497         'tolType'       =>    (defined($opt{tol}) ) ? 'absolute' : 'relative',
 1498         'tol'         =>  .01, # default mode should be relative, to obtain this tol must not be defined
 1499             'relTol'        =>  $functRelPercentTolDefault,
 1500         'numPoints'       =>  $functNumOfPoints,
 1501         'maxConstantOfIntegration'  =>  $functMaxConstantOfIntegration,
 1502         'zeroLevel'       =>  $functZeroLevelDefault,
 1503         'zeroLevelTol'      =>  $functZeroLevelTolDefault,
 1504             'debug'         =>  0,
 1505      );
 1506 
 1507     # allow var => 'x' as an abbreviation for var => ['x']
 1508   my %out_options = %opt;
 1509   unless ( ref($out_options{var}) eq 'ARRAY' || $out_options{var} =~ m/^\d+$/) {
 1510     $out_options{var} = [$out_options{var}];
 1511   }
 1512   # allow params => 'c' as an abbreviation for params => ['c']
 1513   unless ( ref($out_options{params}) eq 'ARRAY' ) {
 1514     $out_options{params} = [$out_options{params}];
 1515   }
 1516   my ($tolType, $tol);
 1517     if ($out_options{tolType} eq 'absolute') {
 1518     $tolType = 'absolute';
 1519     $tol = $out_options{'tol'};
 1520     delete($out_options{'relTol'}) if exists( $out_options{'relTol'} );
 1521   } else {
 1522     $tolType = 'relative';
 1523     $tol = $out_options{'relTol'};
 1524     delete($out_options{'tol'}) if exists( $out_options{'tol'} );
 1525   }
 1526 
 1527   my @output_list = ();
 1528   # thread over lists
 1529   my @ans_list = ();
 1530 
 1531   if ( ref($correctAnswer) eq 'ARRAY' ) {
 1532     @ans_list = @{$correctAnswer};
 1533   }
 1534   else {
 1535     push( @ans_list, $correctAnswer );
 1536   }
 1537 
 1538   # produce answer evaluators
 1539   foreach my $ans (@ans_list) {
 1540     push(@output_list,
 1541       FUNCTION_CMP(
 1542           'correctEqn'    =>  $ans,
 1543           'var'       =>  $out_options{'var'},
 1544           'limits'      =>  $out_options{'limits'},
 1545           'tolerance'     =>  $tol,
 1546           'tolType'     =>  $tolType,
 1547           'numPoints'     =>  $out_options{'numPoints'},
 1548           'test_points' =>  $out_options{'test_points'},
 1549           'mode'        =>  $out_options{'mode'},
 1550           'maxConstantOfIntegration'  =>  $out_options{'maxConstantOfIntegration'},
 1551           'zeroLevel'     =>  $out_options{'zeroLevel'},
 1552           'zeroLevelTol'    =>  $out_options{'zeroLevelTol'},
 1553           'params'      =>  $out_options{'params'},
 1554           'debug'       =>  $out_options{'debug'},
 1555       ),
 1556     );
 1557   }
 1558 
 1559   return (wantarray) ? @output_list : $output_list[0];
 1560 }
 1561 
 1562 =head3 Single-variable Function Comparisons
 1563 
 1564 There are four single-variable function answer evaluators: "normal," absolute
 1565 tolerance, antiderivative, and antiderivative with absolute tolerance. All
 1566 parameters (other than the correct equation) are optional.
 1567 
 1568  function_cmp( $correctEqn ) OR
 1569  function_cmp( $correctEqn, $var ) OR
 1570  function_cmp( $correctEqn, $var, $llimit, $ulimit ) OR
 1571  function_cmp( $correctEqn, $var, $llimit, $ulimit, $relPercentTol ) OR
 1572  function_cmp( $correctEqn, $var, $llimit, $ulimit,
 1573         $relPercentTol, $numPoints ) OR
 1574  function_cmp( $correctEqn, $var, $llimit, $ulimit,
 1575         $relPercentTol, $numPoints, $zeroLevel ) OR
 1576  function_cmp( $correctEqn, $var, $llimit, $ulimit, $relPercentTol, $numPoints,
 1577         $zeroLevel,$zeroLevelTol )
 1578 
 1579   $correctEqn   --  the correct equation, as a string
 1580   $var      --  the string representing the variable (optional)
 1581   $llimit     --  the lower limit of the interval to evaluate the
 1582               variable in (optional)
 1583   $ulimit     --  the upper limit of the interval to evaluate the
 1584               variable in (optional)
 1585   $relPercentTol  --  the error tolerance as a percentage (optional)
 1586   $numPoints    --  the number of points at which to evaluate the
 1587               variable (optional)
 1588   $zeroLevel    --  if the correct answer is this close to zero, then
 1589               zeroLevelTol applies (optional)
 1590   $zeroLevelTol --  absolute tolerance to allow when answer is close to zero
 1591 
 1592   function_cmp() uses standard comparison and relative tolerance. It takes a
 1593   string representing a single-variable function and compares the student
 1594   answer to that function numerically.
 1595 
 1596  function_cmp_up_to_constant( $correctEqn ) OR
 1597  function_cmp_up_to_constant( $correctEqn, $var ) OR
 1598  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit ) OR
 1599  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1600                 $relpercentTol ) OR
 1601  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1602                 $relpercentTol, $numOfPoints ) OR
 1603  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1604                 $relpercentTol, $numOfPoints,
 1605                 $maxConstantOfIntegration ) OR
 1606  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1607                 $relpercentTol, $numOfPoints,
 1608                 $maxConstantOfIntegration, $zeroLevel)  OR
 1609  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1610                 $relpercentTol, $numOfPoints,
 1611                 $maxConstantOfIntegration,
 1612                 $zeroLevel, $zeroLevelTol )
 1613 
 1614   $maxConstantOfIntegration --  the maximum size of the constant of
 1615                   integration
 1616 
 1617   function_cmp_up_to_constant() uses antiderivative compare and relative
 1618   tolerance. All options work exactly like function_cmp(), except of course
 1619   $maxConstantOfIntegration. It will accept as correct any function which
 1620   differs from $correctEqn by at most a constant; that is, if
 1621     $studentEqn = $correctEqn + C
 1622   the answer is correct.
 1623 
 1624  function_cmp_abs( $correctFunction ) OR
 1625  function_cmp_abs( $correctFunction, $var ) OR
 1626  function_cmp_abs( $correctFunction, $var, $llimit, $ulimit ) OR
 1627  function_cmp_abs( $correctFunction, $var, $llimit, $ulimit, $absTol ) OR
 1628  function_cmp_abs( $correctFunction, $var, $llimit, $ulimit, $absTol,
 1629           $numOfPoints )
 1630 
 1631   $absTol --  the tolerance as an absolute value
 1632 
 1633   function_cmp_abs() uses standard compare and absolute tolerance. All
 1634   other options work exactly as for function_cmp().
 1635 
 1636  function_cmp_up_to_constant_abs( $correctFunction ) OR
 1637  function_cmp_up_to_constant_abs( $correctFunction, $var ) OR
 1638  function_cmp_up_to_constant_abs( $correctFunction, $var, $llimit, $ulimit ) OR
 1639  function_cmp_up_to_constant_abs( $correctFunction, $var, $llimit, $ulimit,
 1640                   $absTol ) OR
 1641  function_cmp_up_to_constant_abs( $correctFunction, $var, $llimit, $ulimit,
 1642                   $absTol, $numOfPoints ) OR
 1643  function_cmp_up_to_constant_abs( $correctFunction, $var, $llimit, $ulimit,
 1644                   $absTol, $numOfPoints,
 1645                   $maxConstantOfIntegration )
 1646 
 1647   function_cmp_up_to_constant_abs() uses antiderivative compare
 1648   and absolute tolerance. All other options work exactly as with
 1649   function_cmp_up_to_constant().
 1650 
 1651 Examples:
 1652 
 1653   ANS( function_cmp( "cos(x)" ) ) --  Accepts cos(x), sin(x+pi/2),
 1654     sin(x)^2 + cos(x) + cos(x)^2 -1, etc. This assumes
 1655     $functVarDefault has been set to "x".
 1656   ANS( function_cmp( $answer, "t" ) ) --  Assuming $answer is "cos(t)",
 1657     accepts cos(t), etc.
 1658   ANS( function_cmp_up_to_constant( "cos(x)" ) )  --  Accepts any
 1659     antiderivative of sin(x), e.g. cos(x) + 5.
 1660   ANS( function_cmp_up_to_constant( "cos(z)", "z" ) ) --  Accepts any
 1661     antiderivative of sin(z), e.g. sin(z+pi/2) + 5.
 1662 
 1663 =cut
 1664 
 1665 sub adaptive_function_cmp {
 1666   my $correctEqn = shift;
 1667   my %options = @_;
 1668   set_default_options(  \%options,
 1669       'vars'      =>  [qw( x y )],
 1670                   'params'    =>  [],
 1671                   'limits'    =>  [ [0,1], [0,1]],
 1672                   'reltol'    =>  $functRelPercentTolDefault,
 1673                   'numPoints'   =>  $functNumOfPoints,
 1674                   'zeroLevel'   =>  $functZeroLevelDefault,
 1675                   'zeroLevelTol'  =>  $functZeroLevelTolDefault,
 1676                   'debug'     =>  0,
 1677   );
 1678 
 1679     my $var_ref = $options{'vars'};
 1680     my $ra_params = $options{ 'params'};
 1681     my $limit_ref = $options{'limits'};
 1682     my $relPercentTol= $options{'reltol'};
 1683     my $numPoints = $options{'numPoints'};
 1684     my $zeroLevel = $options{'zeroLevel'};
 1685     my $zeroLevelTol = $options{'zeroLevelTol'};
 1686 
 1687   FUNCTION_CMP( 'correctEqn'          =>  $correctEqn,
 1688       'var'           =>  $var_ref,
 1689       'limits'          =>  $limit_ref,
 1690       'tolerance'         =>  $relPercentTol,
 1691       'tolType'         =>  'relative',
 1692       'numPoints'         =>  $numPoints,
 1693       'mode'            =>  'std',
 1694       'maxConstantOfIntegration'      =>  10**100,
 1695       'zeroLevel'         =>  $zeroLevel,
 1696       'zeroLevelTol'          =>  $zeroLevelTol,
 1697       'scale_norm'                      =>    1,
 1698       'params'                          =>    $ra_params,
 1699       'debug'               =>  $options{debug} ,
 1700   );
 1701 }
 1702 
 1703 sub function_cmp {
 1704   my ($correctEqn,$var,$llimit,$ulimit,$relPercentTol,$numPoints,$zeroLevel,$zeroLevelTol) = @_;
 1705 
 1706   if ( (scalar(@_) == 3) or (scalar(@_) > 8) or (scalar(@_) == 0) ) {
 1707     function_invalid_params( $correctEqn );
 1708   }
 1709   else {
 1710     FUNCTION_CMP( 'correctEqn'          =>  $correctEqn,
 1711         'var'           =>  $var,
 1712         'limits'          =>  [$llimit, $ulimit],
 1713         'tolerance'         =>  $relPercentTol,
 1714         'tolType'         =>  'relative',
 1715         'numPoints'         =>  $numPoints,
 1716         'mode'            =>  'std',
 1717         'maxConstantOfIntegration'      =>  0,
 1718         'zeroLevel'         =>  $zeroLevel,
 1719         'zeroLevelTol'          =>  $zeroLevelTol
 1720           );
 1721   }
 1722 }
 1723 
 1724 sub function_cmp_up_to_constant { ## for antiderivative problems
 1725   my ($correctEqn,$var,$llimit,$ulimit,$relPercentTol,$numPoints,$maxConstantOfIntegration,$zeroLevel,$zeroLevelTol) = @_;
 1726 
 1727   if ( (scalar(@_) == 3) or (scalar(@_) > 9) or (scalar(@_) == 0) ) {
 1728     function_invalid_params( $correctEqn );
 1729   }
 1730   else {
 1731     FUNCTION_CMP( 'correctEqn'          =>  $correctEqn,
 1732         'var'           =>  $var,
 1733         'limits'          =>  [$llimit, $ulimit],
 1734         'tolerance'         =>  $relPercentTol,
 1735         'tolType'         =>  'relative',
 1736         'numPoints'         =>  $numPoints,
 1737         'mode'            =>  'antider',
 1738         'maxConstantOfIntegration'      =>  $maxConstantOfIntegration,
 1739         'zeroLevel'         =>  $zeroLevel,
 1740         'zeroLevelTol'          =>  $zeroLevelTol
 1741           );
 1742   }
 1743 }
 1744 
 1745 sub function_cmp_abs {      ## similar to function_cmp but uses absolute tolerance
 1746   my ($correctEqn,$var,$llimit,$ulimit,$absTol,$numPoints) = @_;
 1747 
 1748   if ( (scalar(@_) == 3) or (scalar(@_) > 6) or (scalar(@_) == 0) ) {
 1749     function_invalid_params( $correctEqn );
 1750   }
 1751   else {
 1752     FUNCTION_CMP( 'correctEqn'      =>  $correctEqn,
 1753         'var'       =>  $var,
 1754         'limits'      =>  [$llimit, $ulimit],
 1755         'tolerance'     =>  $absTol,
 1756         'tolType'     =>  'absolute',
 1757         'numPoints'     =>  $numPoints,
 1758         'mode'        =>  'std',
 1759         'maxConstantOfIntegration'  =>  0,
 1760         'zeroLevel'     =>  0,
 1761         'zeroLevelTol'      =>  0
 1762     );
 1763   }
 1764 }
 1765 
 1766 
 1767 sub function_cmp_up_to_constant_abs  {  ## for antiderivative problems
 1768                     ## similar to function_cmp_up_to_constant
 1769                     ## but uses absolute tolerance
 1770   my ($correctEqn,$var,$llimit,$ulimit,$absTol,$numPoints,$maxConstantOfIntegration) = @_;
 1771 
 1772   if ( (scalar(@_) == 3) or (scalar(@_) > 7) or (scalar(@_) == 0) ) {
 1773     function_invalid_params( $correctEqn );
 1774   }
 1775 
 1776   else {
 1777     FUNCTION_CMP( 'correctEqn'          =>  $correctEqn,
 1778         'var'           =>  $var,
 1779         'limits'          =>  [$llimit, $ulimit],
 1780         'tolerance'         =>  $absTol,
 1781         'tolType'         =>  'absolute',
 1782         'numPoints'         =>  $numPoints,
 1783         'mode'            =>  'antider',
 1784         'maxConstantOfIntegration'      =>  $maxConstantOfIntegration,
 1785         'zeroLevel'         =>  0,
 1786         'zeroLevelTol'          =>  0
 1787     );
 1788   }
 1789 }
 1790 
 1791 ## The following answer evaluator for comparing multivarable functions was
 1792 ## contributed by Professor William K. Ziemer
 1793 ## (Note: most of the multivariable functionality provided by Professor Ziemer
 1794 ## has now been integrated into fun_cmp and FUNCTION_CMP)
 1795 ############################
 1796 # W.K. Ziemer, Sep. 1999
 1797 # Math Dept. CSULB
 1798 # email: wziemer@csulb.edu
 1799 ############################
 1800 
 1801 =head3 multivar_function_cmp
 1802 
 1803 NOTE: this function is maintained for compatibility. fun_cmp() is
 1804     slightly preferred.
 1805 
 1806 usage:
 1807 
 1808   multivar_function_cmp( $answer, $var_reference, options)
 1809     $answer       --  string, represents function of several variables
 1810     $var_reference    --  number (of variables), or list reference (e.g. ["var1","var2"] )
 1811   options:
 1812     $limit_reference  --  reference to list of lists (e.g. [[1,2],[3,4]])
 1813     $relPercentTol    --  relative percent tolerance in answer
 1814     $numPoints      --  number of points to sample in for each variable
 1815     $zeroLevel      --  if the correct answer is this close to zero, then zeroLevelTol applies
 1816     $zeroLevelTol   --  absolute tolerance to allow when answer is close to zero
 1817 
 1818 =cut
 1819 
 1820 sub multivar_function_cmp {
 1821   my ($correctEqn,$var_ref,$limit_ref,$relPercentTol,$numPoints,$zeroLevel,$zeroLevelTol) = @_;
 1822 
 1823   if ( (scalar(@_) > 7) or (scalar(@_) < 2) ) {
 1824     function_invalid_params( $correctEqn );
 1825   }
 1826 
 1827   FUNCTION_CMP( 'correctEqn'      =>  $correctEqn,
 1828       'var'       =>  $var_ref,
 1829       'limits'      =>  $limit_ref,
 1830       'tolerance'     =>  $relPercentTol,
 1831       'tolType'     =>  'relative',
 1832       'numPoints'     =>  $numPoints,
 1833       'mode'        =>  'std',
 1834       'maxConstantOfIntegration'  =>  0,
 1835       'zeroLevel'     =>  $zeroLevel,
 1836       'zeroLevelTol'      =>  $zeroLevelTol
 1837   );
 1838 }
 1839 
 1840 ## LOW-LEVEL ROUTINE -- NOT NORMALLY FOR END USERS -- USE WITH CAUTION
 1841 ## NOTE: PG_answer_eval is used instead of PG_restricted_eval in order to insure that the answer
 1842 ## evaluated within the context of the package the problem was originally defined in.
 1843 ## Includes multivariable modifications contributed by Professor William K. Ziemer
 1844 ##
 1845 ## IN:  a hash consisting of the following keys (error checking to be added later?)
 1846 ##      correctEqn      --  the correct equation as a string
 1847 ##      var       --  the variable name as a string,
 1848 ##                or a reference to an array of variables
 1849 ##      limits        --  reference to an array of arrays of type [lower,upper]
 1850 ##      tolerance     --  the allowable margin of error
 1851 ##      tolType       --  'relative' or 'absolute'
 1852 ##      numPoints     --  the number of points to evaluate the function at
 1853 ##      mode        --  'std' or 'antider'
 1854 ##      maxConstantOfIntegration  --  maximum size of the constant of integration
 1855 ##      zeroLevel     --  if the correct answer is this close to zero,
 1856 ##                        then zeroLevelTol applies
 1857 ##      zeroLevelTol      --  absolute tolerance to allow when answer is close to zero
 1858 ##      test_points     --  user supplied points to use for testing the
 1859 ##                          function, either array of arrays, or optionally
 1860 ##                          reference to single array (for one variable)
 1861 
 1862 
 1863 sub FUNCTION_CMP {
 1864   return ORIGINAL_FUNCTION_CMP(@_)
 1865     if main::PG_restricted_eval(q!$main::useOldAnswerMacros!);
 1866 
 1867   my %func_params = @_;
 1868 
 1869   my $correctEqn               = $func_params{'correctEqn'};
 1870   my $var                      = $func_params{'var'};
 1871   my $ra_limits                = $func_params{'limits'};
 1872   my $tol                      = $func_params{'tolerance'};
 1873   my $tolType                  = $func_params{'tolType'};
 1874   my $numPoints                = $func_params{'numPoints'};
 1875   my $mode                     = $func_params{'mode'};
 1876   my $maxConstantOfIntegration = $func_params{'maxConstantOfIntegration'};
 1877   my $zeroLevel                = $func_params{'zeroLevel'};
 1878   my $zeroLevelTol             = $func_params{'zeroLevelTol'};
 1879   my $testPoints               = $func_params{'test_points'};
 1880 
 1881   #
 1882   #  Check that everything is defined:
 1883   #
 1884   $func_params{debug} = 0 unless defined $func_params{debug};
 1885   $mode = 'std' unless defined $mode;
 1886   my @VARS   = get_var_array($var);
 1887   my @limits = get_limits_array($ra_limits);
 1888   my @PARAMS = @{$func_params{'params'} || []};
 1889 
 1890   if($tolType eq 'relative') {
 1891     $tol = $functRelPercentTolDefault unless defined $tol;
 1892     $tol *= .01;
 1893   } else {
 1894     $tol = $functAbsTolDefault unless defined $tol;
 1895   }
 1896 
 1897   #
 1898   #  Ensure that the number of limits matches number of variables
 1899   #
 1900   foreach my $i (0..scalar(@VARS)-1) {
 1901     $limits[$i][0] = $functLLimitDefault unless defined $limits[$i][0];
 1902     $limits[$i][1] = $functULimitDefault unless defined $limits[$i][1];
 1903   }
 1904 
 1905   #
 1906   #  Check that the test points are array references with the right number of coordinates
 1907   #
 1908   if ($testPoints) {
 1909     my $n = scalar(@VARS); my $s = ($n != 1)? "s": "";
 1910     foreach my $p (@{$testPoints}) {
 1911       $p = [$p] unless ref($p) eq 'ARRAY';
 1912       warn "Test point (".join(',',@{$p}).") should have $n coordiante$s"
 1913         unless scalar(@{$p}) == $n;
 1914     }
 1915   }
 1916 
 1917   $numPoints                = $functNumOfPoints              unless defined $numPoints;
 1918   $maxConstantOfIntegration = $functMaxConstantOfIntegration unless defined $maxConstantOfIntegration;
 1919   $zeroLevel                = $functZeroLevelDefault         unless defined $zeroLevel;
 1920   $zeroLevelTol             = $functZeroLevelTolDefault      unless defined $zeroLevelTol;
 1921 
 1922   $func_params{'var'}                      = \@VARS;
 1923         $func_params{'params'}                   = \@PARAMS;
 1924   $func_params{'limits'}                   = \@limits;
 1925   $func_params{'tolerance'}                = $tol;
 1926   $func_params{'tolType'}                  = $tolType;
 1927   $func_params{'numPoints'}                = $numPoints;
 1928   $func_params{'mode'}                     = $mode;
 1929   $func_params{'maxConstantOfIntegration'} = $maxConstantOfIntegration;
 1930   $func_params{'zeroLevel'}                = $zeroLevel;
 1931   $func_params{'zeroLevelTol'}             = $zeroLevelTol;
 1932 
 1933   ########################################################
 1934   #   End of cleanup of calling parameters
 1935   ########################################################
 1936 
 1937         my %options = (debug => $func_params{'debug'});
 1938 
 1939   #
 1940   #  Initialize the context for the formula
 1941   #
 1942   my $context = $Parser::Context::Default::context{"LegacyNumeric"}->copy;
 1943   $context->flags->set(
 1944     tolerance    => $func_params{'tolerance'},
 1945     tolType      => $func_params{'tolType'},
 1946     zeroLevel    => $func_params{'zeroLevel'},
 1947     zeroLevelTol => $func_params{'zeroLevelTol'},
 1948     num_points   => $func_params{'numPoints'},
 1949   );
 1950   if ($func_params{'mode'} eq 'antider') {
 1951     $context->flags->set(max_adapt => $func_params{'maxConstantOfIntegration'});
 1952     $options{upToConstant} = 1;
 1953   }
 1954 
 1955   #
 1956   #  Add the variables and parameters to the context
 1957   #
 1958   my %variables; my $x;
 1959   foreach $x (@{$func_params{'var'}}) {
 1960     if (length($x) > 1) {
 1961       $context->{_variables}->{pattern} = $context->{_variables}->{namePattern} =
 1962         $x . '|' . $context->{_variables}->{pattern};
 1963       $context->update;
 1964     }
 1965     $variables{$x} = 'Real';
 1966   }
 1967   foreach $x (@{$func_params{'params'}}) {$variables{$x} = 'Parameter'}
 1968   $context->variables->are(%variables);
 1969 
 1970   #
 1971   #  Create the Formula object and get its answer checker
 1972   #
 1973   my $oldContext = &$Context($context);
 1974   my $f = new Value::Formula($correctEqn);
 1975   $f->{limits}      = $func_params{'limits'};
 1976   $f->{test_points} = $func_params{'test_points'};
 1977   my $cmp = $f->cmp(%options);
 1978   $cmp->{debug} = 1 if $func_params{'debug'};
 1979   &$Context($oldContext);
 1980 
 1981   #
 1982   #  Get previous answer from hidden field of form
 1983   #
 1984   $cmp->install_pre_filter(
 1985     sub {
 1986       my $rh_ans = shift;
 1987       $rh_ans->{_filter_name} = "fetch_previous_answer";
 1988       my $prev_ans_label = "previous_".$rh_ans->{ans_label};
 1989       $rh_ans->{prev_ans} =
 1990         (defined $inputs_ref->{$prev_ans_label} and
 1991          $inputs_ref->{$prev_ans_label} =~/\S/) ? $inputs_ref->{$prev_ans_label} : undef;
 1992       $rh_ans;
 1993     }
 1994   );
 1995 
 1996   #
 1997   #  Parse the previous answer, if any
 1998   #
 1999   $cmp->install_pre_filter(
 2000     sub {
 2001       my $rh_ans = shift;
 2002       $rh_ans->{_filter_name} = "parse_previous_answer";
 2003       return $rh_ans unless defined $rh_ans->{prev_ans};
 2004       $rh_ans->{prev_formula} = Parser::Formula($rh_ans->{prev_ans});
 2005       $rh_ans;
 2006     }
 2007   );
 2008 
 2009   #
 2010   #  Check if previous answer equals this current one
 2011   #
 2012   $cmp->install_evaluator(
 2013     sub {
 2014       my $rh_ans = shift;
 2015       $rh_ans->{_filter_name} = "compare_to_previous_answer";
 2016       return $rh_ans unless defined($rh_ans->{prev_formula}) && defined($rh_ans->{student_formula});
 2017       $rh_ans->{prev_equals_current} =
 2018         Value::cmp_compare($rh_ans->{student_formula},$rh_ans->{prev_formula},{});
 2019       $rh_ans;
 2020     }
 2021   );
 2022 
 2023   #
 2024   #  Produce a message if the previous answer equals this one
 2025   #  (and is not correct, and is not specified the same way)
 2026   #
 2027   $cmp->install_post_filter(
 2028     sub {
 2029       my $rh_ans = shift;
 2030       $rh_ans->{_filter_name} = "produce_equivalence_message";
 2031       return $rh_ans unless $rh_ans->{prev_equals_current} && $rh_ans->{score} == 0;
 2032       return $rh_ans if $rh_ans->{prev_ans} eq $rh_ans->{original_student_ans};
 2033       $rh_ans->{ans_message} = "This answer is equivalent to the one you just submitted or previewed.";
 2034       $rh_ans;
 2035     }
 2036   );
 2037 
 2038   return $cmp;
 2039 }
 2040 
 2041 #
 2042 #  The original version, for backward compatibility
 2043 #  (can be removed when the Parser-based version is more fully tested.)
 2044 #
 2045 sub ORIGINAL_FUNCTION_CMP {
 2046   my %func_params = @_;
 2047 
 2048   my $correctEqn               = $func_params{'correctEqn'};
 2049   my $var                      = $func_params{'var'};
 2050   my $ra_limits                = $func_params{'limits'};
 2051   my $tol                      = $func_params{'tolerance'};
 2052   my $tolType                  = $func_params{'tolType'};
 2053   my $numPoints                = $func_params{'numPoints'};
 2054   my $mode                     = $func_params{'mode'};
 2055   my $maxConstantOfIntegration = $func_params{'maxConstantOfIntegration'};
 2056   my $zeroLevel                = $func_params{'zeroLevel'};
 2057   my $zeroLevelTol             = $func_params{'zeroLevelTol'};
 2058   my $ra_test_points           = $func_params{'test_points'};
 2059 
 2060     # Check that everything is defined:
 2061     $func_params{debug} = 0 unless defined $func_params{debug};
 2062     $mode = 'std' unless defined $mode;
 2063     my @VARS = get_var_array($var);
 2064   my @limits = get_limits_array($ra_limits);
 2065   my @PARAMS = ();
 2066   @PARAMS = @{$func_params{'params'}} if defined $func_params{'params'};
 2067 
 2068   my @evaluation_points;
 2069   if(defined $ra_test_points) {
 2070     # see if this is the standard format
 2071     if(ref $ra_test_points->[0] eq 'ARRAY') {
 2072       $numPoints = scalar @{$ra_test_points->[0]};
 2073       # now a little sanity check
 2074       my $j;
 2075       for $j (@{$ra_test_points}) {
 2076         warn "Test points do not give the same number of values for each variable"
 2077           unless(scalar(@{$j}) == $numPoints);
 2078       }
 2079       warn "Test points do not match the number of variables"
 2080         unless scalar @{$ra_test_points} == scalar @VARS;
 2081     } else { # we are got the one-variable format
 2082       $ra_test_points = [$ra_test_points];
 2083       $numPoints = scalar $ra_test_points->[0];
 2084     }
 2085     # The input format for test points is the transpose of what is used
 2086     # internally below, so take care of that now.
 2087     my ($j1, $j2);
 2088     for ($j1 = 0; $j1 < scalar @{$ra_test_points}; $j1++) {
 2089       for ($j2 = 0; $j2 < scalar @{$ra_test_points->[$j1]}; $j2++) {
 2090         $evaluation_points[$j2][$j1] = $ra_test_points->[$j1][$j2];
 2091       }
 2092     }
 2093   } # end of handling of user supplied evaluation points
 2094 
 2095   if ($mode eq 'antider') {
 2096     # doctor the equation to allow addition of a constant
 2097     my $CONSTANT_PARAM = 'Q'; # unfortunately parameters must be single letters.
 2098                               # There is the possibility of conflict here.
 2099                               #  'Q' seemed less dangerous than  'C'.
 2100     $correctEqn = "( $correctEqn ) + $CONSTANT_PARAM";
 2101     push @PARAMS, $CONSTANT_PARAM;
 2102   }
 2103     my $dim_of_param_space = @PARAMS;      # dimension of equivalence space
 2104 
 2105   if($tolType eq 'relative') {
 2106     $tol = $functRelPercentTolDefault unless defined $tol;
 2107     $tol *= .01;
 2108   } else {
 2109     $tol = $functAbsTolDefault unless defined $tol;
 2110   }
 2111 
 2112   #loop ensures that number of limits matches number of variables
 2113   for(my $i = 0; $i < scalar @VARS; $i++) {
 2114     $limits[$i][0] = $functLLimitDefault unless defined $limits[$i][0];
 2115     $limits[$i][1] = $functULimitDefault unless defined $limits[$i][1];
 2116   }
 2117   $numPoints                = $functNumOfPoints              unless defined $numPoints;
 2118   $maxConstantOfIntegration = $functMaxConstantOfIntegration unless defined $maxConstantOfIntegration;
 2119   $zeroLevel                = $functZeroLevelDefault         unless defined $zeroLevel;
 2120   $zeroLevelTol             = $functZeroLevelTolDefault      unless defined $zeroLevelTol;
 2121 
 2122   $func_params{'var'}                      = $var;
 2123   $func_params{'limits'}                   = \@limits;
 2124   $func_params{'tolerance'}                = $tol;
 2125   $func_params{'tolType'}                  = $tolType;
 2126   $func_params{'numPoints'}                = $numPoints;
 2127   $func_params{'mode'}                     = $mode;
 2128   $func_params{'maxConstantOfIntegration'} = $maxConstantOfIntegration;
 2129   $func_params{'zeroLevel'}                = $zeroLevel;
 2130   $func_params{'zeroLevelTol'}             = $zeroLevelTol;
 2131 
 2132   ########################################################
 2133   #   End of cleanup of calling parameters
 2134   ########################################################
 2135 
 2136   my $i; # for use with loops
 2137   my $PGanswerMessage = "";
 2138   my $originalCorrEqn = $correctEqn;
 2139 
 2140   ######################################################################
 2141   # prepare the correct answer and check its syntax
 2142   ######################################################################
 2143 
 2144     my $rh_correct_ans = new AnswerHash;
 2145   $rh_correct_ans->input($correctEqn);
 2146   $rh_correct_ans = check_syntax($rh_correct_ans);
 2147   warn  $rh_correct_ans->{error_message} if $rh_correct_ans->{error_flag};
 2148   $rh_correct_ans->clear_error();
 2149   $rh_correct_ans = function_from_string2($rh_correct_ans,
 2150     ra_vars => [ @VARS, @PARAMS ],
 2151     stdout  => 'rf_correct_ans',
 2152     debug   => $func_params{debug}
 2153   );
 2154   my $correct_eqn_sub = $rh_correct_ans->{rf_correct_ans};
 2155   warn $rh_correct_ans->{error_message} if $rh_correct_ans->{error_flag};
 2156 
 2157   ######################################################################
 2158   # define the points at which the functions are to be evaluated
 2159   ######################################################################
 2160 
 2161   if(not defined $ra_test_points) {
 2162     #create the evaluation points
 2163     my $random_for_answers = new PGrandom($main::PG_original_problemSeed);
 2164     my $NUMBER_OF_STEPS_IN_RANDOM = 1000; # determines the granularity of the random_for_answers number generator
 2165     for(my $count = 0; $count < @PARAMS+1+$numPoints; $count++) {
 2166         my (@vars,$iteration_limit);
 2167       for(my $i = 0; $i < @VARS; $i++) {
 2168         my $iteration_limit = 10;
 2169         while (0 < --$iteration_limit) {  # make sure that the endpoints of the interval are not included
 2170             $vars[$i] = $random_for_answers->random($limits[$i][0], $limits[$i][1], abs($limits[$i][1] - $limits[$i][0])/$NUMBER_OF_STEPS_IN_RANDOM);
 2171             last if $vars[$i]!=$limits[$i][0] and $vars[$i]!=$limits[$i][1];
 2172         }
 2173         warn "Unable to properly choose  evaluation points for this function in the interval ( $limits[$i][0] , $limits[$i][1] )"
 2174           if $iteration_limit == 0;
 2175       }
 2176 
 2177       push @evaluation_points, \@vars;
 2178     }
 2179   }
 2180   my $evaluation_points = Matrix->new_from_array_ref(\@evaluation_points);
 2181 
 2182   #my $COEFFS = determine_param_coeffs($correct_eqn_sub,$evaluation_points[0],$numOfParameters);
 2183   #warn "coeff", join(" | ", @{$COEFFS});
 2184 
 2185   #construct the answer evaluator
 2186     my $answer_evaluator = new AnswerEvaluator;
 2187     $answer_evaluator->{debug} = $func_params{debug};
 2188     $answer_evaluator->ans_hash(
 2189     correct_ans       => $originalCorrEqn,
 2190     rf_correct_ans    => $rh_correct_ans->{rf_correct_ans},
 2191     evaluation_points => \@evaluation_points,
 2192     ra_param_vars     => \@PARAMS,
 2193     ra_vars           => \@VARS,
 2194     type              => 'function',
 2195     score             => 0,
 2196     );
 2197 
 2198     #########################################################
 2199     # Prepare the previous answer for evaluation, discard errors
 2200     #########################################################
 2201 
 2202   $answer_evaluator->install_pre_filter(
 2203     sub {
 2204       my $rh_ans = shift;
 2205       $rh_ans->{_filter_name} = "fetch_previous_answer";
 2206       my $prev_ans_label = "previous_".$rh_ans->{ans_label};
 2207       $rh_ans->{prev_ans} = (defined $inputs_ref->{$prev_ans_label} and $inputs_ref->{$prev_ans_label} =~/\S/)
 2208         ? $inputs_ref->{$prev_ans_label}
 2209         : undef;
 2210       $rh_ans;
 2211     }
 2212   );
 2213 
 2214   $answer_evaluator->install_pre_filter(
 2215     sub {
 2216       my $rh_ans = shift;
 2217       return $rh_ans unless defined $rh_ans->{prev_ans};
 2218       check_syntax($rh_ans,
 2219         stdin          => 'prev_ans',
 2220         stdout         => 'prev_ans',
 2221         error_msg_flag => 0
 2222       );
 2223       $rh_ans->{_filter_name} = "check_syntax_of_previous_answer";
 2224       $rh_ans;
 2225     }
 2226   );
 2227 
 2228   $answer_evaluator->install_pre_filter(
 2229     sub {
 2230       my $rh_ans = shift;
 2231       return $rh_ans unless defined $rh_ans->{prev_ans};
 2232       function_from_string2($rh_ans,
 2233         stdin   => 'prev_ans',
 2234         stdout  => 'rf_prev_ans',
 2235         ra_vars => \@VARS,
 2236         debug   => $func_params{debug}
 2237       );
 2238       $rh_ans->{_filter_name} = "compile_previous_answer";
 2239       $rh_ans;
 2240     }
 2241   );
 2242 
 2243     #########################################################
 2244     # Prepare the current answer for evaluation
 2245     #########################################################
 2246 
 2247   $answer_evaluator->install_pre_filter(\&check_syntax);
 2248   $answer_evaluator->install_pre_filter(\&function_from_string2,
 2249     ra_vars => \@VARS,
 2250     debug   => $func_params{debug}
 2251     ); # @VARS has been guaranteed to be an array, $var might be a single string.
 2252 
 2253     #########################################################
 2254     # Compare the previous and current answer.  Discard errors
 2255     #########################################################
 2256 
 2257   $answer_evaluator->install_evaluator(
 2258     sub {
 2259       my $rh_ans = shift;
 2260       return $rh_ans unless defined $rh_ans->{rf_prev_ans};
 2261       calculate_difference_vector($rh_ans,
 2262         %func_params,
 2263         stdin1         => 'rf_student_ans',
 2264         stdin2         => 'rf_prev_ans',
 2265         stdout         => 'ra_diff_with_prev_ans',
 2266         error_msg_flag => 0,
 2267       );
 2268       $rh_ans->{_filter_name} = "calculate_difference_vector_of_previous_answer";
 2269       $rh_ans;
 2270     }
 2271   );
 2272 
 2273   $answer_evaluator->install_evaluator(
 2274     sub {
 2275       my $rh_ans = shift;
 2276       return $rh_ans unless defined $rh_ans->{ra_diff_with_prev_ans};
 2277       ##
 2278       ## DPVC -- only give the message if the answer is specified differently
 2279       ##
 2280       return $rh_ans if $rh_ans->{prev_ans} eq $rh_ans->{student_ans};
 2281       ##
 2282       ## /DPVC
 2283       ##
 2284       is_zero_array($rh_ans,
 2285         stdin  => 'ra_diff_with_prev_ans',
 2286         stdout => 'ans_equals_prev_ans'
 2287       );
 2288     }
 2289   );
 2290 
 2291     #########################################################
 2292     # Calculate values for approximation parameters and
 2293     # compare the current answer with the correct answer.  Keep errors this time.
 2294     #########################################################
 2295 
 2296     $answer_evaluator->install_pre_filter(\&best_approx_parameters, %func_params, param_vars => \@PARAMS);
 2297     $answer_evaluator->install_evaluator(\&calculate_difference_vector, %func_params);
 2298     $answer_evaluator->install_evaluator(\&is_zero_array, tolerance => $tol );
 2299 
 2300     $answer_evaluator->install_post_filter(
 2301       sub {
 2302         my $rh_ans = shift;
 2303         $rh_ans->clear_error('SYNTAX');
 2304         $rh_ans;
 2305       }
 2306     );
 2307 
 2308   $answer_evaluator->install_post_filter(
 2309     sub {
 2310       my $rh_ans = shift;
 2311       if ($rh_ans->catch_error('EVAL')) {
 2312         $rh_ans->{ans_message} = $rh_ans->{error_message};
 2313         $rh_ans->clear_error('EVAL');
 2314       }
 2315       $rh_ans;
 2316     }
 2317   );
 2318 
 2319   $answer_evaluator->install_post_filter(
 2320     sub {
 2321       my $rh_ans = shift;
 2322       if ( defined($rh_ans->{'ans_equals_prev_ans'}) and $rh_ans->{'ans_equals_prev_ans'} and $rh_ans->{score}==0) {
 2323 ##        $rh_ans->{ans_message} = "This answer is the same as the one you just submitted or previewed.";
 2324         $rh_ans->{ans_message} = "This answer is equivalent to the one you just submitted or previewed."; ## DPVC
 2325       }
 2326       $rh_ans;
 2327     }
 2328   );
 2329 
 2330   $answer_evaluator;
 2331 }
 2332 
 2333 
 2334 ## LOW-LEVEL ROUTINE -- NOT NORMALLY FOR END USERS -- USE WITH CAUTION
 2335 ##
 2336 ## IN:  a hash containing the following items (error-checking to be added later?):
 2337 ##      correctAnswer --  the correct answer
 2338 ##      tolerance   --  the allowable margin of error
 2339 ##      tolType     --  'relative' or 'absolute'
 2340 ##      format      --  the display format of the answer
 2341 ##      mode      --  one of 'std', 'strict', 'arith', or 'frac';
 2342 ##                  determines allowable formats for the input
 2343 ##      zeroLevel   --  if the correct answer is this close to zero, then zeroLevelTol applies
 2344 ##      zeroLevelTol  --  absolute tolerance to allow when answer is close to zero
 2345 
 2346 
 2347 ##########################################################################
 2348 ##########################################################################
 2349 ## String answer evaluators
 2350 
 2351 =head2 String Answer Evaluators
 2352 
 2353 String answer evaluators compare a student string to the correct string.
 2354 Different filters can be applied to allow various degrees of variation.
 2355 Both the student and correct answers are subject to the same filters, to
 2356 ensure that there are no unexpected matches or rejections.
 2357 
 2358 String Filters
 2359 
 2360   remove_whitespace --  Removes all whitespace from the string.
 2361             It applies the following substitution
 2362             to the string:
 2363               $filteredAnswer =~ s/\s+//g;
 2364 
 2365   compress_whitespace --  Removes leading and trailing whitespace, and
 2366             replaces all other blocks of whitespace by a
 2367             single space. Applies the following substitutions:
 2368               $filteredAnswer =~ s/^\s*//;
 2369               $filteredAnswer =~ s/\s*$//;
 2370               $filteredAnswer =~ s/\s+/ /g;
 2371 
 2372   trim_whitespace   --  Removes leading and trailing whitespace.
 2373             Applies the following substitutions:
 2374               $filteredAnswer =~ s/^\s*//;
 2375               $filteredAnswer =~ s/\s*$//;
 2376 
 2377   ignore_case     --  Ignores the case of the string. More accurately,
 2378             it converts the string to uppercase (by convention).
 2379             Applies the following function:
 2380               $filteredAnswer = uc $filteredAnswer;
 2381 
 2382   ignore_order    --  Ignores the order of the letters in the string.
 2383             This is used for problems of the form "Choose all
 2384             that apply." Specifically, it removes all
 2385             whitespace and lexically sorts the letters in
 2386             ascending alphabetical order. Applies the following
 2387             functions:
 2388               $filteredAnswer = join( "", lex_sort(
 2389                 split( /\s*/, $filteredAnswer ) ) );
 2390 
 2391 =cut
 2392 
 2393 ################################
 2394 ## STRING ANSWER FILTERS
 2395 
 2396 ## IN:  --the string to be filtered
 2397 ##    --a list of the filters to use
 2398 ##
 2399 ## OUT: --the modified string
 2400 ##
 2401 ## Use this subroutine instead of the
 2402 ## individual filters below it
 2403 
 2404 sub str_filters {
 2405   my $stringToFilter = shift @_;
 2406   # filters now take an answer hash, so encapsulate the string
 2407   # in the answer hash.
 2408   my $rh_ans = new AnswerHash;
 2409   $rh_ans->{student_ans} = $stringToFilter;
 2410   $rh_ans->{correct_ans}='';
 2411   my @filters_to_use = @_;
 2412   my %known_filters = (
 2413               'remove_whitespace'   =>  \&remove_whitespace,
 2414         'compress_whitespace' =>  \&compress_whitespace,
 2415         'trim_whitespace'   =>  \&trim_whitespace,
 2416         'ignore_case'     =>  \&ignore_case,
 2417         'ignore_order'      =>  \&ignore_order,
 2418   );
 2419 
 2420   #test for unknown filters
 2421   foreach my $filter ( @filters_to_use ) {
 2422     #check that filter is known
 2423     die "Unknown string filter $filter (try checking the parameters to str_cmp() )"
 2424                 unless exists $known_filters{$filter};
 2425     $rh_ans = $known_filters{$filter}($rh_ans);  # apply filter.
 2426   }
 2427 #   foreach $filter (@filters_to_use) {
 2428 #     die "Unknown string filter $filter (try checking the parameters to str_cmp() )"
 2429 #                 unless exists $known_filters{$filter};
 2430 #   }
 2431 #
 2432 #   if( grep( /remove_whitespace/i, @filters_to_use ) ) {
 2433 #     $rh_ans = remove_whitespace( $rh_ans );
 2434 #   }
 2435 #   if( grep( /compress_whitespace/i, @filters_to_use ) ) {
 2436 #     $rh_ans = compress_whitespace( $rh_ans );
 2437 #   }
 2438 #   if( grep( /trim_whitespace/i, @filters_to_use ) ) {
 2439 #     $rh_ans = trim_whitespace( $rh_ans );
 2440 #   }
 2441 #   if( grep( /ignore_case/i, @filters_to_use ) ) {
 2442 #     $rh_ans = ignore_case( $rh_ans );
 2443 #   }
 2444 #   if( grep( /ignore_order/i, @filters_to_use ) ) {
 2445 #     $rh_ans = ignore_order( $rh_ans );
 2446 #   }
 2447 
 2448   return $rh_ans->{student_ans};
 2449 }
 2450 sub remove_whitespace {
 2451   my $rh_ans = shift;
 2452   die "expected an answer hash" unless ref($rh_ans)=~/HASH/i;
 2453   $rh_ans->{_filter_name} = 'remove_whitespace';
 2454   $rh_ans->{student_ans} =~ s/\s+//g;   # remove all whitespace
 2455   $rh_ans->{correct_ans} =~ s/\s+//g;   # remove all whitespace
 2456   return $rh_ans;
 2457 }
 2458 
 2459 sub compress_whitespace {
 2460   my $rh_ans = shift;
 2461   die "expected an answer hash" unless ref($rh_ans)=~/HASH/i;
 2462   $rh_ans->{_filter_name} = 'compress_whitespace';
 2463   $rh_ans->{student_ans} =~ s/^\s*//;   # remove initial whitespace
 2464   $rh_ans->{student_ans} =~ s/\s*$//;   # remove trailing whitespace
 2465   $rh_ans->{student_ans} =~ s/\s+/ /g;    # replace spaces by single space
 2466   $rh_ans->{correct_ans} =~ s/^\s*//;   # remove initial whitespace
 2467   $rh_ans->{correct_ans} =~ s/\s*$//;   # remove trailing whitespace
 2468   $rh_ans->{correct_ans} =~ s/\s+/ /g;    # replace spaces by single space
 2469 
 2470   return $rh_ans;
 2471 }
 2472 
 2473 sub trim_whitespace {
 2474   my $rh_ans = shift;
 2475   die "expected an answer hash" unless ref($rh_ans)=~/HASH/i;
 2476   $rh_ans->{_filter_name} = 'trim_whitespace';
 2477   $rh_ans->{student_ans} =~ s/^\s*//;   # remove initial whitespace
 2478   $rh_ans->{student_ans} =~ s/\s*$//;   # remove trailing whitespace
 2479   $rh_ans->{correct_ans} =~ s/^\s*//;   # remove initial whitespace
 2480   $rh_ans->{correct_ans} =~ s/\s*$//;   # remove trailing whitespace
 2481 
 2482   return $rh_ans;
 2483 }
 2484 
 2485 sub ignore_case {
 2486   my $rh_ans = shift;
 2487   die "expected an answer hash" unless ref($rh_ans)=~/HASH/i;
 2488   $rh_ans->{_filter_name} = 'ignore_case';
 2489   $rh_ans->{student_ans} =~ tr/a-z/A-Z/;
 2490   $rh_ans->{correct_ans} =~ tr/a-z/A-Z/;
 2491   return $rh_ans;
 2492 }
 2493 
 2494 sub ignore_order {
 2495   my $rh_ans = shift;
 2496   die "expected an answer hash" unless ref($rh_ans)=~/HASH/i;
 2497   $rh_ans->{_filter_name} = 'ignore_order';
 2498   $rh_ans->{student_ans} = join( "", lex_sort( split( /\s*/, $rh_ans->{student_ans} ) ) );
 2499   $rh_ans->{correct_ans} = join( "", lex_sort( split( /\s*/, $rh_ans->{correct_ans} ) ) );
 2500 
 2501   return $rh_ans;
 2502 }
 2503 # sub remove_whitespace {
 2504 #   my $filteredAnswer = shift;
 2505 #
 2506 #   $filteredAnswer =~ s/\s+//g;    # remove all whitespace
 2507 #
 2508 #   return $filteredAnswer;
 2509 # }
 2510 #
 2511 # sub compress_whitespace {
 2512 #   my $filteredAnswer = shift;
 2513 #
 2514 #   $filteredAnswer =~ s/^\s*//;    # remove initial whitespace
 2515 #   $filteredAnswer =~ s/\s*$//;    # remove trailing whitespace
 2516 #   $filteredAnswer =~ s/\s+/ /g;   # replace spaces by single space
 2517 #
 2518 #   return $filteredAnswer;
 2519 # }
 2520 #
 2521 # sub trim_whitespace {
 2522 #   my $filteredAnswer = shift;
 2523 #
 2524 #   $filteredAnswer =~ s/^\s*//;    # remove initial whitespace
 2525 #   $filteredAnswer =~ s/\s*$//;    # remove trailing whitespace
 2526 #
 2527 #   return $filteredAnswer;
 2528 # }
 2529 #
 2530 # sub ignore_case {
 2531 #   my $filteredAnswer = shift;
 2532 #   #warn "filtered answer is ", $filteredAnswer;
 2533 #   #$filteredAnswer = uc $filteredAnswer;  # this didn't work on webwork xmlrpc, but does elsewhere ????
 2534 #   $filteredAnswer =~ tr/a-z/A-Z/;
 2535 #
 2536 #   return $filteredAnswer;
 2537 # }
 2538 #
 2539 # sub ignore_order {
 2540 #   my $filteredAnswer = shift;
 2541 #
 2542 #   $filteredAnswer = join( "", lex_sort( split( /\s*/, $filteredAnswer ) ) );
 2543 #
 2544 #   return $filteredAnswer;
 2545 # }
 2546 ################################
 2547 ## END STRING ANSWER FILTERS
 2548 
 2549 
 2550 =head3 str_cmp()
 2551 
 2552 Compares a string or a list of strings, using a named hash of options to set
 2553 parameters. This can make for more readable code than using the "mode"_str_cmp()
 2554 style, but some people find one or the other easier to remember.
 2555 
 2556 ANS( str_cmp( answer or answer_array_ref, options_hash ) );
 2557 
 2558   1. the correct answer or a reference to an array of answers
 2559   2. either a list of filters, or:
 2560      a hash consisting of
 2561     filters - a reference to an array of filters
 2562 
 2563   Returns an answer evaluator, or (if given a reference to an array of answers),
 2564   a list of answer evaluators
 2565 
 2566 FILTERS:
 2567 
 2568   remove_whitespace --  removes all whitespace
 2569   compress_whitespace --  removes whitespace from the beginning and end of the string,
 2570               and treats one or more whitespace characters in a row as a
 2571               single space (true by default)
 2572   trim_whitespace   --  removes whitespace from the beginning and end of the string
 2573   ignore_case   --  ignores the case of the letters (true by default)
 2574   ignore_order    --  ignores the order in which letters are entered
 2575 
 2576 EXAMPLES:
 2577 
 2578   str_cmp( "Hello" )  --  matches "Hello", "  hello" (same as std_str_cmp() )
 2579   str_cmp( ["Hello", "Goodbye"] ) --  same as std_str_cmp_list()
 2580   str_cmp( " hello ", trim_whitespace ) --  matches "hello", " hello  "
 2581   str_cmp( "ABC", filters => 'ignore_order' ) --  matches "ACB", "A B C", but not "abc"
 2582   str_cmp( "D E F", remove_whitespace, ignore_case )  --  matches "def" and "d e f" but not "fed"
 2583 
 2584 
 2585 =cut
 2586 
 2587 sub str_cmp {
 2588   my $correctAnswer = shift @_;
 2589   $correctAnswer = '' unless defined($correctAnswer);
 2590   my @options = @_;
 2591   my %options = ();
 2592   # backward compatibility
 2593   if (grep /filters|debug|filter/, @options) { # see whether we have hash keys in the input.
 2594     %options = @options;
 2595   } elsif (@options) {     # all options are names of filters.
 2596     $options{filters} = [@options];
 2597   }
 2598   my $ra_filters;
 2599   assign_option_aliases( \%options,
 2600         'filter'               =>  'filters',
 2601      );
 2602     set_default_options(  \%options,
 2603           'filters'               =>  [qw(trim_whitespace compress_whitespace ignore_case)],
 2604             'debug'         =>  0,
 2605             'type'                  =>  'str_cmp',
 2606     );
 2607   $options{filters} = (ref($options{filters}))?$options{filters}:[$options{filters}];
 2608   # make sure this is a reference to an array.
 2609   # error-checking for filters occurs in the filters() subroutine
 2610 #   if( not defined( $options[0] ) ) {    # used with no filters as alias for std_str_cmp()
 2611 #     @options = ( 'compress_whitespace', 'ignore_case' );
 2612 #   }
 2613 #
 2614 #   if( $options[0] eq 'filters' ) {    # using filters => [f1, f2, ...] notation
 2615 #     $ra_filters = $options[1];
 2616 #   }
 2617 #   else {            # using a list of filters
 2618 #     $ra_filters = \@options;
 2619 #   }
 2620 
 2621   # thread over lists
 2622   my @ans_list = ();
 2623 
 2624   if ( ref($correctAnswer) eq 'ARRAY' ) {
 2625     @ans_list = @{$correctAnswer};
 2626   }
 2627   else {
 2628     push( @ans_list, $correctAnswer );
 2629   }
 2630 
 2631   # final_answer;
 2632   my @output_list = ();
 2633 
 2634   foreach my $ans (@ans_list) {
 2635     push(@output_list, STR_CMP(
 2636                   'correct_ans' =>  $ans,
 2637             'filters'   =>  $options{filters},
 2638             'type'      =>  $options{type},
 2639             'debug'         =>  $options{debug},
 2640          )
 2641     );
 2642   }
 2643 
 2644   return (wantarray) ? @output_list : $output_list[0] ;
 2645 }
 2646 
 2647 =head3 "mode"_str_cmp functions
 2648 
 2649 The functions of the the form "mode"_str_cmp() use different functions to
 2650 specify which filters to apply. They take no options except the correct
 2651 string. There are also versions which accept a list of strings.
 2652 
 2653  std_str_cmp( $correctString )
 2654  std_str_cmp_list( @correctStringList )
 2655   Filters: compress_whitespace, ignore_case
 2656 
 2657  std_cs_str_cmp( $correctString )
 2658  std_cs_str_cmp_list( @correctStringList )
 2659   Filters: compress_whitespace
 2660 
 2661  strict_str_cmp( $correctString )
 2662  strict_str_cmp_list( @correctStringList )
 2663   Filters: trim_whitespace
 2664 
 2665  unordered_str_cmp( $correctString )
 2666  unordered_str_cmp_list( @correctStringList )
 2667   Filters: ignore_order, ignore_case
 2668 
 2669  unordered_cs_str_cmp( $correctString )
 2670  unordered_cs_str_cmp_list( @correctStringList )
 2671   Filters: ignore_order
 2672 
 2673  ordered_str_cmp( $correctString )
 2674  ordered_str_cmp_list( @correctStringList )
 2675   Filters: remove_whitespace, ignore_case
 2676 
 2677  ordered_cs_str_cmp( $correctString )
 2678  ordered_cs_str_cmp_list( @correctStringList )
 2679   Filters: remove_whitespace
 2680 
 2681 Examples
 2682 
 2683   ANS( std_str_cmp( "W. Mozart" ) ) --  Accepts "W. Mozart", "W. MOZarT",
 2684     and so forth. Case insensitive. All internal spaces treated
 2685     as single spaces.
 2686   ANS( std_cs_str_cmp( "Mozart" ) ) --  Rejects "mozart". Same as
 2687     std_str_cmp() but case sensitive.
 2688   ANS( strict_str_cmp( "W. Mozart" ) )  --  Accepts only the exact string.
 2689   ANS( unordered_str_cmp( "ABC" ) ) --  Accepts "a c B", "CBA" and so forth.
 2690     Unordered, case insensitive, spaces ignored.
 2691   ANS( unordered_cs_str_cmp( "ABC" ) )  --  Rejects "abc". Same as
 2692     unordered_str_cmp() but case sensitive.
 2693   ANS( ordered_str_cmp( "ABC" ) ) --  Accepts "a b C", "A B C" and so forth.
 2694     Ordered, case insensitive, spaces ignored.
 2695   ANS( ordered_cs_str_cmp( "ABC" ) )  --  Rejects "abc", accepts "A BC" and
 2696     so forth. Same as ordered_str_cmp() but case sensitive.
 2697 
 2698 =cut
 2699 
 2700 sub std_str_cmp {         # compare strings
 2701   my $correctAnswer = shift @_;
 2702   my @filters = ( 'compress_whitespace', 'ignore_case' );
 2703   my $type = 'std_str_cmp';
 2704   STR_CMP('correct_ans' =>  $correctAnswer,
 2705       'filters' =>  \@filters,
 2706       'type'    =>  $type
 2707   );
 2708 }
 2709 
 2710 sub std_str_cmp_list {        # alias for std_str_cmp
 2711   my @answerList = @_;
 2712   my @output;
 2713   while (@answerList) {
 2714     push( @output, std_str_cmp(shift @answerList) );
 2715   }
 2716   @output;
 2717 }
 2718 
 2719 sub std_cs_str_cmp {        # compare strings case sensitive
 2720   my $correctAnswer = shift @_;
 2721   my @filters = ( 'compress_whitespace' );
 2722   my $type = 'std_cs_str_cmp';
 2723   STR_CMP(  'correct_ans' =>  $correctAnswer,
 2724       'filters' =>  \@filters,
 2725       'type'    =>  $type
 2726   );
 2727 }
 2728 
 2729 sub std_cs_str_cmp_list {     # alias for std_cs_str_cmp
 2730   my @answerList = @_;
 2731   my @output;
 2732   while (@answerList) {
 2733     push( @output, std_cs_str_cmp(shift @answerList) );
 2734   }
 2735   @output;
 2736 }
 2737 
 2738 sub strict_str_cmp {        # strict string compare
 2739   my $correctAnswer = shift @_;
 2740   my @filters = ( 'trim_whitespace' );
 2741   my $type = 'strict_str_cmp';
 2742   STR_CMP(  'correct_ans' =>  $correctAnswer,
 2743       'filters' =>  \@filters,
 2744       'type'    =>  $type
 2745   );
 2746 }
 2747 
 2748 sub strict_str_cmp_list {     # alias for strict_str_cmp
 2749   my @answerList = @_;
 2750   my @output;
 2751   while (@answerList) {
 2752     push( @output, strict_str_cmp(shift @answerList) );
 2753   }
 2754   @output;
 2755 }
 2756 
 2757 sub unordered_str_cmp {       # unordered, case insensitive, spaces ignored
 2758   my $correctAnswer = shift @_;
 2759   my @filters = ( 'ignore_order', 'ignore_case' );
 2760   my $type = 'unordered_str_cmp';
 2761   STR_CMP(  'correct_ans'   =>  $correctAnswer,
 2762       'filters'   =>  \@filters,
 2763       'type'      =>  $type
 2764   );
 2765 }
 2766 
 2767 sub unordered_str_cmp_list {    # alias for unordered_str_cmp
 2768   my @answerList = @_;
 2769   my @output;
 2770   while (@answerList) {
 2771     push( @output, unordered_str_cmp(shift @answerList) );
 2772   }
 2773   @output;
 2774 }
 2775 
 2776 sub unordered_cs_str_cmp {      # unordered, case sensitive, spaces ignored
 2777   my $correctAnswer = shift @_;
 2778   my @filters = ( 'ignore_order' );
 2779   my $type = 'unordered_cs_str_cmp';
 2780   STR_CMP(  'correct_ans'   =>  $correctAnswer,
 2781       'filters'   =>  \@filters,
 2782       'type'      =>  $type
 2783   );
 2784 }
 2785 
 2786 sub unordered_cs_str_cmp_list {   # alias for unordered_cs_str_cmp
 2787   my @answerList = @_;
 2788   my @output;
 2789   while (@answerList) {
 2790     push( @output, unordered_cs_str_cmp(shift @answerList) );
 2791   }
 2792   @output;
 2793 }
 2794 
 2795 sub ordered_str_cmp {       # ordered, case insensitive, spaces ignored
 2796   my $correctAnswer = shift @_;
 2797   my @filters = ( 'remove_whitespace', 'ignore_case' );
 2798   my $type = 'ordered_str_cmp';
 2799   STR_CMP(  'correct_ans' =>  $correctAnswer,
 2800       'filters' =>  \@filters,
 2801       'type'    =>  $type
 2802   );
 2803 }
 2804 
 2805 sub ordered_str_cmp_list {      # alias for ordered_str_cmp
 2806   my @answerList = @_;
 2807   my @output;
 2808   while (@answerList) {
 2809     push( @output, ordered_str_cmp(shift @answerList) );
 2810   }
 2811   @output;
 2812 }
 2813 
 2814 sub ordered_cs_str_cmp {      # ordered,  case sensitive, spaces ignored
 2815   my $correctAnswer = shift @_;
 2816   my @filters = ( 'remove_whitespace' );
 2817   my $type = 'ordered_cs_str_cmp';
 2818   STR_CMP(  'correct_ans' =>  $correctAnswer,
 2819       'filters' =>  \@filters,
 2820       'type'    =>  $type
 2821   );
 2822 }
 2823 
 2824 sub ordered_cs_str_cmp_list {   # alias for ordered_cs_str_cmp
 2825   my @answerList = @_;
 2826   my @output;
 2827   while (@answerList) {
 2828     push( @output, ordered_cs_str_cmp(shift @answerList) );
 2829   }
 2830   @output;
 2831 }
 2832 
 2833 
 2834 ## LOW-LEVEL ROUTINE -- NOT NORMALLY FOR END USERS -- USE WITH CAUTION
 2835 ##
 2836 ## IN:  a hashtable with the following entries (error-checking to be added later?):
 2837 ##      correctAnswer --  the correct answer, before filtering
 2838 ##      filters     --  reference to an array containing the filters to be applied
 2839 ##      type      --  a string containing the type of answer evaluator in use
 2840 ## OUT: a reference to an answer evaluator subroutine
 2841 sub STR_CMP {
 2842   my %str_params = @_;
 2843   #my $correctAnswer =  str_filters( $str_params{'correct_ans'}, @{$str_params{'filters'}} );
 2844   my $answer_evaluator = new AnswerEvaluator;
 2845   $answer_evaluator->{debug} = $str_params{debug};
 2846   $answer_evaluator->ans_hash(
 2847     correct_ans       => $str_params{correct_ans}||'',
 2848     type              => $str_params{type}||'str_cmp',
 2849     score             => 0,
 2850 
 2851     );
 2852   my %known_filters = (
 2853               'remove_whitespace'   =>  \&remove_whitespace,
 2854         'compress_whitespace' =>  \&compress_whitespace,
 2855         'trim_whitespace'   =>  \&trim_whitespace,
 2856         'ignore_case'     =>  \&ignore_case,
 2857         'ignore_order'      =>  \&ignore_order,
 2858   );
 2859 
 2860   foreach my $filter ( @{$str_params{filters}} ) {
 2861     #check that filter is known
 2862     die "Unknown string filter |$filter|. Known filters are ".
 2863          join(" ", keys %known_filters) .
 2864          "(try checking the parameters to str_cmp() )"
 2865                 unless exists $known_filters{$filter};
 2866     # install related pre_filter
 2867     $answer_evaluator->install_pre_filter( $known_filters{$filter} );
 2868   }
 2869   $answer_evaluator->install_evaluator(sub {
 2870       my $rh_ans = shift;
 2871       $rh_ans->{_filter_name} = "Evaluator: Compare string answers with eq";
 2872       $rh_ans->{score} = ($rh_ans->{student_ans} eq $rh_ans->{correct_ans})?1:0  ;
 2873       $rh_ans;
 2874   });
 2875   $answer_evaluator->install_post_filter(sub {
 2876     my $rh_hash = shift;
 2877     $rh_hash->{_filter_name} = "clean up preview strings";
 2878     $rh_hash->{'preview_text_string'} = $rh_hash->{student_ans};
 2879     $rh_hash->{'preview_latex_string'} = "\\text{ ".$rh_hash->{student_ans}." }";
 2880     $rh_hash;
 2881   });
 2882   return $answer_evaluator;
 2883 }
 2884 
 2885 # sub STR_CMP_old {
 2886 #   my %str_params = @_;
 2887 #   $str_params{'correct_ans'} = str_filters( $str_params{'correct_ans'}, @{$str_params{'filters'}} );
 2888 #   my $answer_evaluator = sub {
 2889 #     my $in = shift @_;
 2890 #     $in = '' unless defined $in;
 2891 #     my $original_student_ans = $in;
 2892 #     $in = str_filters( $in, @{$str_params{'filters'}} );
 2893 #     my $correctQ = ( $in eq $str_params{'correct_ans'} ) ? 1: 0;
 2894 #     my $ans_hash = new AnswerHash(    'score'       =>  $correctQ,
 2895 #               'correct_ans'     =>  $str_params{'correctAnswer'},
 2896 #               'student_ans'     =>  $in,
 2897 #               'ans_message'     =>  '',
 2898 #               'type'        =>  $str_params{'type'},
 2899 #               'preview_text_string'   =>  $in,
 2900 #               'preview_latex_string'    =>  $in,
 2901 #               'original_student_ans'    =>  $original_student_ans
 2902 #     );
 2903 #     return $ans_hash;
 2904 #   };
 2905 #   return $answer_evaluator;
 2906 # }
 2907 
 2908 ##########################################################################
 2909 ##########################################################################
 2910 ## Miscellaneous answer evaluators
 2911 
 2912 =head2 Miscellaneous Answer Evaluators (Checkboxes and Radio Buttons)
 2913 
 2914 These evaluators do not fit any of the other categories.
 2915 
 2916 checkbox_cmp( $correctAnswer )
 2917 
 2918   $correctAnswer  --  a string containing the names of the correct boxes,
 2919             e.g. "ACD". Note that this means that individual
 2920             checkbox names can only be one character. Internally,
 2921             this is largely the same as unordered_cs_str_cmp().
 2922 
 2923 radio_cmp( $correctAnswer )
 2924 
 2925   $correctAnswer  --  a string containing the name of the correct radio
 2926             button, e.g. "Choice1". This is case sensitive and
 2927             whitespace sensitive, so the correct answer must match
 2928             the name of the radio button exactly.
 2929 
 2930 =cut
 2931 
 2932 # added 6/14/2000 by David Etlinger
 2933 # because of the conversion of the answer
 2934 # string to an array, I thought it better not
 2935 # to force STR_CMP() to work with this
 2936 
 2937 #added 2/26/2003 by Mike Gage
 2938 # handled the case where multiple answers are passed as an array reference
 2939 # rather than as a \0 delimited string.
 2940 sub checkbox_cmp {
 2941   my  $correctAnswer = shift @_;
 2942   my %options = @_;
 2943   assign_option_aliases( \%options,
 2944      );
 2945     set_default_options(  \%options,
 2946           'debug'         =>  0,
 2947             'type'                  =>  'checkbox_cmp',
 2948     );
 2949   my $answer_evaluator = new AnswerEvaluator(
 2950     correct_ans      => $correctAnswer,
 2951     type             => $options{type},
 2952   );
 2953   # pass along debug requests
 2954   $answer_evaluator->{debug} = $options{debug};
 2955 
 2956   # join student answer array into a single string if necessary
 2957   $answer_evaluator->install_pre_filter(sub {
 2958     my $rh_ans = shift;
 2959     $rh_ans->{_filter_name} = 'convert student_ans to string';
 2960     $rh_ans->{student_ans} = join("", @{$rh_ans->{student_ans}})
 2961              if ref($rh_ans->{student_ans}) =~/ARRAY/i;
 2962     $rh_ans;
 2963   });
 2964   # ignore order of check boxes
 2965   $answer_evaluator->install_pre_filter(\&ignore_order);
 2966   # compare as strings
 2967   $answer_evaluator->install_evaluator(sub {
 2968     my $rh_ans     = shift;
 2969     $rh_ans->{_filter_name} = 'compare strings generated by checked boxes';
 2970     $rh_ans->{score} = ($rh_ans->{student_ans} eq $rh_ans->{correct_ans}) ? 1 : 0;
 2971     $rh_ans;
 2972   });
 2973   # fix up preview displays
 2974   $answer_evaluator->install_post_filter( sub {
 2975     my $rh_ans      = shift;
 2976     $rh_ans->{_filter_name} = 'adjust preview strings';
 2977     $rh_ans->{type} = $options{type};
 2978     $rh_ans->{preview_text_string}  = '\\text{'.$rh_ans->{student_ans}.'}',
 2979     $rh_ans->{preview_latex_string} = '\\text{'.$rh_ans->{student_ans}.'}',
 2980     $rh_ans;
 2981 
 2982 
 2983   });
 2984 
 2985 #   my  $answer_evaluator = sub {
 2986 #     my $in = shift @_;
 2987 #     $in = '' unless defined $in;      #in case no boxes checked
 2988 #                         # multiple answers could come in two forms
 2989 #                         # either a \0 delimited string or
 2990 #                         # an array reference.  We handle both.
 2991 #         if (ref($in) eq 'ARRAY')   {
 2992 #           $in = join("",@{$in});              # convert array to single no-delimiter string
 2993 #         } else {
 2994 #       my @temp = split( "\0", $in );    #convert "\0"-delimited string to array...
 2995 #       $in = join( "", @temp );      #and then to a single no-delimiter string
 2996 #     }
 2997 #     my $original_student_ans = $in;     #well, almost original
 2998 #     $in = str_filters( $in, 'ignore_order' );
 2999 #
 3000 #     my $correctQ = ($in eq $correctAnswer) ? 1: 0;
 3001 #
 3002 #     my $ans_hash = new AnswerHash(
 3003 #       'score'             =>  $correctQ,
 3004 #       'correct_ans'       =>  "$correctAnswer",
 3005 #       'student_ans'       =>  $in,
 3006 #       'ans_message'       =>  "",
 3007 #       'type'              =>  "checkbox_cmp",
 3008 #       'preview_text_string' =>  $in,
 3009 #       'preview_latex_string'  =>  $in,
 3010 #       'original_student_ans'  =>  $original_student_ans
 3011 #     );
 3012 #     return $ans_hash;
 3013 #
 3014 #   };
 3015   return $answer_evaluator;
 3016 }
 3017 # sub checkbox_cmp {
 3018 #   my  $correctAnswer = shift @_;
 3019 #   $correctAnswer = str_filters( $correctAnswer, 'ignore_order' );
 3020 #
 3021 #   my  $answer_evaluator = sub {
 3022 #     my $in = shift @_;
 3023 #     $in = '' unless defined $in;      #in case no boxes checked
 3024 #                         # multiple answers could come in two forms
 3025 #                         # either a \0 delimited string or
 3026 #                         # an array reference.  We handle both.
 3027 #         if (ref($in) eq 'ARRAY')   {
 3028 #           $in = join("",@{$in});              # convert array to single no-delimiter string
 3029 #         } else {
 3030 #       my @temp = split( "\0", $in );    #convert "\0"-delimited string to array...
 3031 #       $in = join( "", @temp );      #and then to a single no-delimiter string
 3032 #     }
 3033 #     my $original_student_ans = $in;     #well, almost original
 3034 #     $in = str_filters( $in, 'ignore_order' );
 3035 #
 3036 #     my $correctQ = ($in eq $correctAnswer) ? 1: 0;
 3037 #
 3038 #     my $ans_hash = new AnswerHash(
 3039 #       'score'             =>  $correctQ,
 3040 #       'correct_ans'       =>  "$correctAnswer",
 3041 #       'student_ans'       =>  $in,
 3042 #       'ans_message'       =>  "",
 3043 #       'type'              =>  "checkbox_cmp",
 3044 #       'preview_text_string' =>  $in,
 3045 #       'preview_latex_string'  =>  $in,
 3046 #       'original_student_ans'  =>  $original_student_ans
 3047 #     );
 3048 #     return $ans_hash;
 3049 #
 3050 #   };
 3051 #   return $answer_evaluator;
 3052 # }
 3053 
 3054 #added 6/28/2000 by David Etlinger
 3055 #exactly the same as strict_str_cmp,
 3056 #but more intuitive to the user
 3057 
 3058 # check that answer is really a string and not an array
 3059 # also use ordinary string compare
 3060 sub radio_cmp {
 3061   #strict_str_cmp( @_ );
 3062   my $response = shift;  # there should be only one item.
 3063   warn "Multiple choices -- this should not happen with radio buttons. Have
 3064   you used checkboxes perhaps?" if ref($response); #triggered if an ARRAY is passed
 3065   str_cmp($response);
 3066 }
 3067 
 3068 ##########################################################################
 3069 ##########################################################################
 3070 ## Text and e-mail routines
 3071 
 3072 sub store_ans_at {
 3073   my $answerStringRef = shift;
 3074   my %options = @_;
 3075   my $ans_eval= '';
 3076   if ( ref($answerStringRef) eq 'SCALAR' ) {
 3077     $ans_eval= sub {
 3078       my $text = shift;
 3079       $text = '' unless defined($text);
 3080       $$answerStringRef = $$answerStringRef  . $text;
 3081       my $ans_hash = new AnswerHash(
 3082                'score'      =>  1,
 3083                'correct_ans'      =>  '',
 3084                'student_ans'      =>  $text,
 3085                'ans_message'      =>  '',
 3086                'type'       =>  'store_ans_at',
 3087                'original_student_ans'   =>  $text,
 3088                'preview_text_string'    =>  ''
 3089       );
 3090 
 3091     return $ans_hash;
 3092     };
 3093   }
 3094   else {
 3095     die "Syntax error: \n The argument to store_ans_at() must be a pointer to a scalar.\n(e.g.  store_ans_at(~~\$MSG) )\n\n";
 3096   }
 3097 
 3098   return $ans_eval;
 3099 }
 3100 
 3101 #### subroutines used in producing a questionnaire
 3102 #### these are at least good models for other answers of this type
 3103 
 3104 # my $QUESTIONNAIRE_ANSWERS=''; #  stores the answers until it is time to send them
 3105        #  this must be initialized before the answer evaluators are run
 3106        #  but that happens long after all of the text in the problem is
 3107        #  evaluated.
 3108 # this is a utility script for cleaning up the answer output for display in
 3109 #the answers.
 3110 
 3111 sub DUMMY_ANSWER {
 3112   my $num = shift;
 3113   qq{<INPUT TYPE="HIDDEN" NAME="answer$num" VALUE="">}
 3114 }
 3115 
 3116 sub escapeHTML {
 3117   my $string = shift;
 3118   $string =~ s/\n/$BR/ge;
 3119   $string;
 3120 }
 3121 
 3122 # these next three subroutines show how to modify the "store_ans_at()" answer
 3123 # evaluator to add extra information before storing the info
 3124 # They provide a good model for how to tweak answer evaluators in special cases.
 3125 
 3126 sub anstext {
 3127   my $num = shift;
 3128   my $ans_eval_template = store_ans_at(\$QUESTIONNAIRE_ANSWERS);
 3129   my $psvnNumber  = PG_restricted_eval(q!$main::psvnNumber!);
 3130   my $probNum     = PG_restricted_eval(q!$main::probNum!);
 3131   my $ans_eval    = sub {
 3132          my $text = shift;
 3133          $text = '' unless defined($text);
 3134          my $new_text = "\n$psvnNumber-Problem-$probNum-Question-$num:\n $text "; # modify entered text
 3135          my $out = &$ans_eval_template($new_text);       # standard evaluator
 3136          #warn "$QUESTIONNAIRE_ANSWERS";
 3137          $out->{student_ans} = escapeHTML($text);  #  restore original entered text
 3138          $out->{correct_ans} = "Question  $num answered";
 3139          $out->{original_student_ans} = escapeHTML($text);
 3140          $out;
 3141     };
 3142    $ans_eval;
 3143 }
 3144 
 3145 
 3146 sub ansradio {
 3147   my $num = shift;
 3148   my $psvnNumber  = PG_restricted_eval(q!$main::psvnNumber!);
 3149   my $probNum  = PG_restricted_eval(q!$main::probNum!);
 3150 
 3151   my $ans_eval_template = store_ans_at(\$QUESTIONNAIRE_ANSWERS);
 3152   my $ans_eval = sub {
 3153          my $text = shift;
 3154          $text = '' unless defined($text);
 3155          my $new_text = "\n$psvnNumber-Problem-$probNum-RADIO-$num:\n $text ";       # modify entered text
 3156          my $out = $ans_eval_template->($new_text);       # standard evaluator
 3157          $out->{student_ans} =escapeHTML($text);  # restore original entered text
 3158          $out->{original_student_ans} = escapeHTML($text);
 3159          $out;
 3160    };
 3161 
 3162    $ans_eval;
 3163 }
 3164 
 3165 sub anstext_non_anonymous {
 3166   ## this emails identifying information
 3167   my $num          = shift;
 3168     my $psvnNumber   = PG_restricted_eval(q!$main::psvnNumber!);
 3169   my $probNum      = PG_restricted_eval(q!$main::probNum!);
 3170     my $studentLogin = PG_restricted_eval(q!$main::studentLogin!);
 3171   my $studentID    = PG_restricted_eval(q!$main::studentID!);
 3172     my $studentName  = PG_restricted_eval(q!$main::studentName!);
 3173 
 3174 
 3175   my $ans_eval_template = store_ans_at(\$QUESTIONNAIRE_ANSWERS);
 3176   my $ans_eval = sub {
 3177          my $text = shift;
 3178          $text = '' unless defined($text);
 3179          my $new_text = "\n$psvnNumber-Problem-$probNum-Question-$num:\n$studentLogin $main::studentID $studentName\n$text "; # modify entered text
 3180          my $out = &$ans_eval_template($new_text);       # standard evaluator
 3181          #warn "$QUESTIONNAIRE_ANSWERS";
 3182          $out->{student_ans} = escapeHTML($text);  #  restore original entered text
 3183          $out->{correct_ans} = "Question  $num answered";
 3184          $out->{original_student_ans} = escapeHTML($text);
 3185          $out;
 3186     };
 3187    $ans_eval;
 3188 }
 3189 
 3190 
 3191 #  This is another example of how to modify an  answer evaluator to obtain
 3192 #  the desired behavior in a special case.  Here the object is to have
 3193 #  have the last answer trigger the send_mail_to subroutine which mails
 3194 #  all of the answers to the designated address.
 3195 #  (This address must be listed in PG_environment{'ALLOW_MAIL_TO'} or an error occurs.)
 3196 
 3197 # Fix me?? why is the body hard wired to the string QUESTIONNAIRE_ANSWERS?
 3198 
 3199 sub mail_answers_to {  #accepts the last answer and mails off the result
 3200   my $user_address = shift;
 3201   my $ans_eval = sub {
 3202 
 3203     # then mail out all of the answers, including this last one.
 3204 
 3205     send_mail_to( $user_address,
 3206           'subject'       =>  "$main::courseName WeBWorK questionnaire",
 3207           'body'          =>  $QUESTIONNAIRE_ANSWERS,
 3208           'ALLOW_MAIL_TO'   =>  $rh_envir->{ALLOW_MAIL_TO}
 3209     );
 3210 
 3211     my $ans_hash = new AnswerHash(  'score'   =>  1,
 3212             'correct_ans' =>  '',
 3213             'student_ans' =>  'Answer recorded',
 3214             'ans_message' =>  '',
 3215             'type'    =>  'send_mail_to',
 3216     );
 3217 
 3218     return $ans_hash;
 3219   };
 3220 
 3221   return $ans_eval;
 3222 }
 3223 
 3224 sub save_answer_to_file {  #accepts the last answer and mails off the result
 3225   my $fileID = shift;
 3226   my $ans_eval = new AnswerEvaluator;
 3227   $ans_eval->install_evaluator(
 3228       sub {
 3229          my $rh_ans = shift;
 3230 
 3231              unless ( defined( $rh_ans->{student_ans} ) ) {
 3232               $rh_ans->throw_error("save_answers_to_file","{student_ans} field not defined");
 3233               return $rh_ans;
 3234             }
 3235 
 3236         my $error;
 3237         my $string = '';
 3238         $string = qq![[<$main::studentLogin> $main::studentName /!. time() . qq!/]]\n!.
 3239           $rh_ans->{student_ans}. qq!\n\n============================\n\n!;
 3240 
 3241         if ($error = AnswerIO::saveAnswerToFile('preflight',$string) ) {
 3242           $rh_ans->throw_error("save_answers_to_file","Error:  $error");
 3243         } else {
 3244           $rh_ans->{'student_ans'} = 'Answer saved';
 3245           $rh_ans->{'score'} = 1;
 3246         }
 3247         $rh_ans;
 3248       }
 3249   );
 3250 
 3251   return $ans_eval;
 3252 }
 3253 
 3254 sub mail_answers_to2 {  #accepts the last answer and mails off the result
 3255   my $user_address         = shift;
 3256   my $subject              = shift;
 3257   my $ra_allow_mail_to     = shift;
 3258   $subject = "$main::courseName WeBWorK questionnaire" unless defined $subject;
 3259   send_mail_to($user_address,
 3260       'subject'     => $subject,
 3261       'body'        => $QUESTIONNAIRE_ANSWERS,
 3262       'ALLOW_MAIL_TO'   => $rh_envir->{ALLOW_MAIL_TO},
 3263   );
 3264 }
 3265 
 3266 ##########################################################################
 3267 ##########################################################################
 3268 
 3269 
 3270 ###########################################################################
 3271 ### THE FOLLOWING ARE LOCAL SUBROUTINES THAT ARE MEANT TO BE CALLED ONLY FROM THIS SCRIPT.
 3272 
 3273 ## Internal routine that converts variables into the standard array format
 3274 ##
 3275 ## IN:  one of the following:
 3276 ##      an undefined value (i.e., no variable was specified)
 3277 ##      a reference to an array of variable names -- [var1, var2]
 3278 ##      a number (the number of variables desired) -- 3
 3279 ##      one or more variable names -- (var1, var2)
 3280 ## OUT: an array of variable names
 3281 
 3282 sub get_var_array {
 3283   my $in = shift @_;
 3284   my @out;
 3285 
 3286   if( not defined($in) ) {      #if nothing defined, build default array and return
 3287     @out = ( $functVarDefault );
 3288     return @out;
 3289   }
 3290   elsif( ref( $in ) eq 'ARRAY' ) {  #if given an array ref, dereference and return
 3291     return @{$in};
 3292   }
 3293   elsif( $in =~ /^\d+/ ) {      #if given a number, set up the array and return
 3294     if( $in == 1 ) {
 3295       $out[0] = 'x';
 3296     }
 3297     elsif( $in == 2 ) {
 3298       $out[0] = 'x';
 3299       $out[1] = 'y';
 3300     }
 3301     elsif( $in == 3 ) {
 3302       $out[0] = 'x';
 3303       $out[1] = 'y';
 3304       $out[2] = 'z';
 3305     }
 3306     else {  #default to the x_1, x_2, ... convention
 3307       my ($i, $tag);
 3308       for($i = 0; $i < $in; $i++) {$out[$i] = "${functVarDefault}_".($i+1)}
 3309     }
 3310     return @out;
 3311   }
 3312   else {            #if given one or more names, return as an array
 3313     unshift( @_, $in );
 3314     return @_;
 3315   }
 3316 }
 3317 
 3318 ## Internal routine that converts limits into the standard array of arrays format
 3319 ##  Some of the cases are probably unneccessary, but better safe than sorry
 3320 ##
 3321 ## IN:  one of the following:
 3322 ##      an undefined value (i.e., no limits were specified)
 3323 ##      a reference to an array of arrays of limits -- [[llim,ulim], [llim,ulim]]
 3324 ##      a reference to an array of limits -- [llim, ulim]
 3325 ##      an array of array references -- ([llim,ulim], [llim,ulim])
 3326 ##      an array of limits -- (llim,ulim)
 3327 ## OUT: an array of array references -- ([llim,ulim], [llim,ulim]) or ([llim,ulim])
 3328 
 3329 sub get_limits_array {
 3330   my $in = shift @_;
 3331   my @out;
 3332 
 3333   if( not defined($in) ) {        #if nothing defined, build default array and return
 3334     @out = ( [$functLLimitDefault, $functULimitDefault] );
 3335     return @out;
 3336   }
 3337   elsif( ref($in) eq 'ARRAY' ) {        #$in is either ref to array, or ref to array of refs
 3338     my @deref = @{$in};
 3339 
 3340     if( ref( $in->[0] ) eq 'ARRAY' ) {    #$in is a ref to an array of array refs
 3341       return @deref;
 3342     }
 3343     else {            #$in was just a ref to an array of numbers
 3344       @out = ( $in );
 3345       return @out;
 3346     }
 3347   }
 3348   else {              #$in was an array of references or numbers
 3349     unshift( @_, $in );
 3350 
 3351     if( ref($_[0]) eq 'ARRAY' ) {     #$in was an array of references, so just return it
 3352       return @_;
 3353     }
 3354     else {            #$in was an array of numbers
 3355       @out = ( \@_ );
 3356       return @out;
 3357     }
 3358   }
 3359 }
 3360 
 3361 #sub check_option_list {
 3362 # my $size = scalar(@_);
 3363 # if( ( $size % 2 ) != 0 ) {
 3364 #   warn "ERROR in answer evaluator generator:\n" .
 3365 #     "Usage: <CODE>str_cmp([\$ans1,  \$ans2],%options)</CODE>
 3366 #     or <CODE> num_cmp([\$num1, \$num2], %options)</CODE><BR>
 3367 #     A list of inputs must be inclosed in square brackets <CODE>[\$ans1, \$ans2]</CODE>";
 3368 # }
 3369 #}
 3370 
 3371 # simple subroutine to display an error message when
 3372 # function compares are called with invalid parameters
 3373 sub function_invalid_params {
 3374   my $correctEqn = shift @_;
 3375   my $error_response = sub {
 3376     my $PGanswerMessage = "Tell your professor that there is an error with the parameters " .
 3377             "to the function answer evaluator";
 3378     return ( 0, $correctEqn, "", $PGanswerMessage );
 3379   };
 3380   return $error_response;
 3381 }
 3382 
 3383 sub clean_up_error_msg {
 3384   my $msg = $_[0];
 3385   $msg =~ s/^\[[^\]]*\][^:]*://;
 3386   $msg =~ s/Unquoted string//g;
 3387   $msg =~ s/may\s+clash.*/does not make sense here/;
 3388   $msg =~ s/\sat.*line [\d]*//g;
 3389   $msg = 'Error: '. $msg;
 3390 
 3391   return $msg;
 3392 }
 3393 
 3394 #formats the student and correct answer as specified
 3395 #format must be of a form suitable for sprintf (e.g. '%0.5g'),
 3396 #with the exception that a '#' at the end of the string
 3397 #will cause trailing zeros in the decimal part to be removed
 3398 sub prfmt {
 3399   my($number,$format) = @_;  # attention, the order of format and number are reversed
 3400   my $out;
 3401   if ($format) {
 3402     warn "Incorrect format used: $format. <BR> Format should look something like %4.5g<BR>"
 3403                 unless $format =~ /^\s*%\d*\.?\d*\w#?\s*$/;
 3404 
 3405     if( $format =~ s/#\s*$// ) {  # remove trailing zeros in the decimal
 3406       $out = sprintf( $format, $number );
 3407       $out =~ s/(\.\d*?)0+$/$1/;
 3408       $out =~ s/\.$//;      # in case all decimal digits were zero, remove the decimal
 3409       $out =~ s/e/E/g;        # only use capital E's for exponents. Little e is for 2.71828...
 3410     } elsif (is_a_number($number) ){
 3411       $out = sprintf( $format, $number );
 3412       $out =~ s/e/E/g;        # only use capital E's for exponents. Little e is for 2.71828...
 3413     } else { # number is probably a string representing an arithmetic expression
 3414       $out = $number;
 3415     }
 3416 
 3417   } else {
 3418     if (is_a_number($number)) {# only use capital E's for exponents. Little e is for 2.71828...
 3419       $out = $number;
 3420       $out =~ s/e/E/g;
 3421     } else { # number is probably a string representing an arithmetic expression
 3422       $out = $number;
 3423     }
 3424   }
 3425   return $out;
 3426 }
 3427 #########################################################################
 3428 # Filters for answer evaluators
 3429 #########################################################################
 3430 
 3431 =head2 Filters
 3432 
 3433 =pod
 3434 
 3435 A filter is a short subroutine with the following structure.  It accepts an
 3436 AnswerHash, followed by a hash of options.  It returns an AnswerHash
 3437 
 3438   $ans_hash = filter($ans_hash, %options);
 3439 
 3440 See the AnswerHash.pm file for a list of entries which can be expected to be found
 3441 in an AnswerHash, such as 'student_ans', 'score' and so forth.  Other entries
 3442 may be present for specialized answer evaluators.
 3443 
 3444 The hope is that a well designed set of filters can easily be combined to form
 3445 a new answer_evaluator and that this method will produce answer evaluators which are
 3446 are more robust than the method of copying existing answer evaluators and modifying them.
 3447 
 3448 Here is an outline of how a filter is constructed:
 3449 
 3450   sub filter{
 3451     my $rh_ans = shift;
 3452     my %options = @_;
 3453     assign_option_aliases(\%options,
 3454         'alias1'  => 'option5'
 3455         'alias2'  => 'option7'
 3456     );
 3457     set_default_options(\%options,
 3458         '_filter_name'  =>  'filter',
 3459         'option5'   =>  .0001,
 3460         'option7'   =>  'ascii',
 3461         'allow_unknown_options  =>  0,
 3462     }
 3463     .... body code of filter .......
 3464       if ($error) {
 3465         $rh_ans->throw_error("FILTER_ERROR", "Something went wrong");
 3466         # see AnswerHash.pm for details on using the throw_error method.
 3467 
 3468     $rh_ans;  #reference to an AnswerHash object is returned.
 3469   }
 3470 
 3471 =cut
 3472 
 3473 =head4 compare_numbers
 3474 
 3475 
 3476 =cut
 3477 
 3478 
 3479 sub compare_numbers {
 3480   my ($rh_ans, %options) = @_;
 3481   my ($inVal,$PG_eval_errors,$PG_full_error_report) = PG_answer_eval($rh_ans->{student_ans});
 3482   if ($PG_eval_errors) {
 3483     $rh_ans->throw_error('EVAL','There is a syntax error in your answer');
 3484     $rh_ans->{ans_message} = clean_up_error_msg($PG_eval_errors);
 3485     # return $rh_ans;
 3486   } else {
 3487     $rh_ans->{student_ans} = prfmt($inVal,$options{format});
 3488   }
 3489 
 3490   my $permitted_error;
 3491 
 3492   if ($rh_ans->{tolType} eq 'absolute') {
 3493     $permitted_error = $rh_ans->{tolerance};
 3494   }
 3495   elsif ( abs($rh_ans->{correct_ans}) <= $options{zeroLevel}) {
 3496       $permitted_error = $options{zeroLevelTol};  ## want $tol to be non zero
 3497   }
 3498   else {
 3499     $permitted_error = abs($rh_ans->{tolerance}*$rh_ans->{correct_ans});
 3500   }
 3501 
 3502   my $is_a_number = is_a_number($inVal);
 3503   $rh_ans->{score} = 1 if ( ($is_a_number) and
 3504       (abs( $inVal - $rh_ans->{correct_ans} ) <= $permitted_error) );
 3505   if (not $is_a_number) {
 3506     $rh_ans->{error_message} = "$rh_ans->{error_message}". 'Your answer does not evaluate to a number ';
 3507   }
 3508 
 3509   $rh_ans;
 3510 }
 3511 
 3512 =head4 std_num_filter
 3513 
 3514   std_num_filter($rh_ans, %options)
 3515   returns $rh_ans
 3516 
 3517 Replaces some constants using math_constants, then evaluates a perl expression.
 3518 
 3519 
 3520 =cut
 3521 
 3522 sub std_num_filter {
 3523   my $rh_ans = shift;
 3524   my %options = @_;
 3525   my $in = $rh_ans->input();
 3526   $in = math_constants($in);
 3527   $rh_ans->{type} = 'std_number';
 3528   my ($inVal,$PG_eval_errors,$PG_full_error_report);
 3529   if ($in =~ /\S/) {
 3530     ($inVal,$PG_eval_errors,$PG_full_error_report) = PG_answer_eval($in);
 3531   } else {
 3532     $PG_eval_errors = '';
 3533   }
 3534 
 3535   if ($PG_eval_errors) {        ##error message from eval or above
 3536     $rh_ans->{ans_message} = 'There is a syntax error in your answer';
 3537     $rh_ans->{student_ans} =
 3538     clean_up_error_msg($PG_eval_errors);
 3539   } else {
 3540     $rh_ans->{student_ans} = $inVal;
 3541   }
 3542   $rh_ans;
 3543 }
 3544 
 3545 =head std_num_array_filter
 3546 
 3547   std_num_array_filter($rh_ans, %options)
 3548   returns $rh_ans
 3549 
 3550 Assumes the {student_ans} field is a numerical  array, and applies BOTH check_syntax and std_num_filter
 3551 to each element of the array.  Does it's best to generate sensible error messages for syntax errors.
 3552 A typical error message displayed in {studnet_ans} might be ( 56, error message, -4).
 3553 
 3554 =cut
 3555 
 3556 sub std_num_array_filter {
 3557   my $rh_ans= shift;
 3558   my %options = @_;
 3559   set_default_options(  \%options,
 3560         '_filter_name'  =>  'std_num_array_filter',
 3561     );
 3562   my @in = @{$rh_ans->{student_ans}};
 3563   my $temp_hash = new AnswerHash;
 3564   my @out=();
 3565   my $PGanswerMessage = '';
 3566   foreach my $item (@in)   {  # evaluate each number in the vector
 3567     $temp_hash->input($item);
 3568     $temp_hash = check_syntax($temp_hash);
 3569     if (defined($temp_hash->{error_flag}) and $temp_hash->{error_flag} eq 'SYNTAX') {
 3570       $PGanswerMessage .= $temp_hash->{ans_message};
 3571       $temp_hash->{ans_message} = undef;
 3572     } else {
 3573       #continue processing
 3574       $temp_hash = std_num_filter($temp_hash);
 3575       if (defined($temp_hash->{ans_message}) and $temp_hash->{ans_message} ) {
 3576         $PGanswerMessage .= $temp_hash->{ans_message};
 3577         $temp_hash->{ans_message} = undef;
 3578       }
 3579     }
 3580     push(@out, $temp_hash->input());
 3581 
 3582   }
 3583   if ($PGanswerMessage) {
 3584     $rh_ans->input( "( " . join(", ", @out ) . " )" );
 3585         $rh_ans->throw_error('SYNTAX', 'There is a syntax error in your answer.');
 3586   } else {
 3587     $rh_ans->input( [@out] );
 3588   }
 3589   $rh_ans;
 3590 }
 3591 
 3592 =head4 function_from_string2
 3593 
 3594 
 3595 
 3596 =cut
 3597 
 3598 sub function_from_string2 {
 3599     my $rh_ans = shift;
 3600     my %options = @_;
 3601   assign_option_aliases(\%options,
 3602         'vars'      => 'ra_vars',
 3603         'var'           => 'ra_vars',
 3604         'store_in'      => 'stdout',
 3605   );
 3606   set_default_options(  \%options,
 3607         'stdin'         =>  'student_ans',
 3608               'stdout'    =>  'rf_student_ans',
 3609           'ra_vars'   =>  [qw( x y )],
 3610           'debug'     =>  0,
 3611           '_filter_name'  =>  'function_from_string2',
 3612     );
 3613     # initialize
 3614     $rh_ans->{_filter_name} = $options{_filter_name};
 3615 
 3616     my $eqn         = $rh_ans->{ $options{stdin} };
 3617     my @VARS        = @{ $options{ 'ra_vars'}    };
 3618     #warn "VARS = ", join("<>", @VARS) if defined($options{debug}) and $options{debug} ==1;
 3619     my $originalEqn = $eqn;
 3620     $eqn            = &math_constants($eqn);
 3621     for( my $i = 0; $i < @VARS; $i++ ) {
 3622         #  This next line is a hack required for 5.6.0 -- it doesn't appear to be needed in 5.6.1
 3623         my ($temp,$er1,$er2) = PG_restricted_eval('"'. $VARS[$i] . '"');
 3624     #$eqn =~ s/\b$VARS[$i]\b/\$VARS[$i]/g;
 3625         $eqn  =~ s/\b$temp\b/\$VARS[$i]/g;
 3626 
 3627   }
 3628   #warn "equation evaluated = $eqn",$rh_ans->pretty_print(), "<br>\noptions<br>\n",
 3629   #     pretty_print(\%options)
 3630   #     if defined($options{debug}) and $options{debug} ==1;
 3631     my ($function_sub,$PG_eval_errors, $PG_full_errors) = PG_answer_eval( q!
 3632       sub {
 3633         my @VARS = @_;
 3634         my $input_str = '';
 3635         for( my $i=0; $i<@VARS; $i++ ) {
 3636           $input_str .= "\$VARS[$i] = $VARS[$i]; ";
 3637         }
 3638         my $PGanswerMessage;
 3639         $input_str .= '! . $eqn . q!';  # need the single quotes to keep the contents of $eqn from being
 3640                                         # evaluated when it is assigned to $input_str;
 3641         my ($out, $PG_eval_errors, $PG_full_errors) = PG_answer_eval($input_str); #Finally evaluated
 3642 
 3643         if ( defined($PG_eval_errors) and $PG_eval_errors =~ /\S/ ) {
 3644             $PGanswerMessage  = clean_up_error_msg($PG_eval_errors);
 3645 # This message seemed too verbose, but it does give extra information, we'll see if it is needed.
 3646 #                    "<br> There was an error in evaluating your function <br>
 3647 #           !. $originalEqn . q! <br>
 3648 #           at ( " . join(', ', @VARS) . " ) <br>
 3649 #            $PG_eval_errors
 3650 #           ";   # this message appears in the answer section which is not process by Latex2HTML so it must
 3651 #                # be in HTML.  That is why $BR is NOT used.
 3652 
 3653       }
 3654       (wantarray) ? ($out, $PGanswerMessage): $out;   # PGanswerMessage may be undefined.
 3655       };
 3656   !);
 3657 
 3658   if (defined($PG_eval_errors) and $PG_eval_errors =~/\S/ ) {
 3659         $PG_eval_errors = clean_up_error_msg($PG_eval_errors);
 3660 
 3661     my $PGanswerMessage = "There was an error in converting the expression
 3662       $BR $originalEqn $BR into a function.
 3663       $BR $PG_eval_errors.";
 3664     $rh_ans->{rf_student_ans} = $function_sub;
 3665     $rh_ans->{ans_message} = $PGanswerMessage;
 3666     $rh_ans->{error_message} = $PGanswerMessage;
 3667     $rh_ans->{error_flag} = 1;
 3668      # we couldn't compile the equation, we'll return an error message.
 3669   } else {
 3670 #     if (defined($options{stdout} )) {
 3671 #       $rh_ans ->{$options{stdout}} = $function_sub;
 3672 #     } else {
 3673 #         $rh_ans->{rf_student_ans} = $function_sub;
 3674 #       }
 3675       $rh_ans ->{$options{stdout}} = $function_sub;
 3676   }
 3677 
 3678     $rh_ans;
 3679 }
 3680 
 3681 =head4 is_zero_array
 3682 
 3683 
 3684 =cut
 3685 
 3686 
 3687 sub is_zero_array {
 3688     my $rh_ans = shift;
 3689     my %options = @_;
 3690     set_default_options(  \%options,
 3691         '_filter_name'  =>  'is_zero_array',
 3692         'tolerance'     =>  0.000001,
 3693         'stdin'         => 'ra_differences',
 3694         'stdout'        => 'score',
 3695     );
 3696     #intialize
 3697     $rh_ans->{_filter_name} = $options{_filter_name};
 3698 
 3699     my $array = $rh_ans -> {$options{stdin}};  # default ra_differences
 3700   my $num = @$array;
 3701   my $i;
 3702   my $max = 0; my $mm;
 3703   for ($i=0; $i< $num; $i++) {
 3704     $mm = $array->[$i] ;
 3705     if  (not is_a_number($mm) ) {
 3706       $max = $mm;  # break out if one of the elements is not a number
 3707       last;
 3708     }
 3709     $max = abs($mm) if abs($mm) > $max;
 3710   }
 3711   if (not is_a_number($max)) {
 3712     $rh_ans->{score} = 0;
 3713       my $error = "WeBWorK was unable evaluate your function. Please check that your
 3714                 expression doesn't take roots of negative numbers, or divide by zero.";
 3715     $rh_ans->throw_error('EVAL',$error);
 3716   } else {
 3717       $rh_ans->{$options{stdout}} = ($max < $options{tolerance} ) ? 1: 0;       # set 'score' to 1 if the array is close to 0;
 3718   }
 3719   $rh_ans;
 3720 }
 3721 
 3722 =head4 best_approx_parameters
 3723 
 3724   best_approx_parameters($rh_ans,%options);   #requires the following fields in $rh_ans
 3725                         {rf_student_ans}      # reference to the test answer
 3726                         {rf_correct_ans}      # reference to the comparison answer
 3727                         {evaluation_points},  # an array of row vectors indicating the points
 3728                                       # to evaluate when comparing the functions
 3729 
 3730                          %options       # debug => 1   gives more error answers
 3731                                     # param_vars => ['']  additional parameters used to adapt to function
 3732                          )
 3733 
 3734 
 3735 The parameters for the comparison function which best approximates the test_function are stored
 3736 in the field {ra_parameters}.
 3737 
 3738 
 3739 The last $dim_of_parms_space variables are assumed to be parameters, and it is also
 3740 assumed that the function \&comparison_fun
 3741 depends linearly on these variables.  This function finds the  values for these parameters which minimizes the
 3742 Euclidean distance (L2 distance) between the test function and the comparison function and the test points specified
 3743 by the array reference  \@rows_of_test_points.  This is assumed to be an array of arrays, with the inner arrays
 3744 determining a test point.
 3745 
 3746 The comparison function should have $dim_of_params_space more input variables than the test function.
 3747 
 3748 
 3749 
 3750 
 3751 
 3752 =cut
 3753 
 3754 # Used internally:
 3755 #
 3756 #   &$determine_param_coeff( $rf_comparison_function # a reference to the correct answer function
 3757 #                    $ra_variables                   # an array of the active input variables to the functions
 3758 #                    $dim_of_params_space            # indicates the number of parameters upon which the
 3759 #                                                    # the comparison function depends linearly.  These are assumed to
 3760 #                                                    # be the last group of inputs to the comparison function.
 3761 #
 3762 #                    %options                        # $options{debug} gives more error messages
 3763 #
 3764 #                                                    # A typical function might look like
 3765 #                                                    # f(x,y,z,a,b) = x^2+a*cos(xz) + b*sin(x) with a parameter
 3766 #                                                    # space of dimension 2 and a variable space of dimension 3.
 3767 #                   )
 3768 #         # returns a list of coefficients
 3769 
 3770 sub best_approx_parameters {
 3771     my $rh_ans = shift;
 3772     my %options = @_;
 3773     set_default_options(\%options,
 3774         '_filter_name'      =>  'best_approx_paramters',
 3775         'allow_unknown_options' =>  1,
 3776     );
 3777     my $errors = undef;
 3778     # This subroutine for the determining the coefficents of the parameters at a given point
 3779     # is pretty specialized, so it is included here as a sub-subroutine.
 3780     my $determine_param_coeffs  = sub {
 3781     my ($rf_fun, $ra_variables, $dim_of_params_space, %options) =@_;
 3782     my @zero_params=();
 3783     for(my $i=1;$i<=$dim_of_params_space;$i++){push(@zero_params,0); }
 3784     my @vars = @$ra_variables;
 3785     my @coeff = ();
 3786     my @inputs = (@vars,@zero_params);
 3787     my ($f0, $f1, $err);
 3788     ($f0, $err) = &{$rf_fun}(@inputs);
 3789     if (defined($err) ) {
 3790       $errors .= "$err ";
 3791     } else {
 3792       for (my $i=@vars;$i<@inputs;$i++) {
 3793         $inputs[$i]=1;  # set one parameter to 1;
 3794         my($f1,$err) = &$rf_fun(@inputs);
 3795         if (defined($err) ) {
 3796           $errors .= " $err ";
 3797         } else {
 3798           push(@coeff, $f1-$f0);
 3799         }
 3800         $inputs[$i]=0;  # set it back
 3801       }
 3802     }
 3803     (\@coeff, $errors);
 3804   };
 3805     my $rf_fun = $rh_ans->{rf_student_ans};
 3806     my $rf_correct_fun = $rh_ans->{rf_correct_ans};
 3807     my $ra_vars_matrix = $rh_ans->{evaluation_points};
 3808     my $dim_of_param_space = @{$options{param_vars}};
 3809     # Short cut.  Bail if there are no param_vars
 3810     unless ($dim_of_param_space >0) {
 3811     $rh_ans ->{ra_parameters} = [];
 3812     return $rh_ans;
 3813     }
 3814     # inputs are row arrays in this case.
 3815     my @zero_params=();
 3816 
 3817     for(my $i=1;$i<=$dim_of_param_space;$i++){push(@zero_params,0); }
 3818     my @rows_of_vars = @$ra_vars_matrix;
 3819     warn "input rows ", pretty_print(\@rows_of_vars) if defined($options{debug}) and $options{debug};
 3820     my $rows = @rows_of_vars;
 3821     my $matrix =new Matrix($rows,$dim_of_param_space);
 3822     my $rhs_vec = new Matrix($rows, 1);
 3823     my $row_num = 1;
 3824     my ($ra_coeff,$val2, $val1, $err1,$err2,@inputs,@vars);
 3825     my $number_of_data_points = $dim_of_param_space +2;
 3826     while (@rows_of_vars and $row_num <= $number_of_data_points) {
 3827      # get one set of data points from the test function;
 3828       @vars = @{ shift(@rows_of_vars) };
 3829       ($val2, $err1) = &{$rf_fun}(@vars);
 3830       $errors .= " $err1 "  if defined($err1);
 3831       @inputs = (@vars,@zero_params);
 3832       ($val1, $err2) = &{$rf_correct_fun}(@inputs);
 3833       $errors .= " $err2 " if defined($err2);
 3834 
 3835       unless (defined($err1) or defined($err2) ) {
 3836           $rhs_vec->assign($row_num,1, $val2-$val1 );
 3837 
 3838     # warn "rhs data  val1=$val1, val2=$val2, val2 - val1 = ", $val2 - $val1 if $options{debug};
 3839     # warn "vars ", join(" | ", @vars) if $options{debug};
 3840 
 3841       ($ra_coeff, $err1) = &{$determine_param_coeffs}($rf_correct_fun,\@vars,$dim_of_param_space,%options);
 3842       if (defined($err1) ) {
 3843         $errors .= " $err1 ";
 3844       } else {
 3845         my @coeff = @$ra_coeff;
 3846         my $col_num=1;
 3847           while(@coeff) {
 3848             $matrix->assign($row_num,$col_num, shift(@coeff) );
 3849             $col_num++;
 3850           }
 3851         }
 3852       }
 3853       $row_num++;
 3854       last if $errors;  # break if there are any errors.
 3855                       # This cuts down on the size of error messages.
 3856                       # However it impossible to check for equivalence at 95% of points
 3857             # which might be useful for functions that are not defined at some points.
 3858   }
 3859     warn "<br> best_approx_parameters: matrix1 <br>  ", " $matrix " if $options{debug};
 3860     warn "<br> best_approx_parameters: vector <br>  ", " $rhs_vec " if $options{debug};
 3861 
 3862    # we have   Matrix * parameter = data_vec + perpendicular vector
 3863    # where the matrix has column vectors defining the span of the parameter space
 3864    # multiply both sides by Matrix_transpose and solve for the parameters
 3865    # This is exactly what the method proj_coeff method does.
 3866    my @array;
 3867    if (defined($errors) ) {
 3868     @array = ();   #     new Matrix($dim_of_param_space,1);
 3869    } else {
 3870     @array = $matrix->proj_coeff($rhs_vec)->list();
 3871    }
 3872   # check size (hack)
 3873   my $max = 0;
 3874   foreach my $val (@array ) {
 3875     $max = abs($val) if  $max < abs($val);
 3876     if (not is_a_number($val) ) {
 3877       $max = "NaN: $val";
 3878       last;
 3879     }
 3880   }
 3881   if ($max =~/NaN/) {
 3882     $errors .= "WeBWorK was unable evaluate your function. Please check that your
 3883                 expression doesn't take roots of negative numbers, or divide by zero.";
 3884   } elsif ($max > $options{maxConstantOfIntegration} ) {
 3885     $errors .= "At least one of the adapting parameters
 3886              (perhaps the constant of integration) is too large: $max,
 3887              ( the maximum allowed is $options{maxConstantOfIntegration} )";
 3888   }
 3889 
 3890     $rh_ans->{ra_parameters} = \@array;
 3891     $rh_ans->throw_error('EVAL', $errors) if defined($errors);
 3892     $rh_ans;
 3893 }
 3894 
 3895 =head4 calculate_difference_vector
 3896 
 3897   calculate_difference_vector( $ans_hash, %options);
 3898 
 3899                 {rf_student_ans},     # a reference to the test function
 3900                                {rf_correct_ans},      # a reference to the correct answer function
 3901                                {evaluation_points},   # an array of row vectors indicating the points
 3902                                           # to evaluate when comparing the functions
 3903                                {ra_parameters}        # these are the (optional) additional inputs to
 3904                                                       # the comparison function which adapt it properly
 3905                                                       # to the problem at hand.
 3906 
 3907                                %options               # mode => 'rel'  specifies that each element in the
 3908                                                       # difference matrix is divided by the correct answer.
 3909                                                       # unless the correct answer is nearly 0.
 3910                               )
 3911 
 3912 =cut
 3913 
 3914 sub calculate_difference_vector {
 3915   my $rh_ans = shift;
 3916   my %options = @_;
 3917   assign_option_aliases( \%options,
 3918     );
 3919     set_default_options(  \%options,
 3920         allow_unknown_options  =>  1,
 3921       stdin1               => 'rf_student_ans',
 3922       stdin2                 => 'rf_correct_ans',
 3923       stdout                 => 'ra_differences',
 3924     debug                  =>  0,
 3925     tolType                => 'absolute',
 3926     error_msg_flag         =>  1,
 3927      );
 3928   # initialize
 3929   $rh_ans->{_filter_name} = 'calculate_difference_vector';
 3930   my $rf_fun              = $rh_ans -> {$options{stdin1}};        # rf_student_ans by default
 3931   my $rf_correct_fun      = $rh_ans -> {$options{stdin2}};        # rf_correct_ans by default
 3932   my $ra_parameters       = $rh_ans -> {ra_parameters};
 3933   my @evaluation_points   = @{$rh_ans->{evaluation_points} };
 3934   my @parameters          = ();
 3935   @parameters             = @$ra_parameters if defined($ra_parameters) and ref($ra_parameters) eq 'ARRAY';
 3936   my $errors              = undef;
 3937   my @zero_params         = ();
 3938   for (my $i=1;$i<=@{$ra_parameters};$i++) {
 3939     push(@zero_params,0);
 3940   }
 3941   my @differences         = ();
 3942   my @student_values;
 3943   my @adjusted_student_values;
 3944   my @instructorVals;
 3945   my ($diff,$instructorVal);
 3946   # calculate the vector of differences between the test function and the comparison function.
 3947   while (@evaluation_points) {
 3948     my ($err1, $err2,$err3);
 3949     my @vars = @{ shift(@evaluation_points) };
 3950     my @inputs = (@vars, @parameters);
 3951     my ($inVal,  $correctVal);
 3952     ($inVal, $err1) = &{$rf_fun}(@vars);
 3953     $errors .= " $err1 "  if defined($err1);
 3954     $errors .= " Error detected evaluating student input at (".join(' , ',@vars) ." ) " if  defined($options{debug}) and $options{debug}==1 and defined($err1);
 3955     ($correctVal, $err2) =&{$rf_correct_fun}(@inputs);
 3956     $errors .= " There is an error in WeBWorK's answer to this problem, please alert your instructor.<br> $err2 " if defined($err2);
 3957     $errors .= " Error detected evaluating correct adapted answer  at (".join(' , ',@inputs) ." ) " if defined($options{debug}) and $options{debug}=1 and defined($err2);
 3958     ($instructorVal,$err3)= &$rf_correct_fun(@vars, @zero_params);
 3959     $errors .= " There is an error in WeBWorK's answer to this problem, please alert your instructor.<br> $err3 " if defined($err3);
 3960     $errors .= " Error detected evaluating instructor answer  at (".join(' , ',@vars, @zero_params) ." ) " if defined($options{debug}) and $options{debug}=1 and defined($err3);
 3961     unless (defined($err1) or defined($err2) or defined($err3) ) {
 3962       $diff = ( $inVal - ($correctVal -$instructorVal ) ) - $instructorVal;  #prevents entering too high a number?
 3963       #warn "taking the difference of ", $inVal, " and ", $correctVal, " is ", $diff;
 3964       if ( $options{tolType} eq 'relative' ) {  #relative tolerance
 3965         #warn "diff = $diff";
 3966         #$diff = ( $inVal - ($correctVal-$instructorVal ) )/abs($instructorVal) -1    if abs($instructorVal) > $options{zeroLevel};
 3967         $diff = ( $inVal - ($correctVal-$instructorVal ) )/$instructorVal -1    if abs($instructorVal) > $options{zeroLevel};
 3968         #$diff = ( $inVal - ($correctVal-$instructorVal- $instructorVal ) )/abs($instructorVal)    if abs($instructorVal) > $options{zeroLevel};
 3969         #warn "diff = $diff,   ", abs( &$rf_correct_fun(@inputs) ) , "-- $correctVal";
 3970       }
 3971     }
 3972     last if $errors;  # break if there are any errors.
 3973                   # This cuts down on the size of error messages.
 3974                   # However it impossible to check for equivalence at 95% of points
 3975                   # which might be useful for functions that are not defined at some points.
 3976         push(@student_values,$inVal);
 3977         push(@adjusted_student_values,(  $inVal - ($correctVal -$instructorVal) ) );
 3978     push(@differences, $diff);
 3979     push(@instructorVals,$instructorVal);
 3980   }
 3981   if (( not defined($errors) )  or $errors eq '' or $options{error_msg_flag} ) {
 3982       $rh_ans ->{$options{stdout}} = \@differences;
 3983     $rh_ans ->{ra_student_values} = \@student_values;
 3984     $rh_ans ->{ra_adjusted_student_values} = \@adjusted_student_values;
 3985     $rh_ans->{ra_instructor_values}=\@instructorVals;
 3986     $rh_ans->throw_error('EVAL', $errors) if defined($errors);
 3987   } else {
 3988 
 3989   }      # no output if error_msg_flag is set to 0.
 3990 
 3991   $rh_ans;
 3992 }
 3993 
 3994 =head4 fix_answer_for_display
 3995 
 3996 =cut
 3997 
 3998 sub fix_answers_for_display {
 3999   my ($rh_ans, %options) = @_;
 4000   if ( $rh_ans->{answerIsString} ==1) {
 4001     $rh_ans = evaluatesToNumber ($rh_ans, %options);
 4002   }
 4003   if (defined ($rh_ans->{student_units})) {
 4004     $rh_ans->{student_ans} = $rh_ans->{student_ans}. ' '. $rh_ans->{student_units};
 4005 
 4006   }
 4007   if ( $rh_ans->catch_error('UNITS')  ) {  # create preview latex string for expressions even if the units are incorrect
 4008       my $rh_temp = new AnswerHash;
 4009       $rh_temp->{student_ans} = $rh_ans->{student_ans};
 4010       $rh_temp = check_syntax($rh_temp);
 4011       $rh_ans->{preview_latex_string} = $rh_temp->{preview_latex_string};
 4012   }
 4013   $rh_ans->{correct_ans} = $rh_ans->{original_correct_ans};
 4014 
 4015   $rh_ans;
 4016 }
 4017 
 4018 =head4 evaluatesToNumber
 4019 
 4020 =cut
 4021 
 4022 sub evaluatesToNumber {
 4023   my ($rh_ans, %options) = @_;
 4024   if (is_a_numeric_expression($rh_ans->{student_ans})) {
 4025     my ($inVal,$PG_eval_errors,$PG_full_error_report) = PG_answer_eval($rh_ans->{student_ans});
 4026     if ($PG_eval_errors) { # this if statement should never be run
 4027       # change nothing
 4028     } else {
 4029       # change this
 4030       $rh_ans->{student_ans} = prfmt($inVal,$options{format});
 4031     }
 4032   }
 4033   $rh_ans;
 4034 }
 4035 
 4036 =head4 is_numeric_expression
 4037 
 4038 =cut
 4039 
 4040 sub is_a_numeric_expression {
 4041   my $testString = shift;
 4042   my $is_a_numeric_expression = 0;
 4043   my ($inVal,$PG_eval_errors,$PG_full_error_report) = PG_answer_eval($testString);
 4044   if ($PG_eval_errors) {
 4045     $is_a_numeric_expression = 0;
 4046   } else {
 4047     $is_a_numeric_expression = 1;
 4048   }
 4049   $is_a_numeric_expression;
 4050 }
 4051 
 4052 =head4 is_a_number
 4053 
 4054 =cut
 4055 
 4056 sub is_a_number {
 4057   my ($num,%options) =  @_;
 4058   my $process_ans_hash = ( ref( $num ) eq 'AnswerHash' ) ? 1 : 0 ;
 4059   my ($rh_ans);
 4060   if ($process_ans_hash) {
 4061     $rh_ans = $num;
 4062     $num = $rh_ans->{student_ans};
 4063   }
 4064 
 4065   my $is_a_number = 0;
 4066   return $is_a_number unless defined($num);
 4067   $num =~ s/^\s*//; ## remove initial spaces
 4068   $num =~ s/\s*$//; ## remove trailing spaces
 4069 
 4070   ## the following is copied from the online perl manual
 4071   if ($num =~ /^([+-]?)(?=\d|\.\d)\d*(\.\d*)?([Ee]([+-]?\d+))?$/){
 4072     $is_a_number = 1;
 4073   }
 4074 
 4075   if ($process_ans_hash)   {
 4076         if ($is_a_number == 1 ) {
 4077           $rh_ans->{student_ans}=$num;
 4078           return $rh_ans;
 4079         } else {
 4080           $rh_ans->{student_ans} = "Incorrect number format:  You must enter a number, e.g. -6, 5.3, or 6.12E-3";
 4081           $rh_ans->throw_error('NUMBER', 'You must enter a number, e.g. -6, 5.3, or 6.12E-3');
 4082           return $rh_ans;
 4083         }
 4084   } else {
 4085     return $is_a_number;
 4086   }
 4087 }
 4088 
 4089 =head4 is_a_fraction
 4090 
 4091 =cut
 4092 
 4093 sub is_a_fraction {
 4094   my ($num,%options) =  @_;
 4095   my $process_ans_hash = ( ref( $num ) eq 'AnswerHash' ) ? 1 : 0 ;
 4096   my ($rh_ans);
 4097   if ($process_ans_hash) {
 4098     $rh_ans = $num;
 4099     $num = $rh_ans->{student_ans};
 4100   }
 4101 
 4102   my $is_a_fraction = 0;
 4103   return $is_a_fraction unless defined($num);
 4104   $num =~ s/^\s*//; ## remove initial spaces
 4105   $num =~ s/\s*$//; ## remove trailing spaces
 4106 
 4107   if ($num =~ /^\s*\-?\s*[\/\d\.Ee\s]*$/) {
 4108     $is_a_fraction = 1;
 4109   }
 4110 
 4111     if ($process_ans_hash)   {
 4112       if ($is_a_fraction == 1 ) {
 4113         $rh_ans->{student_ans}=$num;
 4114         return $rh_ans;
 4115       } else {
 4116         $rh_ans->{student_ans} = "Not a number of fraction: You must enter a number or fraction, e.g. -6 or 7/13";
 4117         $rh_ans->throw_error('NUMBER', 'You must enter a number, e.g. -6, 5.3, or 6.12E-3');
 4118         return $rh_ans;
 4119       }
 4120 
 4121       } else {
 4122     return $is_a_fraction;
 4123   }
 4124 }
 4125 
 4126 =head4 phase_pi
 4127   I often discovered that the answers I was getting, when using the arctan function would be off by phases of
 4128   pi, which for the tangent function, were equivalent values. This method allows for this.
 4129 =cut
 4130 
 4131 sub phase_pi {
 4132   my ($num,%options) =  @_;
 4133   my $process_ans_hash = ( ref( $num ) eq 'AnswerHash' ) ? 1 : 0 ;
 4134   my ($rh_ans);
 4135   if ($process_ans_hash) {
 4136     $rh_ans = $num;
 4137     $num = $rh_ans->{correct_ans};
 4138   }
 4139   while( ($rh_ans->{correct_ans}) >  3.14159265358979/2 ){
 4140     $rh_ans->{correct_ans} -= 3.14159265358979;
 4141   }
 4142   while( ($rh_ans->{correct_ans}) <= -3.14159265358979/2 ){
 4143     $rh_ans->{correct_ans} += 3.14159265358979;
 4144   }
 4145   $rh_ans;
 4146 }
 4147 
 4148 =head4 is_an_arithemetic_expression
 4149 
 4150 =cut
 4151 
 4152 sub is_an_arithmetic_expression {
 4153   my ($num,%options) =  @_;
 4154   my $process_ans_hash = ( ref( $num ) eq 'AnswerHash' ) ? 1 : 0 ;
 4155   my ($rh_ans);
 4156   if ($process_ans_hash) {
 4157     $rh_ans = $num;
 4158     $num = $rh_ans->{student_ans};
 4159   }
 4160 
 4161   my $is_an_arithmetic_expression = 0;
 4162   return $is_an_arithmetic_expression unless defined($num);
 4163   $num =~ s/^\s*//; ## remove initial spaces
 4164   $num =~ s/\s*$//; ## remove trailing spaces
 4165 
 4166   if ($num =~ /^[+\-*\/\^\(\)\[\]\{\}\s\d\.Ee]*$/) {
 4167     $is_an_arithmetic_expression =  1;
 4168   }
 4169 
 4170     if ($process_ans_hash)   {
 4171       if ($is_an_arithmetic_expression == 1 ) {
 4172         $rh_ans->{student_ans}=$num;
 4173         return $rh_ans;
 4174       } else {
 4175 
 4176     $rh_ans->{student_ans} = "Not an arithmetic expression: You must enter an arithmetic expression, e.g. -6 or (2.3*4+5/3)^2";
 4177         $rh_ans->throw_error('NUMBER', 'You must enter an arithmetic expression, e.g. -6 or (2.3*4+5/3)^2');
 4178         return $rh_ans;
 4179       }
 4180 
 4181       } else {
 4182     return $is_an_arithmetic_expression;
 4183   }
 4184 }
 4185 
 4186 #
 4187 
 4188 =head4 math_constants
 4189 
 4190 replaces pi, e, and ^ with their Perl equivalents
 4191 if useBaseTenLog is non-zero, convert log to logten
 4192 
 4193 =cut
 4194 
 4195 sub math_constants {
 4196   my($in,%options) = @_;
 4197   my $rh_ans;
 4198   my $process_ans_hash = ( ref( $in ) eq 'AnswerHash' ) ? 1 : 0 ;
 4199   if ($process_ans_hash) {
 4200     $rh_ans = $in;
 4201     $in = $rh_ans->{student_ans};
 4202   }
 4203   # The code fragment above allows this filter to be used when the input is simply a string
 4204   # as well as when the input is an AnswerHash, and options.
 4205   $in =~s/\bpi\b/(4*atan2(1,1))/ge;
 4206   $in =~s/\be\b/(exp(1))/ge;
 4207   $in =~s/\^/**/g;
 4208   if($useBaseTenLog) {
 4209     $in =~ s/\blog\b/logten/g;
 4210   }
 4211 
 4212   if ($process_ans_hash)   {
 4213       $rh_ans->{student_ans}=$in;
 4214       return $rh_ans;
 4215     } else {
 4216     return $in;
 4217   }
 4218 }
 4219 
 4220 
 4221 
 4222 =head4 is_array
 4223 
 4224   is_array($rh_ans)
 4225     returns: $rh_ans.   Throws error "NOTARRAY" if this is not an array
 4226 
 4227 =cut
 4228 
 4229 sub is_array  {
 4230   my $rh_ans = shift;
 4231     # return if the result is an array
 4232   return($rh_ans) if  ref($rh_ans->{student_ans}) eq 'ARRAY' ;
 4233   $rh_ans->throw_error("NOTARRAY","The answer is not an array");
 4234   $rh_ans;
 4235 }
 4236 
 4237 =head4 check_syntax
 4238 
 4239   check_syntax( $rh_ans, %options)
 4240     returns an answer hash.
 4241 
 4242 latex2html preview code are installed in the answer hash.
 4243 The input has been transformed, changing 7pi to 7*pi  or 7x to 7*x.
 4244 Syntax error messages may be generated and stored in student_ans
 4245 Additional syntax error messages are stored in {ans_message} and duplicated in {error_message}
 4246 
 4247 
 4248 =cut
 4249 
 4250 sub check_syntax {
 4251         my $rh_ans = shift;
 4252         my %options = @_;
 4253         assign_option_aliases(\%options,
 4254     );
 4255     set_default_options(  \%options,
 4256           'stdin'         =>  'student_ans',
 4257           'stdout'    =>  'student_ans',
 4258           'ra_vars'   =>  [qw( x y )],
 4259           'debug'     =>  0,
 4260           '_filter_name'  =>  'check_syntax',
 4261           error_msg_flag  =>  1,
 4262     );
 4263     #initialize
 4264     $rh_ans->{_filter_name}     = $options{_filter_name};
 4265         unless ( defined( $rh_ans->{$options{stdin}} ) ) {
 4266           warn "Check_syntax requires an equation in the field '$options{stdin}' or input";
 4267           $rh_ans->throw_error("1","'$options{stdin}' field not defined");
 4268           return $rh_ans;
 4269         }
 4270         my $in     = $rh_ans->{$options{stdin}};
 4271     my $parser = new AlgParserWithImplicitExpand;
 4272     my $ret    = $parser -> parse($in);     #for use with loops
 4273 
 4274     if ( ref($ret) )  {   ## parsed successfully
 4275       # $parser -> tostring();   # FIXME?  was this needed for some reason?????
 4276       $parser -> normalize();
 4277       $rh_ans -> {$options{stdout}}     = $parser -> tostring();
 4278       $rh_ans -> {preview_text_string}  = $in;
 4279       $rh_ans -> {preview_latex_string} = $parser -> tolatex();
 4280 
 4281     } elsif ($options{error_msg_flag} ) {         ## error in parsing
 4282 
 4283       $rh_ans->{$options{stdout}}     = 'syntax error:'. $parser->{htmlerror},
 4284       $rh_ans->{'ans_message'}      = $parser -> {error_msg},
 4285       $rh_ans->{'preview_text_string'}  = '',
 4286       $rh_ans->{'preview_latex_string'} = '',
 4287       $rh_ans->throw_error('SYNTAX',  'syntax error in answer:'. $parser->{htmlerror} . "$BR" .$parser -> {error_msg});
 4288     }   # no output is produced if there is an error and the error_msg_flag is set to zero
 4289        $rh_ans;
 4290 
 4291 }
 4292 
 4293 =head4 check_strings
 4294 
 4295   check_strings ($rh_ans, %options)
 4296     returns $rh_ans
 4297 
 4298 =cut
 4299 
 4300 sub check_strings {
 4301   my ($rh_ans, %options) = @_;
 4302 
 4303   # if the student's answer is a number, simply return the answer hash (unchanged).
 4304 
 4305   #  we allow constructions like -INF to be treated as a string. Thus we ignore an initial
 4306   # - in deciding whether the student's answer is a number or string
 4307 
 4308   my $temp_ans = $rh_ans->{student_ans};
 4309   $temp_ans =~ s/^\s*\-//;   # remove an initial -
 4310 
 4311   if  ( $temp_ans =~ m/[\d+\-*\/^(){}\[\]]|^\s*e\s*$|^\s*pi\s*$/)   {
 4312   # if ( $rh_ans->{answerIsString} == 1) {
 4313   #     #$rh_ans->throw_error('STRING','Incorrect Answer'); # student's answer is a number
 4314   # }
 4315     return $rh_ans;
 4316   }
 4317   # the student's answer is recognized as a string
 4318   my $ans = $rh_ans->{student_ans};
 4319 
 4320 # OVERVIEW of reminder of function:
 4321 # if answer is correct, return correct.  (adjust score to 1)
 4322 # if answer is incorect:
 4323 # 1) determine if the answer is sensible.  if it is, return incorrect.
 4324 # 2) if the answer is not sensible (and incorrect), then return an error message indicating so.
 4325 # no matter what:  throw a 'STRING' error to skip numerical evaluations.  (error flag skips remainder of pre_filters and evaluators)
 4326 # last: 'STRING' post_filter will clear the error (avoiding pink screen.)
 4327 
 4328   my $sensibleAnswer = 0;
 4329   $ans = str_filters( $ans, 'compress_whitespace' );  # remove trailing, leading, and double spaces.
 4330   my ($ans_eval) = str_cmp($rh_ans->{correct_ans});
 4331   my $temp_ans_hash = $ans_eval->evaluate($ans);
 4332   $rh_ans->{test} = $temp_ans_hash;
 4333 
 4334   if ($temp_ans_hash->{score} ==1 ) {     # students answer matches the correct answer.
 4335     $rh_ans->{score} = 1;
 4336     $sensibleAnswer = 1;
 4337   } else {            # students answer does not match the correct answer.
 4338     my $legalString = '';       # find out if string makes sense
 4339     my @legalStrings = @{$options{strings}};
 4340     foreach $legalString (@legalStrings) {
 4341       if ( uc($ans) eq uc($legalString) ) {
 4342         $sensibleAnswer = 1;
 4343         last;
 4344         }
 4345       }
 4346     $sensibleAnswer = 1 unless $ans =~ /\S/;  ## empty answers are sensible
 4347     $rh_ans->throw_error('EVAL', "Your answer is not a recognized answer") unless ($sensibleAnswer);
 4348     # $temp_ans_hash -> setKeys( 'ans_message' => 'Your answer is not a recognized answer' ) unless ($sensibleAnswer);
 4349     # $temp_ans_hash -> setKeys( 'student_ans' => uc($ans) );
 4350   }
 4351 
 4352   $rh_ans->{student_ans} = $ans;
 4353 
 4354   if ($sensibleAnswer) {
 4355     $rh_ans->throw_error('STRING', "The student's answer $rh_ans->{student_ans} is interpreted as a string.");
 4356   }
 4357 
 4358   $rh_ans->{'preview_text_string'}  = $ans,
 4359   $rh_ans->{'preview_latex_string'} = $ans,
 4360 
 4361   # warn ("\$rh_ans->{answerIsString} = $rh_ans->{answerIsString}");
 4362   $rh_ans;
 4363 }
 4364 
 4365 =head4 check_units
 4366 
 4367   check_strings ($rh_ans, %options)
 4368     returns $rh_ans
 4369 
 4370 
 4371 =cut
 4372 
 4373 sub check_units {
 4374   my ($rh_ans, %options) = @_;
 4375   my %correct_units = %{$rh_ans-> {rh_correct_units}};
 4376   my $ans = $rh_ans->{student_ans};
 4377   # $ans = '' unless defined ($ans);
 4378   $ans = str_filters ($ans, 'trim_whitespace');
 4379   my $original_student_ans = $ans;
 4380   $rh_ans->{original_student_ans} = $original_student_ans;
 4381 
 4382   # it surprises me that the match below works since the first .* is greedy.
 4383   my ($num_answer, $units) = $ans =~ /^(.*)\s+([^\s]*)$/;
 4384 
 4385   unless ( defined($num_answer) && $units ) {
 4386     # there is an error reading the input
 4387     if ( $ans =~ /\S/ )  {  # the answer is not blank
 4388       $rh_ans -> setKeys( 'ans_message' => "The answer \"$ans\" could not be interpreted " .
 4389         "as a number or an arithmetic expression followed by a unit specification. " .
 4390         "Your answer must contain units." );
 4391       $rh_ans->throw_error('UNITS', "The answer \"$ans\" could not be interpreted " .
 4392         "as a number or an arithmetic expression followed by a unit specification. " .
 4393         "Your answer must contain units." );
 4394     }
 4395     return $rh_ans;
 4396   }
 4397 
 4398   # we have been able to parse the answer into a numerical part and a unit part
 4399 
 4400   # $num_answer = $1;   #$1 and $2 from the regular expression above
 4401   # $units    = $2;
 4402 
 4403   my %units = Units::evaluate_units($units);
 4404   if ( defined( $units{'ERROR'} ) ) {
 4405      # handle error condition
 4406           $units{'ERROR'} = clean_up_error_msg($units{'ERROR'});
 4407     $rh_ans -> setKeys( 'ans_message' => "$units{'ERROR'}" );
 4408     $rh_ans -> throw_error('UNITS', "$units{'ERROR'}");
 4409     return $rh_ans;
 4410   }
 4411 
 4412   my $units_match = 1;
 4413   my $fund_unit;
 4414   foreach $fund_unit (keys %correct_units) {
 4415     next if $fund_unit eq 'factor';
 4416     $units_match = 0 unless $correct_units{$fund_unit} == $units{$fund_unit};
 4417   }
 4418 
 4419   if ( $units_match ) {
 4420         # units are ok.  Evaluate the numerical part of the answer
 4421     $rh_ans->{'tolerance'} = $rh_ans->{'tolerance'}* $correct_units{'factor'}/$units{'factor'}  if
 4422           $rh_ans->{'tolType'} eq 'absolute'; # the tolerance is in the units specified by the instructor.
 4423     $rh_ans->{correct_ans} =  prfmt($rh_ans->{correct_ans}*$correct_units{'factor'}/$units{'factor'});
 4424     $rh_ans->{student_units} = $units;
 4425     $rh_ans->{student_ans} = $num_answer;
 4426 
 4427   } else {
 4428         $rh_ans -> setKeys( ans_message => 'There is an error in the units for this answer.' );
 4429         $rh_ans -> throw_error ( 'UNITS', 'There is an error in the units for this answer.' );
 4430   }
 4431 
 4432   return $rh_ans;
 4433 }
 4434 
 4435 
 4436 
 4437 =head2 Filter utilities
 4438 
 4439 These two subroutines can be used in filters to set default options.  They
 4440 help make filters perform in uniform, predictable ways, and also make it
 4441 easy to recognize from the code which options a given filter expects.
 4442 
 4443 
 4444 =head4 assign_option_aliases
 4445 
 4446 Use this to assign aliases for the standard options.  It must come before set_default_options
 4447 within the subroutine.
 4448 
 4449     assign_option_aliases(\%options,
 4450         'alias1'  => 'option5'
 4451         'alias2'  => 'option7'
 4452     );
 4453 
 4454 
 4455 If the subroutine is called with an option  " alias1 => 23 " it will behave as if it had been
 4456 called with the option " option5 => 23 "
 4457 
 4458 =cut
 4459 
 4460 
 4461 
 4462 sub assign_option_aliases {
 4463   my $rh_options = shift;
 4464   warn "The first entry to set_default_options must be a reference to the option hash" unless ref($rh_options) eq 'HASH';
 4465   my @option_aliases = @_;
 4466   while (@option_aliases) {
 4467     my $alias = shift @option_aliases;
 4468     my $option_key = shift @option_aliases;
 4469 
 4470     if (defined($rh_options->{$alias} )) {                       # if the alias appears in the option list
 4471       if (not defined($rh_options->{$option_key}) ) {          # and the option itself is not defined,
 4472         $rh_options->{$option_key} = $rh_options->{$alias};  # insert the value defined by the alias into the option value
 4473                                                              # the FIRST alias for a given option takes precedence
 4474                                                              # (after the option itself)
 4475       } else {
 4476         warn "option $option_key is already defined as", $rh_options->{$option_key}, "<br>\n",
 4477              "The attempt to override this option with the alias $alias with value ", $rh_options->{$alias},
 4478              " was ignored.";
 4479       }
 4480     }
 4481     delete($rh_options->{$alias});                               # remove the alias from the initial list
 4482   }
 4483 
 4484 }
 4485 
 4486 =head4 set_default_options
 4487 
 4488     set_default_options(\%options,
 4489         '_filter_name'  =>  'filter',
 4490         'option5'   =>  .0001,
 4491         'option7'   =>  'ascii',
 4492         'allow_unknown_options  =>  0,
 4493     }
 4494 
 4495 Note that the first entry is a reference to the options with which the filter was called.
 4496 
 4497 The option5 is set to .0001 unless the option is explicitly set when the subroutine is called.
 4498 
 4499 The B<'_filter_name'> option should always be set, although there is no error if it is missing.
 4500 It is used mainly for debugging answer evaluators and allows
 4501 you to keep track of which filter is currently processing the answer.
 4502 
 4503 If B<'allow_unknown_options'> is set to 0 then if the filter is called with options which do NOT appear in the
 4504 set_default_options list an error will be signaled and a warning message will be printed out.  This provides
 4505 error checking against misspelling an option and is generally what is desired for most filters.
 4506 
 4507 Occasionally one wants to write a filter which accepts a long list of options, not all of which are known in advance,
 4508 but only uses a subset of the options
 4509 provided.  In this case, setting 'allow_unkown_options' to 1 prevents the error from being signaled.
 4510 
 4511 =cut
 4512 
 4513 sub set_default_options {
 4514   my $rh_options = shift;
 4515   warn "The first entry to set_default_options must be a reference to the option hash" unless ref($rh_options) eq 'HASH';
 4516   my %default_options = @_;
 4517   unless ( defined($default_options{allow_unknown_options}) and $default_options{allow_unknown_options} == 1 ) {
 4518     foreach  my $key1 (keys %$rh_options) {
 4519       warn "This option |$key1| is not recognized in this subroutine<br> ", pretty_print($rh_options) unless exists($default_options{$key1});
 4520     }
 4521   }
 4522   foreach my $key (keys %default_options) {
 4523     if  ( not defined($rh_options->{$key} ) and defined( $default_options{$key} )  ) {
 4524       $rh_options->{$key} = $default_options{$key};  #this allows     tol   => undef to allow the tol option, but doesn't define
 4525                                                      # this key unless tol is explicitly defined.
 4526     }
 4527   }
 4528 }
 4529 
 4530 =head2 Problem Grader Subroutines
 4531 
 4532 =cut
 4533 
 4534 ## Problem Grader Subroutines
 4535 
 4536 #####################################
 4537 # This is a model for plug-in problem graders
 4538 #####################################
 4539 sub install_problem_grader {
 4540   my $rf_problem_grader = shift;
 4541   my $rh_flags = PG_restricted_eval(q!\\%main::PG_FLAGS!);
 4542   $rh_flags->{PROBLEM_GRADER_TO_USE} = $rf_problem_grader;
 4543 }
 4544 
 4545 =head4 std_problem_grader
 4546 
 4547 This is an all-or-nothing grader.  A student must get all parts of the problem write
 4548 before receiving credit.  You should make sure to use this grader on multiple choice
 4549 and true-false questions, otherwise students will be able to deduce how many
 4550 answers are correct by the grade reported by webwork.
 4551 
 4552 
 4553   install_problem_grader(~~&std_problem_grader);
 4554 
 4555 =cut
 4556 
 4557 sub std_problem_grader {
 4558   my $rh_evaluated_answers = shift;
 4559   my $rh_problem_state = shift;
 4560   my %form_options = @_;
 4561   my %evaluated_answers = %{$rh_evaluated_answers};
 4562   #  The hash $rh_evaluated_answers typically contains:
 4563   #    'answer1' => 34, 'answer2'=> 'Mozart', etc.
 4564 
 4565   # By default the  old problem state is simply passed back out again.
 4566   my %problem_state = %$rh_problem_state;
 4567 
 4568   # %form_options might include
 4569   # The user login name
 4570   # The permission level of the user
 4571   # The studentLogin name for this psvn.
 4572   # Whether the form is asking for a refresh or is submitting a new answer.
 4573 
 4574   # initial setup of the answer
 4575   my %problem_result = ( score    => 0,
 4576                errors   => '',
 4577              type   => 'std_problem_grader',
 4578              msg    => '',
 4579   );
 4580   # Checks
 4581 
 4582   my $ansCount = keys %evaluated_answers;  # get the number of answers
 4583 
 4584   unless ($ansCount > 0 ) {
 4585 
 4586     $problem_result{msg} = "This problem did not ask any questions.";
 4587     return(\%problem_result,\%problem_state);
 4588   }
 4589 
 4590   if ($ansCount > 1 ) {
 4591     $problem_result{msg} = 'In order to get credit for this problem all answers must be correct.' ;
 4592   }
 4593 
 4594   unless ($form_options{answers_submitted} == 1) {
 4595     return(\%problem_result,\%problem_state);
 4596   }
 4597 
 4598   my $allAnswersCorrectQ=1;
 4599   foreach my $ans_name (keys %evaluated_answers) {
 4600   # I'm not sure if this check is really useful.
 4601     if ( ( ref($evaluated_answers{$ans_name} ) eq 'HASH' ) or ( ref($evaluated_answers{$ans_name}) eq 'AnswerHash' ) )  {
 4602       $allAnswersCorrectQ = 0 unless( 1 == $evaluated_answers{$ans_name}->{score} );
 4603     }
 4604     else {
 4605       die "Error at file ",__FILE__,"line ", __LINE__,":  Answer |$ans_name| is not a hash reference\n".
 4606          $evaluated_answers{$ans_name} .
 4607          "This probably means that the answer evaluator for this answer\n" .
 4608          "is not working correctly.";
 4609       $problem_result{error} = "Error: Answer $ans_name is not a hash: $evaluated_answers{$ans_name}";
 4610     }
 4611   }
 4612   # report the results
 4613   $problem_result{score} = $allAnswersCorrectQ;
 4614 
 4615   # I don't like to put in this bit of code.
 4616   # It makes it hard to construct error free problem graders
 4617   # I would prefer to know that the problem score was numeric.
 4618   unless (defined($problem_state{recorded_score}) and $problem_state{recorded_score} =~ /^([+-]?)(?=\d|\.\d)\d*(\.\d*)?([Ee]([+-]?\d+))?$/ ) {
 4619     $problem_state{recorded_score} = 0;  # This gets rid of non-numeric scores
 4620   }
 4621   #
 4622   if ($allAnswersCorrectQ == 1 or $problem_state{recorded_score} == 1) {
 4623     $problem_state{recorded_score} = 1;
 4624   }
 4625   else {
 4626     $problem_state{recorded_score} = 0;
 4627   }
 4628 
 4629   $problem_state{num_of_correct_ans}++ if $allAnswersCorrectQ == 1;
 4630   $problem_state{num_of_incorrect_ans}++ if $allAnswersCorrectQ == 0;
 4631 
 4632   $problem_state{state_summary_msg} = '';  # an HTML formatted message printed at the bottom of the problem page
 4633 
 4634   (\%problem_result, \%problem_state);
 4635 }
 4636 
 4637 =head4 std_problem_grader2
 4638 
 4639 This is an all-or-nothing grader.  A student must get all parts of the problem write
 4640 before receiving credit.  You should make sure to use this grader on multiple choice
 4641 and true-false questions, otherwise students will be able to deduce how many
 4642 answers are correct by the grade reported by webwork.
 4643 
 4644 
 4645   install_problem_grader(~~&std_problem_grader2);
 4646 
 4647 The only difference between the two versions
 4648 is at the end of the subroutine, where std_problem_grader2
 4649 records the attempt only if there have been no syntax errors,
 4650 whereas std_problem_grader records it regardless.
 4651 
 4652 =cut
 4653 
 4654 
 4655 
 4656 sub std_problem_grader2 {
 4657   my $rh_evaluated_answers = shift;
 4658   my $rh_problem_state = shift;
 4659   my %form_options = @_;
 4660   my %evaluated_answers = %{$rh_evaluated_answers};
 4661   #  The hash $rh_evaluated_answers typically contains:
 4662   #    'answer1' => 34, 'answer2'=> 'Mozart', etc.
 4663 
 4664   # By default the  old problem state is simply passed back out again.
 4665   my %problem_state = %$rh_problem_state;
 4666 
 4667   # %form_options might include
 4668   # The user login name
 4669   # The permission level of the user
 4670   # The studentLogin name for this psvn.
 4671   # Whether the form is asking for a refresh or is submitting a new answer.
 4672 
 4673   # initial setup of the answer
 4674   my %problem_result = ( score        => 0,
 4675              errors       => '',
 4676              type       => 'std_problem_grader',
 4677              msg        => '',
 4678   );
 4679 
 4680   # syntax errors are not counted.
 4681   my $record_problem_attempt = 1;
 4682   # Checks
 4683 
 4684   my $ansCount = keys %evaluated_answers;  # get the number of answers
 4685   unless ($ansCount > 0 ) {
 4686     $problem_result{msg} = "This problem did not ask any questions.";
 4687     return(\%problem_result,\%problem_state);
 4688   }
 4689 
 4690   if ($ansCount > 1 ) {
 4691     $problem_result{msg} = 'In order to get credit for this problem all answers must be correct.' ;
 4692   }
 4693 
 4694   unless ($form_options{answers_submitted} == 1) {
 4695     return(\%problem_result,\%problem_state);
 4696   }
 4697 
 4698   my  $allAnswersCorrectQ=1;
 4699   foreach my $ans_name (keys %evaluated_answers) {
 4700   # I'm not sure if this check is really useful.
 4701     if ( ( ref($evaluated_answers{$ans_name} ) eq 'HASH' ) or ( ref($evaluated_answers{$ans_name}) eq 'AnswerHash' ) )  {
 4702       $allAnswersCorrectQ = 0 unless( 1 == $evaluated_answers{$ans_name}->{score} );
 4703     }
 4704     else {
 4705       die "Error at file ",__FILE__,"line ", __LINE__,":  Answer |$ans_name| is not a hash reference\n".
 4706          $evaluated_answers{$ans_name} .
 4707          "This probably means that the answer evaluator for this answer\n" .
 4708          "is not working correctly.";
 4709       $problem_result{error} = "Error: Answer $ans_name is not a hash: $evaluated_answers{$ans_name}";
 4710     }
 4711   }
 4712   # report the results
 4713   $problem_result{score} = $allAnswersCorrectQ;
 4714 
 4715   # I don't like to put in this bit of code.
 4716   # It makes it hard to construct error free problem graders
 4717   # I would prefer to know that the problem score was numeric.
 4718   unless ($problem_state{recorded_score} =~ /^([+-]?)(?=\d|\.\d)\d*(\.\d*)?([Ee]([+-]?\d+))?$/ ) {
 4719     $problem_state{recorded_score} = 0;  # This gets rid of non-numeric scores
 4720   }
 4721   #
 4722   if ($allAnswersCorrectQ == 1 or $problem_state{recorded_score} == 1) {
 4723     $problem_state{recorded_score} = 1;
 4724   }
 4725   else {
 4726     $problem_state{recorded_score} = 0;
 4727   }
 4728   # record attempt only if there have been no syntax errors.
 4729 
 4730   if ($record_problem_attempt == 1) {
 4731     $problem_state{num_of_correct_ans}++ if $allAnswersCorrectQ == 1;
 4732     $problem_state{num_of_incorrect_ans}++ if $allAnswersCorrectQ == 0;
 4733     $problem_state{state_summary_msg} = '';  # an HTML formatted message printed at the bottom of the problem page
 4734 
 4735   }
 4736   else {
 4737     $problem_result{show_partial_correct_answers} = 0 ;  # prevent partial correct answers from being shown for syntax errors.
 4738   }
 4739   (\%problem_result, \%problem_state);
 4740 }
 4741 
 4742 =head4 avg_problem_grader
 4743 
 4744 This grader gives a grade depending on how many questions from the problem are correct.  (The highest
 4745 grade is the one that is kept.  One can never lower the recorded grade on a problem by repeating it.)
 4746 Many professors (and almost all students :-)  ) prefer this grader.
 4747 
 4748 
 4749   install_problem_grader(~~&avg_problem_grader);
 4750 
 4751 =cut
 4752 
 4753 
 4754 sub avg_problem_grader {
 4755     my $rh_evaluated_answers = shift;
 4756   my $rh_problem_state = shift;
 4757   my %form_options = @_;
 4758   my %evaluated_answers = %{$rh_evaluated_answers};
 4759   #  The hash $rh_evaluated_answers typically contains:
 4760   #    'answer1' => 34, 'answer2'=> 'Mozart', etc.
 4761 
 4762   # By default the  old problem state is simply passed back out again.
 4763   my %problem_state = %$rh_problem_state;
 4764 
 4765 
 4766   # %form_options might include
 4767   # The user login name
 4768   # The permission level of the user
 4769   # The studentLogin name for this psvn.
 4770   # Whether the form is asking for a refresh or is submitting a new answer.
 4771 
 4772   # initial setup of the answer
 4773   my  $total=0;
 4774   my %problem_result = ( score        => 0,
 4775              errors       => '',
 4776              type       => 'avg_problem_grader',
 4777              msg        => '',
 4778   );
 4779   my $count = keys %evaluated_answers;
 4780   $problem_result{msg} = 'You can earn partial credit on this problem.' if $count >1;
 4781   # Return unless answers have been submitted
 4782   unless ($form_options{answers_submitted} == 1) {
 4783     return(\%problem_result,\%problem_state);
 4784   }
 4785 
 4786   # Answers have been submitted -- process them.
 4787   foreach my $ans_name (keys %evaluated_answers) {
 4788     # I'm not sure if this check is really useful.
 4789     if ( ( ref($evaluated_answers{$ans_name} ) eq 'HASH' ) or ( ref($evaluated_answers{$ans_name}) eq 'AnswerHash' ) )  {
 4790       $total += $evaluated_answers{$ans_name}->{score};
 4791     }
 4792     else {
 4793       die "Error: Answer |$ans_name| is not a hash reference\n".
 4794          $evaluated_answers{$ans_name} .
 4795          "This probably means that the answer evaluator for this answer\n" .
 4796          "is not working correctly.";
 4797       $problem_result{error} = "Error: Answer $ans_name is not a hash: $evaluated_answers{$ans_name}";
 4798     }
 4799   }
 4800   # Calculate score rounded to three places to avoid roundoff problems
 4801   $problem_result{score} = $total/$count if $count;
 4802   # increase recorded score if the current score is greater.
 4803   $problem_state{recorded_score} = $problem_result{score} if $problem_result{score} > $problem_state{recorded_score};
 4804 
 4805 
 4806   $problem_state{num_of_correct_ans}++ if $total == $count;
 4807   $problem_state{num_of_incorrect_ans}++ if $total < $count ;
 4808 
 4809   $problem_state{state_summary_msg} = '';  # an HTML formatted message printed at the bottom of the problem page
 4810 
 4811   warn "Error in grading this problem the total $total is larger than $count" if $total > $count;
 4812   (\%problem_result, \%problem_state);
 4813 }
 4814 
 4815 =head2 Utility subroutines
 4816 
 4817 =head4
 4818 
 4819   warn pretty_print( $rh_hash_input)
 4820 
 4821 This can be very useful for printing out messages about objects while debugging
 4822 
 4823 =cut
 4824 
 4825 sub pretty_print {
 4826     my $r_input = shift;
 4827     my $out = '';
 4828     if ( not ref($r_input) ) {
 4829       $out = $r_input;    # not a reference
 4830     } elsif ("$r_input" =~/hash/i) {  # this will pick up objects whose '$self' is hash and so works better than ref($r_iput).
 4831       local($^W) = 0;
 4832     $out .= "$r_input " ."<TABLE border = \"2\" cellpadding = \"3\" BGCOLOR = \"#FFFFFF\">";
 4833     foreach my $key (lex_sort( keys %$r_input )) {
 4834       $out .= "<tr><TD> $key</TD><TD>=&gt;</td><td>&nbsp;".pretty_print($r_input->{$key}) . "</td></tr>";
 4835     }
 4836     $out .="</table>";
 4837   } elsif (ref($r_input) eq 'ARRAY' ) {
 4838     my @array = @$r_input;
 4839     $out .= "( " ;
 4840     while (@array) {
 4841       $out .= pretty_print(shift @array) . " , ";
 4842     }
 4843     $out .= " )";
 4844   } elsif (ref($r_input) eq 'CODE') {
 4845     $out = "$r_input";
 4846   } else {
 4847     $out = $r_input;
 4848   }
 4849     $out;
 4850 }
 4851 
 4852 1;

aubreyja at gmail dot com
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