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Revision 1050 - (download) (as text) (annotate)
Fri Jun 6 21:39:42 2003 UTC (16 years, 7 months ago) by sh002i
File size: 146178 byte(s)
moved PG modules and macro files from webwork-modperl to pg
-sam

    1 #!/usr/local/bin/webwork-perl
    2 
    3 # This file is PGanswermacros.pl
    4 # This includes the subroutines for the ANS macros, that
    5 # is, macros allowing a more flexible answer checking
    6 ####################################################################
    7 # Copyright @ 1995-2000 University of Rochester
    8 # All Rights Reserved
    9 ####################################################################
   10 #$Id$
   11 
   12 =head1 NAME
   13 
   14   PGanswermacros.pl -- located in the courseScripts directory
   15 
   16 =head1 SYNPOSIS
   17 
   18   Number Answer Evaluators:
   19     num_cmp() --  uses an input hash to determine parameters
   20 
   21     std_num_cmp(), std_num_cmp_list(), std_num_cmp_abs, std_num_cmp_abs_list()
   22     frac_num_cmp(), frac_num_cmp_list(), frac_num_cmp_abs, frac_num_cmp_abs_list()
   23     arith_num_cmp(), arith_num_cmp_list(), arith_num_cmp_abs, arith_num_cmp_abs_list()
   24     strict_num_cmp(), strict_num_cmp_list(), strict_num_cmp_abs, strict_num_cmp_abs_list()
   25     numerical_compare_with_units()  --  requires units as part of the answer
   26     std_num_str_cmp() --  also accepts a set of strings as possible answers
   27 
   28   Function Answer Evaluators:
   29     fun_cmp() --  uses an input hash to determine parameters
   30 
   31     function_cmp(), function_cmp_abs()
   32     function_cmp_up_to_constant(), function_cmp_up_to_constant_abs()
   33     multivar_function_cmp()
   34 
   35   String Answer Evaluators:
   36     str_cmp() --  uses an input hash to determine parameters
   37 
   38     std_str_cmp(), std_str_cmp_list(), std_cs_str_cmp(), std_cs_str_cmp_list()
   39     strict_str_cmp(), strict_str_cmp_list()
   40     ordered_str_cmp(), ordered_str_cmp_list(), ordered_cs_str_cmp(), ordered_cs_str_cmp_list()
   41     unordered_str_cmp(), unordered_str_cmp_list(), unordered_cs_str_cmp(), unordered_cs_str_cmp_list()
   42 
   43   Miscellaneous Answer Evaluators:
   44     checkbox_cmp()
   45     radio_cmp()
   46 
   47 =cut
   48 
   49 =head1 DESCRIPTION
   50 
   51 This file adds subroutines which create "answer evaluators" for checking
   52 answers. Each answer evaluator accepts a single input from a student answer,
   53 checks it and creates an output hash %ans_hash with seven or eight entries
   54 (the preview_latex_string is optional). The output hash is now being created
   55 with the AnswerHash package "class", which is located at the end of this file.
   56 This class is currently just a wrapper for the hash, but this might change in
   57 the future as new capabilities are added.
   58 
   59           score     =>  $correctQ,
   60           correct_ans   =>  $originalCorrEqn,
   61           student_ans   =>  $modified_student_ans
   62           original_student_ans  =>  $original_student_answer,
   63           ans_message   =>  $PGanswerMessage,
   64           type      =>  'typeString',
   65           preview_text_string =>  $preview_text_string,
   66           preview_latex_string  =>  $preview_latex_string
   67 
   68 
   69   $ans_hash{score}      --  a number between 0 and 1 indicating
   70                     whether the answer is correct. Fractions
   71                     allow the implementation of partial
   72                     credit for incorrect answers.
   73   $ans_hash{correct_ans}      --  The correct answer, as supplied by the
   74                     instructor and then formatted. This can
   75                     be viewed by the student after the answer date.
   76   $ans_hash{student_ans}      --  This is the student answer, after reformatting;
   77                     for example the answer might be forced
   78                     to capital letters for comparison with
   79                     the instructors answer. For a numerical
   80                     answer, it gives the evaluated answer.
   81                     This is displayed in the section reporting
   82                     the results of checking the student answers.
   83   $ans_hash{original_student_ans}   --  This is the original student answer. This is displayed
   84                     on the preview page and may be used for sticky answers.
   85   $ans_hash{ans_message}      --  Any error message, or hint provided by the answer evaluator.
   86                     This is also displayed in the section reporting
   87                     the results of checking the student answers.
   88   $ans_hash{type}       --  A string indicating the type of answer evaluator. This
   89                     helps in preprocessing the student answer for errors.
   90                     Some examples:
   91                       'number_with_units'
   92                       'function'
   93                       'frac_number'
   94                       'arith_number'
   95   $ans_hash{preview_text_string}    --  This typically shows how the student answer was parsed. It is
   96                     displayed on the preview page. For a student answer of 2sin(3x)
   97                     this would be 2*sin(3*x). For string answers it is typically the
   98                     same as $ans_hash{student_ans}.
   99   $ans_hash{preview_latex_string}   --  THIS IS OPTIONAL. This is latex version of the student answer
  100                     which is used to show a typeset view on the answer on the preview
  101                     page. For a student answer of 2/3, this would be \frac{2}{3}.
  102 
  103 Technical note: the routines in this file are not actually answer evaluators. Instead, they create
  104 answer evaluators. An answer evaluator is an anonymous subroutine, referenced by a named scalar. The
  105 routines in this file build the subroutine and return a reference to it. Later, when the student
  106 actually enters an answer, the problem processor feeds that answer to the referenced subroutine, which
  107 evaluates it and returns a score (usually 0 or 1). For most users, this distinction is unimportant, but
  108 if you plan on writing your own answer evaluators, you should understand this point.
  109 
  110 =cut
  111 
  112 BEGIN {
  113   be_strict(); # an alias for use strict.  This means that all global variable must contain main:: as a prefix.
  114 }
  115 
  116 
  117 my ($BR           ,   # convenient localizations.
  118   $PAR          ,
  119   $numRelPercentTolDefault    ,
  120   $numZeroLevelDefault      ,
  121   $numZeroLevelTolDefault     ,
  122   $numAbsTolDefault     ,
  123   $numFormatDefault     ,
  124   $functRelPercentTolDefault    ,
  125   $functZeroLevelDefault      ,
  126   $functZeroLevelTolDefault   ,
  127   $functAbsTolDefault     ,
  128   $functNumOfPoints     ,
  129   $functVarDefault      ,
  130   $functLLimitDefault     ,
  131   $functULimitDefault     ,
  132   $functMaxConstantOfIntegration    ,
  133   $CA
  134 );
  135 
  136 
  137 
  138 
  139 sub _PGanswermacros_init {
  140 
  141      $BR  = $main::BR;    # convenient localizations.
  142      $PAR = $main::PAR;
  143 
  144     # import defaults
  145     # these are now imported from the %envir variable
  146      $numRelPercentTolDefault     = $main::numRelPercentTolDefault;
  147      $numZeroLevelDefault       = $main::numZeroLevelDefault;
  148      $numZeroLevelTolDefault      = $main::numZeroLevelTolDefault;
  149      $numAbsTolDefault        = $main::numAbsTolDefault;
  150      $numFormatDefault        = $main::numFormatDefault;
  151      $functRelPercentTolDefault     = $main::functRelPercentTolDefault;
  152      $functZeroLevelDefault       = $main::functZeroLevelDefault;
  153      $functZeroLevelTolDefault      = $main::functZeroLevelTolDefault;
  154      $functAbsTolDefault        = $main::functAbsTolDefault;
  155      $functNumOfPoints        = $main::functNumOfPoints;
  156      $functVarDefault       = $main::functVarDefault;
  157      $functLLimitDefault        = $main::functLLimitDefault;
  158      $functULimitDefault        = $main::functULimitDefault;
  159      $functMaxConstantOfIntegration     = $main::functMaxConstantOfIntegration;
  160 
  161 
  162 
  163 }
  164 
  165 ##########################################################################
  166 ##########################################################################
  167 ## Number answer evaluators
  168 
  169 =head2 Number Answer Evaluators
  170 
  171 Number answer evaluators take in a numerical answer, compare it to the correct answer,
  172 and return a score. In addition, they can choose to accept or reject an answer based on
  173 its format, closeness to the correct answer, and other criteria. There are two types
  174 of numerical answer evaluators: num_cmp(), which takes a hash of named options as parameters,
  175 and the "mode"_num_cmp() variety, which use different functions to access different sets of
  176 options. In addition, there is the special case of std_num_str_cmp(), which can evaluate
  177 both numbers and strings.
  178 
  179 Numerical Comparison Options
  180 
  181   correctAnswer   --  This is the correct answer that the student answer will
  182             be compared to. However, this does not mean that the
  183             student answer must match this exactly. How close the
  184             student answer must be is determined by the other
  185             options, especially tolerance and format.
  186 
  187   tolerance   --  These options determine how close the student answer
  188             must be to the correct answer to qualify. There are two
  189             types of tolerance: relative and absolute. Relative
  190             tolerances are given in percentages. A relative
  191             tolerance of 1 indicates that the student answer must
  192             be within 1% of the correct answer to qualify as correct.
  193             In other words, a student answer is correct when
  194               abs(studentAnswer - correctAnswer) <= abs(.01*relpercentTol*correctAnswer)
  195             Using absolute tolerance, the student answer must be a
  196             fixed distance from the correct answer to qualify.
  197             For example, an absolute tolerance of 5 means that any
  198             number which is +-5 of the correct answer qualifies as correct.
  199               Final (rarely used) tolerance options are zeroLevel
  200             and zeroLevelTol, used in conjunction with relative
  201             tolerance. if correctAnswer has absolute value less than
  202             or equal to zeroLevel, then the student answer must be,
  203             in absolute terms, within zeroLevelTol of correctAnswer, i.e.,
  204               abs(studentAnswer - correctAnswer) <= zeroLevelTol.
  205             In other words, if the correct answer is very near zero,
  206             an absolute tolerance will be used. One must do this to
  207             handle floating point answers very near zero, because of
  208             the inaccuracy of floating point arithmetic. However, the
  209             default values are almost always adequate.
  210 
  211   mode      --  This determines the allowable methods for entering an
  212             answer. Answers which do not meet this requirement will
  213             be graded as incorrect, regardless of their numerical
  214             value. The recognized modes are:
  215               'std' (default) --  allows any expression which evaluates
  216                         to a number, including those using
  217                         elementary functions like sin() and
  218                         exp(), as well as the operations of
  219                         arithmetic (+, -, *, /, ^)
  220               'strict'  --  only decimal numbers are allowed
  221               'frac'    --  whole numbers and fractions are allowed
  222               'arith'   --  arithmetic expressions are allowed, but
  223                         no functions
  224             Note that all modes allow the use of "pi" and "e" as
  225             constants, and also the use of "E" to represent scientific
  226             notation.
  227 
  228   format      --  The format to use when displaying the correct and
  229             submitted answers. This has no effect on how answers are
  230             evaluated; it is only for cosmetic purposes. The
  231             formatting syntax is the same as Perl uses for the sprintf()
  232             function. Format strings are of the form '%m.nx' or '%m.nx#',
  233             where m and n are described below, and x is a formatter.
  234               Esentially, m is the minimum length of the field
  235             (make this negative to left-justify). Note that the decimal
  236             point counts as a character when determining the field width.
  237             If m begins with a zero, the number will be padded with zeros
  238             instead of spaces to fit the field.
  239               The precision specifier (n) works differently, depending
  240             on which formatter you are using. For d, i, o, u, x and X
  241             formatters (non-floating point formatters), n is the minimum
  242             number of digits to display. For e and f, it is the number of
  243             digits that appear after the decimal point (extra digits will
  244             be rounded; insufficient digits will be padded with spaces--see
  245             '#' below). For g, it is the number of significant digits to
  246             display.
  247               The full list of formatters can be found in the manpage
  248             for printf(3), or by typing "perldoc -f sprintf" at a
  249             terminal prompt. The following is a brief summary of the
  250             most frequent formatters:
  251               d --  decimal number
  252               ld  --  long decimal number
  253               u --  unsigned decimal number
  254               lu  --  long unsigned decimal number
  255               x --  hexadecimal number
  256               o --  octal number
  257               e --  floating point number in scientific notation
  258               f --  floating point number
  259               g --  either e or f, whichever takes less space
  260             Technically, g will use e if the exponent is less than -4 or
  261             greater than or equal to the precision. Trailing zeros are
  262             removed in this mode.
  263               If the format string ends in '#', trailing zeros will be
  264             removed in the decimal part. Note that this is not a standard
  265             syntax; it is handled internally by WeBWorK and not by Perl
  266             (although this should not be a concern to end users).
  267             The default format is '%0.5f#', which displays as a floating
  268             point number with 5 digits of precision and no trailing zeros.
  269             Other useful format strings might be '%0.2f' for displaying
  270             dollar amounts, or '%010d' to display an integer with leading
  271             zeros. Setting format to an empty string ( '' ) means no
  272             formatting will be used; this will show 'arbitrary' precision
  273             floating points.
  274 
  275 Default Values (As of 7/24/2000) (Option -- Variable Name -- Value)
  276 
  277   Format          --  $numFormatDefault   --  "%0.5f#"
  278   Relative Tolerance    --  $numRelPercentTolDefault  --  .1
  279   Absolute Tolerance    --  $numAbsTolDefault   --  .001
  280   Zero Level        --  $numZeroLevelDefault    --  1E-14
  281   Zero Level Tolerance  --  $numZeroLevelTolDefault   --  1E-12
  282 
  283 =cut
  284 
  285 
  286 =head3 num_cmp()
  287 
  288 Compares a number or a list of numbers, using a named hash of options to set
  289 parameters. This can make for more readable code than using the "mode"_num_cmp()
  290 style, but some people find one or the other easier to remember.
  291 
  292 ANS( num_cmp( answer or answer_array_ref, options_hash ) );
  293 
  294   1. the correct answer, or a reference to an array of correct answers
  295   2. a hash with the following keys (all optional):
  296     mode      --  'std' (default) (allows any expression evaluating to
  297                 a number)
  298               'strict' (only numbers are allowed)
  299               'frac' (fractions are allowed)
  300               'arith' (arithmetic expressions allowed)
  301     format      --  '%0.5f#' (default); defines formatting for the
  302                 correct answer
  303     tol       --  an absolute tolerance, or
  304     relTol      --  a relative tolerance
  305     units     --  the units to use for the answer(s)
  306     strings     --  a reference to an array of strings which are valid
  307                 answers (works like std_num_str_cmp() )
  308     zeroLevel   --  if the correct answer is this close to zero,
  309                  then zeroLevelTol applies
  310     zeroLevelTol  --  absolute tolerance to allow when answer is close
  311                  to zero
  312 
  313     debug     --  if set to 1, provides verbose listing of
  314                 hash entries throughout fliters.
  315 
  316   Returns an answer evaluator, or (if given a reference to an array of
  317   answers), a list of answer evaluators. Note that a reference to an array of
  318   answers results is just a shortcut for writing a separate <code>num_cmp()</code> for each
  319   answer.
  320 
  321 EXAMPLES:
  322 
  323   num_cmp( 5 )          --  correct answer is 5, using defaults
  324                   for all options
  325   num_cmp( [5,6,7] )        --  correct answers are 5, 6, and 7,
  326                   using defaults for all options
  327   num_cmp( 5, mode => 'strict' )  --  correct answer is 5, mode is strict
  328   num_cmp( [5,6], relTol => 5 ) --  correct answers are 5 and 6,
  329                     both with 5% relative tolerance
  330   num_cmp( 6, strings => ["Inf", "Minf", "NaN"] )
  331                   --  correct answer is 6, "Inf", "Minf",
  332                    and "NaN" recognized as valid, but
  333                    incorrect answers.
  334   num_cmp( "-INF", strings => ["INF", "-INF"] )
  335                   --  correct answer is "-INF", "INF" and
  336                    numerical expressions recognized as valid,
  337                    but incorrect answers.
  338 
  339 
  340 =cut
  341 
  342 sub num_cmp {
  343   my $correctAnswer = shift @_;
  344   $CA = $correctAnswer;
  345   my @opt = @_;
  346   my %out_options;
  347 
  348 #########################################################################
  349 # Retain this first check for backword compatibility.  Allows input of the form
  350 # num_cmp($ans, 1, '%0.5f') but warns against it
  351 #########################################################################
  352   my %known_options = (
  353           'mode'      =>  'std',
  354           'format'    =>  $numFormatDefault,
  355           'tol'     =>  $numAbsTolDefault,
  356           'relTol'    =>  $numRelPercentTolDefault,
  357           'units'     =>  undef,
  358           'strings'   =>  undef,
  359           'zeroLevel'   =>  $numZeroLevelDefault,
  360           'zeroLevelTol'  =>  $numZeroLevelTolDefault,
  361           'tolType'       =>  'relative',
  362           'tolerance'     =>  1,
  363           'reltol'    =>  undef,      #alternate spelling
  364           'unit'      =>  undef,      #alternate spelling
  365           'debug'     =>  0
  366         );
  367 
  368   my @output_list;
  369   my( $relPercentTol, $format, $zeroLevel, $zeroLevelTol) = @opt;
  370 
  371   unless( ref($correctAnswer) eq 'ARRAY' || scalar( @opt ) == 0 ||
  372         ( defined($opt[0]) and exists $known_options{$opt[0]} ) ) {
  373     # unless the first parameter is a list of arrays
  374     # or the second parameter is a known option or
  375     # no options were used,
  376     # use the old num_cmp which does not use options, but has inputs
  377     # $relPercentTol,$format,$zeroLevel,$zeroLevelTol
  378     warn "This method of using num_cmp() is deprecated. Please rewrite this" .
  379           " problem using the options style of parameter passing (or" .
  380           " check that your first option is spelled correctly).";
  381 
  382     %out_options = (  'relTol'    => $relPercentTol,
  383           'format'    => $format,
  384           'zeroLevel'   => $zeroLevel,
  385           'zeroLevelTol'  => $zeroLevelTol,
  386           'mode'      => 'std'
  387     );
  388   }
  389 
  390 #########################################################################
  391 # Now handle the options assuming they are entered in the form
  392 # num_cmp($ans, relTol=>1, format=>'%0.5f')
  393 #########################################################################
  394   %out_options = @opt;
  395   assign_option_aliases( \%out_options,
  396         'reltol'    =>      'relTol',
  397         'unit'      =>      'units',
  398         'abstol'  =>    'tol',
  399         );
  400 
  401   set_default_options( \%out_options,
  402            'tolType'    =>  (defined($out_options{'tol'}) ) ? 'absolute' : 'relative',  # the existence of "tol" means that we use absolute tolerance mode
  403            'tolerance'    =>  (defined($out_options{'tolType'}) && $out_options{'tolType'} eq 'absolute' ) ? $numAbsTolDefault : $numRelPercentTolDefault,  # relative tolerance is the default
  404            'mode'       =>  'std',
  405            'format'   =>  $numFormatDefault,
  406            'tol'        =>  undef,
  407            'relTol'   =>  undef,
  408            'units'    =>  undef,
  409            'strings'    =>  undef,
  410            'zeroLevel'  =>  $numZeroLevelDefault,
  411            'zeroLevelTol' =>  $numZeroLevelTolDefault,
  412            'debug'    =>  0,
  413   );
  414 
  415   # can't use both units and strings
  416   if( defined( $out_options{'units'} ) && defined( $out_options{'strings'} ) ) {
  417     warn "Can't use both 'units' and 'strings' in the same problem " .
  418     "(check your parameters to num_cmp() )";
  419   }
  420 
  421   # absolute tolType and relTol are incompatible. So are relative tolType and tol
  422   if( defined( $out_options{'relTol'} ) &&  $out_options{'tolType'} eq 'absolute' )  {
  423     warn "The 'tolType' 'absolute' is not compatible with 'relTol' " .
  424     "(check your parameters to num_cmp() )";
  425   }
  426   if( defined( $out_options{'tol'} ) &&  $out_options{'tolType'} eq 'relative' )  {
  427     warn "The 'tolType' 'relative' is not compatible with 'tol' " .
  428     "(check your parameters to num_cmp() )";
  429   }
  430 
  431 
  432   # Handle legacy options
  433     if ($out_options{tolType} eq 'absolute')   {
  434     $out_options{'tolerance'}=$out_options{'tol'} if defined($out_options{'tol'});
  435     delete($out_options{'relTol'}) if exists( $out_options{'relTol'} );
  436   } else {
  437     $out_options{'tolerance'}=$out_options{'relTol'} if defined($out_options{'relTol'});
  438     # delete($out_options{'tol'}) if exists( $out_options{'tol'} );
  439   }
  440   # end legacy options
  441 
  442   # thread over lists
  443   my @ans_list = ();
  444 
  445   if ( ref($correctAnswer) eq 'ARRAY' ) {
  446     @ans_list = @{$correctAnswer};
  447   }
  448   else { push( @ans_list, $correctAnswer );
  449   }
  450 
  451   # produce answer evaluators
  452   foreach my $ans (@ans_list) {
  453     if( defined( $out_options{'units'} ) ) {
  454       $ans = "$ans $out_options{'units'}";
  455 
  456       push( @output_list, NUM_CMP(  'correctAnswer'       =>  $ans,
  457               'tolerance'   =>  $out_options{'tolerance'},
  458               'tolType'   =>  $out_options{'tolType'},
  459               'format'    =>  $out_options{'format'},
  460               'mode'      =>  $out_options{'mode'},
  461               'zeroLevel'   =>  $out_options{'zeroLevel'},
  462               'zeroLevelTol'  =>  $out_options{'zeroLevelTol'},
  463               'debug'     =>  $out_options{'debug'},
  464               'units'     =>  $out_options{'units'},
  465             )
  466       );
  467     } elsif( defined( $out_options{'strings'} ) ) {
  468 
  469 
  470       push( @output_list, NUM_CMP(  'correctAnswer' =>  $ans,
  471               'tolerance' =>  $out_options{tolerance},
  472               'tolType' =>  $out_options{tolType},
  473               'format'  =>  $out_options{'format'},
  474               'mode'    =>  $out_options{'mode'},
  475               'zeroLevel' =>  $out_options{'zeroLevel'},
  476               'zeroLevelTol'  =>  $out_options{'zeroLevelTol'},
  477               'debug'   =>  $out_options{'debug'},
  478               'strings' =>  $out_options{'strings'},
  479          )
  480          );
  481     } else {
  482       push(@output_list,
  483         NUM_CMP(  'correctAnswer'       =>  $ans,
  484           'tolerance'   =>  $out_options{tolerance},
  485           'tolType'   =>  $out_options{tolType},
  486           'format'    =>  $out_options{'format'},
  487           'mode'      =>  $out_options{'mode'},
  488           'zeroLevel'   =>  $out_options{'zeroLevel'},
  489           'zeroLevelTol'        =>  $out_options{'zeroLevelTol'},
  490           'debug'     =>  $out_options{'debug'},
  491         ),
  492       );
  493       }
  494   }
  495 
  496   return (wantarray) ? @output_list : $output_list[0];
  497 }
  498 
  499 #legacy code for compatability purposes
  500 sub num_rel_cmp {   # compare numbers
  501     std_num_cmp( @_ );
  502 }
  503 
  504 
  505 =head3 "mode"_num_cmp() functions
  506 
  507 There are 16 functions total, 4 for each mode (std, frac, strict, arith). Each mode has
  508 one "normal" function, one which accepts a list of answers, one which uses absolute
  509 rather than relative tolerance, and one which uses absolute tolerance and accepts a list.
  510 The "std" family is documented below; all others work precisely the same.
  511 
  512  std_num_cmp($correctAnswer) OR
  513  std_num_cmp($correctAnswer, $relPercentTol) OR
  514  std_num_cmp($correctAnswer, $relPercentTol, $format) OR
  515  std_num_cmp($correctAnswer, $relPercentTol, $format, $zeroLevel) OR
  516  std_num_cmp($correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol)
  517 
  518   $correctAnswer  --  the correct answer
  519   $relPercentTol  --  the tolerance, as a percentage (optional)
  520   $format   --  the format of the displayed answer (optional)
  521   $zeroLevel  --  if the correct answer is this close to zero, then zeroLevelTol applies (optional)
  522   $zeroLevelTol --  absolute tolerance to allow when correct answer is close to zero (optional)
  523 
  524   std_num_cmp() uses standard mode (arithmetic operations and elementary
  525   functions allowed) and relative tolerance. Options are specified by
  526   one or more parameters. Note that if you wish to set an option which
  527   is later in the parameter list, you must set all previous options.
  528 
  529  std_num_cmp_abs($correctAnswer) OR
  530  std_num_cmp_abs($correctAnswer, $absTol) OR
  531  std_num_cmp_abs($correctAnswer, $absTol, $format)
  532 
  533   $correctAnswer    --  the correct answer
  534   $absTol     --  an absolute tolerance (optional)
  535   $format     --  the format of the displayed answer (optional)
  536 
  537   std_num_cmp_abs() uses standard mode and absolute tolerance. Options
  538   are set as with std_num_cmp(). Note that $zeroLevel and $zeroLevelTol
  539   do not apply with absolute tolerance.
  540 
  541  std_num_cmp_list($relPercentTol, $format, @answerList)
  542 
  543   $relPercentTol    --  the tolerance, as a percentage
  544   $format     --  the format of the displayed answer(s)
  545   @answerList   --  a list of one or more correct answers
  546 
  547   std_num_cmp_list() uses standard mode and relative tolerance. There
  548   is no way to set $zeroLevel or $zeroLevelTol. Note that no
  549   parameters are optional. All answers in the list will be
  550   evaluated with the same set of parameters.
  551 
  552  std_num_cmp_abs_list($absTol, $format, @answerList)
  553 
  554   $absTol   --  an absolute tolerance
  555   $format   --  the format of the displayed answer(s)
  556   @answerList --  a list of one or more correct answers
  557 
  558   std_num_cmp_abs_list() uses standard mode and absolute tolerance.
  559   Note that no parameters are optional. All answers in the list will be
  560   evaluated with the same set of parameters.
  561 
  562  arith_num_cmp(), arith_num_cmp_list(), arith_num_cmp_abs(), arith_num_cmp_abs_list()
  563  strict_num_cmp(), strict_num_cmp_list(), strict_num_cmp_abs(), strict_num_cmp_abs_list()
  564  frac_num_cmp(), frac_num_cmp_list(), frac_num_cmp_abs(), frac_num_cmp_abs_list()
  565 
  566 Examples:
  567 
  568   ANS( strict_num_cmp( 3.14159 ) )  --  The student answer must be a number
  569     in decimal or scientific notation which is within .1 percent of 3.14159.
  570     This assumes $numRelPercentTolDefault has been set to .1.
  571   ANS( strict_num_cmp( $answer, .01 ) ) --  The student answer must be a
  572     number within .01 percent of $answer (e.g. 3.14159 if $answer is 3.14159
  573     or $answer is "pi" or $answer is 4*atan(1)).
  574   ANS( frac_num_cmp( $answer) ) or ANS( frac_num_cmp( $answer,.01 ))  --
  575     The student answer can be a number or fraction, e.g. 2/3.
  576   ANS( arith_num_cmp( $answer) ) or ANS( arith_num_cmp( $answer,.01 ))  --
  577     The student answer can be an arithmetic expression, e.g. (2+3)/7-2^.5 .
  578   ANS( std_num_cmp( $answer) ) or ANS( std_num_cmp( $answer,.01 ))  --
  579     The student answer can contain elementary functions, e.g. sin(.3+pi/2)
  580 
  581 =cut
  582 
  583 sub std_num_cmp {           # compare numbers allowing use of elementary functions
  584     my ( $correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
  585 
  586   my %options = ( 'relTol'    =>  $relPercentTol,
  587         'format'    =>  $format,
  588         'zeroLevel'   =>  $zeroLevel,
  589         'zeroLevelTol'  =>  $zeroLevelTol
  590     );
  591 
  592     set_default_options( \%options,
  593        'tolType'  =>      'relative',
  594        'tolerance'    =>      $numRelPercentTolDefault,
  595        'mode'   =>  'std',
  596        'format' =>  $numFormatDefault,
  597        'relTol' =>  $numRelPercentTolDefault,
  598        'zeroLevel'    =>      $numZeroLevelDefault,
  599        'zeroLevelTol' =>      $numZeroLevelTolDefault,
  600        'debug'        =>      0,
  601     );
  602 
  603     num_cmp([$correctAnswer], %options);
  604 }
  605 
  606 ##  Similar to std_num_cmp but accepts a list of numbers in the form
  607 ##  std_num_cmp_list(relpercentTol,format,ans1,ans2,ans3,...)
  608 ##  format is of the form "%10.3g" or "", i.e., a format suitable for sprintf(). Use "" for default
  609 ##  You must enter a format and tolerance
  610 
  611 sub std_num_cmp_list {
  612   my ( $relPercentTol, $format, @answerList) = @_;
  613 
  614   my %options = ( 'relTol'  =>      $relPercentTol,
  615       'format'        =>      $format,
  616   );
  617 
  618   set_default_options( \%options,
  619            'tolType'      =>      'relative',
  620            'tolerance'    =>      $numRelPercentTolDefault,
  621            'mode'         =>      'std',
  622            'format'       =>      $numFormatDefault,
  623            'relTol'       =>      $numRelPercentTolDefault,
  624            'zeroLevel'    =>      $numZeroLevelDefault,
  625            'zeroLevelTol' =>      $numZeroLevelTolDefault,
  626            'debug'        =>      0,
  627   );
  628 
  629   num_cmp(\@answerList, %options);
  630 
  631 }
  632 
  633 sub std_num_cmp_abs {     # compare numbers allowing use of elementary functions with absolute tolerance
  634   my ( $correctAnswer, $absTol, $format) = @_;
  635   my %options = ( 'tolerance'  => $absTol,
  636             'format'     => $format
  637   );
  638 
  639   set_default_options (\%options,
  640            'tolType'      =>      'absolute',
  641            'tolerance'    =>      $absTol,
  642            'mode'         =>      'std',
  643            'format'       =>      $numFormatDefault,
  644            'zeroLevel'    =>      0,
  645            'zeroLevelTol' =>      0,
  646            'debug'        =>      0,
  647   );
  648 
  649   num_cmp([$correctAnswer], %options);
  650 }
  651 
  652 ##  See std_num_cmp_list for usage
  653 
  654 sub std_num_cmp_abs_list {
  655   my ( $absTol, $format, @answerList ) = @_;
  656 
  657         my %options = ( 'tolerance'         =>      $absTol,
  658                         'format'            =>      $format,
  659   );
  660 
  661         set_default_options( \%options,
  662                              'tolType'      =>      'absolute',
  663                              'tolerance'    =>      $absTol,
  664                              'mode'         =>      'std',
  665                              'format'       =>      $numFormatDefault,
  666                              'zeroLevel'    =>      0,
  667                              'zeroLevelTol' =>      0,
  668                              'debug'        =>      0,
  669         );
  670 
  671         num_cmp(\@answerList, %options);
  672 }
  673 
  674 sub frac_num_cmp {            # only allow fractions and numbers as submitted answer
  675 
  676   my ( $correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
  677 
  678   my %options = ( 'relTol'   =>   $relPercentTol,
  679           'format'     =>   $format,
  680           'zeroLevel'  =>   $zeroLevel,
  681           'zeroLevelTol'   =>   $zeroLevelTol
  682   );
  683 
  684   set_default_options( \%options,
  685          'tolType'     =>   'relative',
  686          'tolerance'   =>   $relPercentTol,
  687          'mode'      =>   'frac',
  688          'format'    =>   $numFormatDefault,
  689          'zeroLevel'   =>   $numZeroLevelDefault,
  690          'zeroLevelTol'  =>   $numZeroLevelTolDefault,
  691          'relTol'    =>   $numRelPercentTolDefault,
  692          'debug'     =>   0,
  693    );
  694 
  695   num_cmp([$correctAnswer], %options);
  696 }
  697 
  698 ##  See std_num_cmp_list for usage
  699 sub frac_num_cmp_list {
  700   my ( $relPercentTol, $format, @answerList ) = @_;
  701 
  702   my %options = (      'relTol'  =>   $relPercentTol,
  703                'format'    =>   $format
  704   );
  705 
  706   set_default_options( \%options,
  707        'tolType'     =>   'relative',
  708        'tolerance'   =>   $relPercentTol,
  709        'mode'      =>   'frac',
  710        'format'    =>   $numFormatDefault,
  711        'zeroLevel'   =>   $numZeroLevelDefault,
  712        'zeroLevelTol'  =>   $numZeroLevelTolDefault,
  713        'relTol'    =>   $numRelPercentTolDefault,
  714        'debug'     =>   0,
  715   );
  716 
  717   num_cmp(\@answerList, %options);
  718 }
  719 
  720 sub frac_num_cmp_abs {      # only allow fraction expressions as submitted answer with absolute tolerance
  721     my ( $correctAnswer, $absTol, $format ) = @_;
  722 
  723     my %options = (             'tolerance'    =>     $absTol,
  724               'format'       =>     $format
  725     );
  726 
  727   set_default_options (\%options,
  728       'tolType'    =>   'absolute',
  729       'tolerance'    =>   $absTol,
  730       'mode'       =>   'frac',
  731       'format'     =>   $numFormatDefault,
  732       'zeroLevel'    =>   0,
  733       'zeroLevelTol' =>   0,
  734       'debug'      =>   0,
  735   );
  736 
  737     num_cmp([$correctAnswer], %options);
  738 }
  739 
  740 ##  See std_num_cmp_list for usage
  741 
  742 sub frac_num_cmp_abs_list {
  743     my ( $absTol, $format, @answerList ) = @_;
  744 
  745     my %options = (             'tolerance'    =>     $absTol,
  746               'format'       =>     $format
  747     );
  748 
  749     set_default_options (\%options,
  750        'tolType'      =>     'absolute',
  751        'tolerance'    =>     $absTol,
  752        'mode'         =>     'frac',
  753        'format'       =>     $numFormatDefault,
  754        'zeroLevel'    =>     0,
  755        'zeroLevelTol' =>     0,
  756        'debug'        =>     0,
  757     );
  758 
  759     num_cmp(\@answerList, %options);
  760 }
  761 
  762 
  763 sub arith_num_cmp {           # only allow arithmetic expressions as submitted answer
  764 
  765     my ( $correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
  766 
  767     my %options = (     'relTol'      =>     $relPercentTol,
  768       'format'         =>     $format,
  769       'zeroLevel'      =>     $zeroLevel,
  770       'zeroLevelTol'   =>     $zeroLevelTol
  771     );
  772 
  773     set_default_options( \%options,
  774                         'tolType'       =>     'relative',
  775                         'tolerance'     =>     $relPercentTol,
  776                         'mode'          =>     'arith',
  777                         'format'        =>     $numFormatDefault,
  778                         'zeroLevel'     =>     $numZeroLevelDefault,
  779                         'zeroLevelTol'  =>     $numZeroLevelTolDefault,
  780                         'relTol'        =>     $numRelPercentTolDefault,
  781                         'debug'         =>     0,
  782     );
  783 
  784     num_cmp([$correctAnswer], %options);
  785 }
  786 
  787 ##  See std_num_cmp_list for usage
  788 sub arith_num_cmp_list {
  789     my ( $relPercentTol, $format, @answerList ) = @_;
  790 
  791     my %options = (     'relTol'     =>     $relPercentTol,
  792                         'format'        =>     $format,
  793     );
  794 
  795     set_default_options( \%options,
  796                          'tolType'       =>     'relative',
  797                          'tolerance'     =>     $relPercentTol,
  798                          'mode'          =>     'arith',
  799                          'format'        =>     $numFormatDefault,
  800                          'zeroLevel'     =>     $numZeroLevelDefault,
  801                          'zeroLevelTol'  =>     $numZeroLevelTolDefault,
  802                          'relTol'        =>     $numRelPercentTolDefault,
  803                          'debug'         =>     0,
  804     );
  805 
  806     num_cmp(\@answerList, %options);
  807 }
  808 
  809 sub arith_num_cmp_abs {     # only allow arithmetic expressions as submitted answer with absolute tolerance
  810     my ( $correctAnswer, $absTol, $format ) = @_;
  811 
  812     my %options = (      'tolerance'    =>     $absTol,
  813                          'format'       =>     $format
  814     );
  815 
  816     set_default_options (\%options,
  817                          'tolType'      =>     'absolute',
  818                          'tolerance'    =>     $absTol,
  819                          'mode'         =>     'arith',
  820                          'format'       =>     $numFormatDefault,
  821                          'zeroLevel'    =>     0,
  822                          'zeroLevelTol' =>     0,
  823                          'debug'        =>     0,
  824     );
  825 
  826     num_cmp([$correctAnswer], %options);
  827 }
  828 
  829 ##  See std_num_cmp_list for usage
  830 sub arith_num_cmp_abs_list {
  831     my ( $absTol, $format, @answerList ) = @_;
  832 
  833     my %options = (      'tolerance'    =>     $absTol,
  834                          'format'       =>     $format
  835     );
  836 
  837     set_default_options (\%options,
  838                          'tolType'      =>     'absolute',
  839                          'tolerance'    =>     $absTol,
  840                          'mode'         =>     'arith',
  841                          'format'       =>     $numFormatDefault,
  842                          'zeroLevel'    =>     0,
  843                          'zeroLevelTol' =>     0,
  844                          'debug'        =>     0,
  845     );
  846 
  847     num_cmp(\@answerList, %options);
  848 }
  849 
  850 sub strict_num_cmp {          # only allow numbers as submitted answer
  851     my ( $correctAnswer, $relPercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
  852 
  853     my %options = (      'relTol'     =>     $relPercentTol,
  854                          'format'        =>     $format,
  855                          'zeroLevel'     =>     $zeroLevel,
  856                          'zeroLevelTol'  =>     $zeroLevelTol
  857     );
  858 
  859     set_default_options( \%options,
  860                          'tolType'       =>     'relative',
  861                          'tolerance'     =>     $relPercentTol,
  862                          'mode'          =>     'strict',
  863                          'format'        =>     $numFormatDefault,
  864                          'zeroLevel'     =>     $numZeroLevelDefault,
  865                          'zeroLevelTol'  =>     $numZeroLevelTolDefault,
  866                          'relTol'        =>     $numRelPercentTolDefault,
  867                          'debug'         =>     0,
  868     );
  869     num_cmp([$correctAnswer], %options);
  870 
  871 }
  872 
  873 ##  See std_num_cmp_list for usage
  874 sub strict_num_cmp_list {       # compare numbers
  875     my ( $relPercentTol, $format, @answerList ) = @_;
  876 
  877     my %options = (    'relTol'     =>     $relPercentTol,
  878        'format'        =>     $format,
  879     );
  880 
  881     set_default_options( \%options,
  882                          'tolType'       =>     'relative',
  883                          'tolerance'     =>     $relPercentTol,
  884                          'mode'          =>     'strict',
  885                          'format'        =>     $numFormatDefault,
  886                          'zeroLevel'     =>     $numZeroLevelDefault,
  887                          'zeroLevelTol'  =>     $numZeroLevelTolDefault,
  888                          'relTol'        =>     $numRelPercentTolDefault,
  889                          'debug'         =>     0,
  890     );
  891 
  892     num_cmp(\@answerList, %options);
  893 }
  894 
  895 
  896 sub strict_num_cmp_abs {        # only allow numbers as submitted answer with absolute tolerance
  897     my ( $correctAnswer, $absTol, $format ) = @_;
  898 
  899     my %options = (       'tolerance'    =>     $absTol,
  900                     'format'       =>     $format
  901     );
  902 
  903     set_default_options (\%options,
  904                          'tolType'      =>     'absolute',
  905                          'tolerance'    =>     $absTol,
  906                          'mode'         =>     'strict',
  907                          'format'       =>     $numFormatDefault,
  908                          'zeroLevel'    =>     0,
  909                          'zeroLevelTol' =>     0,
  910                          'debug'        =>     0,
  911     );
  912     num_cmp([$correctAnswer], %options);
  913 
  914 }
  915 
  916 ##  See std_num_cmp_list for usage
  917 sub strict_num_cmp_abs_list {     # compare numbers
  918     my ( $absTol, $format, @answerList ) = @_;
  919 
  920     my %options = (      'tolerance'    =>     $absTol,
  921                          'format'       =>     $format
  922     );
  923 
  924     set_default_options (\%options,
  925                          'tolType'      =>     'absolute',
  926                          'tolerance'    =>     $absTol,
  927                          'mode'         =>     'strict',
  928                          'format'       =>     $numFormatDefault,
  929                          'zeroLevel'    =>     0,
  930                          'zeroLevelTol' =>     0,
  931                          'debug'        =>     0,
  932     );
  933 
  934     num_cmp(\@answerList, %options);
  935 }
  936 
  937 ## sub numerical_compare_with_units
  938 ## Compares a number with units
  939 ## Deprecated; use num_cmp()
  940 ##
  941 ## IN:  a string which includes the numerical answer and the units
  942 ##    a hash with the following keys (all optional):
  943 ##      mode    --  'std', 'frac', 'arith', or 'strict'
  944 ##      format    --  the format to use when displaying the answer
  945 ##      tol   --  an absolute tolerance, or
  946 ##      relTol    --  a relative tolerance
  947 ##      zeroLevel --  if the correct answer is this close to zero, then zeroLevelTol applies
  948 ##      zeroLevelTol  --  absolute tolerance to allow when correct answer is close to zero
  949 
  950 # This mode is depricated.  send input through num_cmp -- it can handle units.
  951 
  952 sub numerical_compare_with_units {
  953   my $correct_answer = shift;  # the answer is a string which includes both the numerical answer and the units.
  954   my %options = @_;    # all of the other inputs are (key value) pairs
  955 
  956   # Prepare the correct answer
  957   $correct_answer = str_filters( $correct_answer, 'trim_whitespace' );
  958 
  959   # it surprises me that the match below works since the first .* is greedy.
  960   my ($correct_num_answer, $correct_units) = $correct_answer =~ /^(.*)\s+([^\s]*)$/;
  961   $options{units} = $correct_units;
  962 
  963   num_cmp($correct_num_answer, %options);
  964 }
  965 
  966 
  967 =head3 std_num_str_cmp()
  968 
  969 NOTE: This function is maintained for compatibility. num_cmp() with the
  970     'strings' parameter is slightly preferred.
  971 
  972 std_num_str_cmp() is used when the correct answer could be either a number or a
  973 string. For example, if you wanted the student to evaluate a function at number
  974 of points, but write "Inf" or "Minf" if the function is unbounded. This routine
  975 will provide error messages that do not give a hint as to whether the correct
  976 answer is a string or a number. For numerical comparisons, std_num_cmp() is
  977 used internally; for string comparisons, std_str_cmp() is used.  String answers
  978 must consist entirely of letters except that an initial minus sign is allowed.
  979 E.g. "inf" and "-inf" are valid strings where as "too-big" is not.
  980 
  981  std_num_str_cmp( $correctAnswer ) OR
  982  std_num_str_cmp( $correctAnswer, $ra_legalStrings ) OR
  983  std_num_str_cmp( $correctAnswer, $ra_legalStrings, $relPercentTol ) OR
  984  std_num_str_cmp( $correctAnswer, $ra_legalStrings, $relPercentTol, $format ) OR
  985  std_num_str_cmp( $correctAnswer, $ra_legalStrings, $relPercentTol, $format, $zeroLevel ) OR
  986  std_num_str_cmp( $correctAnswer, $ra_legalStrings, $relPercentTol, $format,
  987           $zeroLevel, $zeroLevelTol )
  988 
  989   $correctAnswer    --  the correct answer
  990   $ra_legalStrings  --  a reference to an array of legal strings, e.g. ["str1", "str2"]
  991   $relPercentTol    --  the error tolerance as a percentage
  992   $format     --  the display format
  993   $zeroLevel    --  if the correct answer is this close to zero, then zeroLevelTol applies
  994   $zeroLevelTol   --  absolute tolerance to allow when correct answer is close to zero
  995 
  996 Examples:
  997   ANS( std_num_str_cmp( $ans, ["Inf", "Minf", "NaN"] ) );
  998   ANS( std_num_str_cmp( $ans, ["INF", "-INF"] ) );
  999 
 1000 =cut
 1001 
 1002 sub std_num_str_cmp {
 1003   my ( $correctAnswer, $ra_legalStrings, $relpercentTol, $format, $zeroLevel, $zeroLevelTol ) = @_;
 1004   # warn ('This method is depreciated.  Use num_cmp instead.');
 1005   return num_cmp ($correctAnswer, strings=>$ra_legalStrings, relTol=>$relpercentTol, format=>$format,
 1006     zeroLevel=>$zeroLevel, zeroLevelTol=>$zeroLevelTol);
 1007 }
 1008 
 1009 sub NUM_CMP {   # low level numeric compare
 1010   my %num_params = @_;
 1011 
 1012   my @keys = qw ( correctAnswer tolerance tolType format mode zeroLevel zeroLevelTol debug );
 1013   foreach my $key (@keys) {
 1014       warn "$key must be defined in options when calling NUM_CMP" unless defined ($num_params{$key});
 1015   }
 1016 
 1017   my $correctAnswer = $num_params{'correctAnswer'};
 1018   my $format    = $num_params{'format'};
 1019   my $mode    = $num_params{'mode'};
 1020 
 1021   if( $num_params{tolType} eq 'relative' ) {
 1022     $num_params{'tolerance'} = .01*$num_params{'tolerance'};
 1023   }
 1024 
 1025   my $formattedCorrectAnswer;
 1026   my $correct_units;
 1027   my $correct_num_answer;
 1028   my %correct_units;
 1029   my $corrAnswerIsString = 0;
 1030 
 1031 
 1032   if (defined($num_params{units}) && $num_params{units}) {
 1033     $correctAnswer  = str_filters( $correctAnswer, 'trim_whitespace' );
 1034             # units are in form stuff space units where units contains no spaces.
 1035 
 1036     ($correct_num_answer, $correct_units) = $correctAnswer =~ /^(.*)\s+([^\s]*)$/;
 1037     %correct_units = Units::evaluate_units($correct_units);
 1038     if ( defined( $correct_units{'ERROR'} ) ) {
 1039        warn ("ERROR: The answer \"$correctAnswer\" in the problem definition cannot be parsed:\n" .
 1040         "$correct_units{'ERROR'}\n");
 1041     }
 1042     # $formattedCorrectAnswer = spf($correct_num_answer,$num_params{'format'}) . " $correct_units";
 1043     $formattedCorrectAnswer = prfmt($correct_num_answer,$num_params{'format'}) . " $correct_units";
 1044 
 1045   } elsif (defined($num_params{strings}) && $num_params{strings}) {
 1046     my $legalString = '';
 1047     my @legalStrings = @{$num_params{strings}};
 1048     $correct_num_answer = $correctAnswer;
 1049     $formattedCorrectAnswer = $correctAnswer;
 1050     foreach $legalString (@legalStrings) {
 1051       if ( uc($correctAnswer) eq uc($legalString) ) {
 1052         $corrAnswerIsString = 1;
 1053 
 1054         last;
 1055       }
 1056     }     ## at this point $corrAnswerIsString = 0 iff correct answer is numeric
 1057   } else {
 1058     $correct_num_answer = $correctAnswer;
 1059     $formattedCorrectAnswer = prfmt( $correctAnswer, $num_params{'format'} );
 1060   }
 1061 
 1062   $correct_num_answer = math_constants($correct_num_answer);
 1063 
 1064   my $PGanswerMessage = '';
 1065 
 1066   my ($inVal,$correctVal,$PG_eval_errors,$PG_full_error_report);
 1067 
 1068   if (defined($correct_num_answer) && $correct_num_answer =~ /\S/ && $corrAnswerIsString == 0 ) {
 1069       ($correctVal, $PG_eval_errors,$PG_full_error_report) = PG_answer_eval($correct_num_answer);
 1070   } else { # case of a string answer
 1071     $PG_eval_errors = ' ';
 1072     $correctVal = $correctAnswer;
 1073   }
 1074 
 1075   if ( ($PG_eval_errors && $corrAnswerIsString == 0) or ((not is_a_number($correctVal)) && $corrAnswerIsString == 0)) {
 1076         ##error message from eval or above
 1077     warn "Error in 'correct' answer: $PG_eval_errors<br>
 1078           The answer $correctAnswer evaluates to $correctVal,
 1079           which cannot be interpreted as a number.  ";
 1080 
 1081   }
 1082   #########################################################################
 1083 
 1084   #construct the answer evaluator
 1085       my $answer_evaluator = new AnswerEvaluator;
 1086       $answer_evaluator->{debug} = $num_params{debug};
 1087       $answer_evaluator->ans_hash(
 1088                 correct_ans       =>  $correctVal,
 1089                 type          =>  "${mode}_number",
 1090                 tolerance       =>  $num_params{tolerance},
 1091               tolType         =>  $num_params{tolType},
 1092               units         =>  $correct_units,
 1093                 original_correct_ans  =>  $formattedCorrectAnswer,
 1094                 rh_correct_units    =>      \%correct_units,
 1095                 answerIsString      =>  $corrAnswerIsString,
 1096       );
 1097       my ($in, $formattedSubmittedAnswer);
 1098   $answer_evaluator->install_pre_filter(sub {my $rh_ans = shift;
 1099     $rh_ans->{original_student_ans} = $rh_ans->{student_ans}; $rh_ans;}
 1100   );
 1101   if (defined($num_params{units}) && $num_params{units}) {
 1102       $answer_evaluator->install_pre_filter(\&check_units);
 1103   }
 1104   if (defined($num_params{strings}) && $num_params{strings}) {
 1105       $answer_evaluator->install_pre_filter(\&check_strings, %num_params);
 1106   }
 1107 
 1108   $answer_evaluator->install_pre_filter(\&check_syntax);
 1109 
 1110   $answer_evaluator->install_pre_filter(\&math_constants);
 1111 
 1112   if ($mode eq 'std') {
 1113         # do nothing
 1114   } elsif ($mode eq 'strict') {
 1115     $answer_evaluator->install_pre_filter(\&is_a_number);
 1116   } elsif ($mode eq 'arith') {
 1117       $answer_evaluator->install_pre_filter(\&is_an_arithmetic_expression);
 1118     } elsif ($mode eq 'frac') {
 1119       $answer_evaluator->install_pre_filter(\&is_a_fraction);
 1120 
 1121     } elsif ($mode eq 'phase_pi') {
 1122       $answer_evaluator->install_pre_filter(\&phase_pi);
 1123 
 1124     } else {
 1125       $PGanswerMessage = 'Tell your professor that there is an error in his or her answer mechanism. No mode was specified.';
 1126       $formattedSubmittedAnswer = $in;
 1127     }
 1128 
 1129   if ($corrAnswerIsString == 0 ){   # avoiding running compare_numbers when correct answer is a string.
 1130     $answer_evaluator->install_evaluator(\&compare_numbers, %num_params);
 1131    }
 1132 
 1133 
 1134 ###############################################################################
 1135 # We'll leave these next lines out for now, so that the evaluated versions of the student's and professor's
 1136 # can be displayed in the answer message.  This may still cause a few anomolies when strings are used
 1137 #
 1138 ###############################################################################
 1139 
 1140   $answer_evaluator->install_post_filter(\&fix_answers_for_display);
 1141 
 1142       $answer_evaluator->install_post_filter(sub {my $rh_ans = shift;
 1143           return $rh_ans unless $rh_ans->catch_error('EVAL');
 1144           $rh_ans->{student_ans} = $rh_ans->{original_student_ans}. ' '. $rh_ans->{error_message};
 1145           $rh_ans->clear_error('EVAL'); } );
 1146       $answer_evaluator->install_post_filter(sub {my $rh_ans = shift; $rh_ans->clear_error('SYNTAX'); } );
 1147       $answer_evaluator->install_post_filter(sub {my $rh_ans = shift; $rh_ans->clear_error('UNITS'); } );
 1148       $answer_evaluator->install_post_filter(sub {my $rh_ans = shift; $rh_ans->clear_error('NUMBER'); } );
 1149   $answer_evaluator->install_post_filter(sub {my $rh_ans = shift; $rh_ans->clear_error('STRING'); } );
 1150       $answer_evaluator;
 1151 }
 1152 
 1153 
 1154 
 1155 ##########################################################################
 1156 ##########################################################################
 1157 ## Function answer evaluators
 1158 
 1159 =head2 Function Answer Evaluators
 1160 
 1161 Function answer evaluators take in a function, compare it numerically to a
 1162 correct function, and return a score. They can require an exactly equivalent
 1163 function, or one that is equal up to a constant. They can accept or reject an
 1164 answer based on specified tolerances for numerical deviation.
 1165 
 1166 Function Comparison Options
 1167 
 1168   correctEqn  --  The correct equation, specified as a string. It may include
 1169           all basic arithmetic operations, as well as elementary
 1170           functions. Variable usage is described below.
 1171 
 1172   Variables --  The independent variable(s). When comparing the correct
 1173           equation to the student equation, each variable will be
 1174           replaced by a certain number of numerical values. If
 1175           the student equation agrees numerically with the correct
 1176           equation, they are considered equal. Note that all
 1177           comparison is numeric; it is possible (although highly
 1178           unlikely and never a practical concern) for two unequal
 1179           functions to yield the same numerical results.
 1180 
 1181   Limits    --  The limits of evaluation for the independent variables.
 1182           Each variable is evaluated only in the half-open interval
 1183           [lower_limit, upper_limit). This is useful if the function
 1184           has a singularity or is not defined in a certain range.
 1185           For example, the function "sqrt(-1-x)" could be evaluated
 1186           in [-2,-1).
 1187 
 1188   Tolerance --  Tolerance in function comparisons works exactly as in
 1189           numerical comparisons; see the numerical comparison
 1190           documentation for a complete description. Note that the
 1191           tolerance does applies to the function as a whole, not
 1192           each point individually.
 1193 
 1194   Number of --  Specifies how many points to evaluate each variable at. This
 1195   Points      is typically 3, but can be set higher if it is felt that
 1196           there is a strong possibility of "false positives."
 1197 
 1198   Maximum   --  Sets the maximum size of the constant of integration. For
 1199   Constant of   technical reasons concerning floating point arithmetic, if
 1200   Integration   the additive constant, i.e., the constant of integration, is
 1201           greater (in absolute value) than maxConstantOfIntegration
 1202           AND is greater than maxConstantOfIntegration times the
 1203           correct value, WeBWorK will give an error message saying
 1204           that it can not handle such a large constant of integration.
 1205           This is to prevent e.g. cos(x) + 1E20 or even 1E20 as being
 1206           accepted as a correct antiderivatives of sin(x) since
 1207           floating point arithmetic cannot tell the difference
 1208           between cos(x) + 1E20, 1E20, and -cos(x) + 1E20.
 1209 
 1210 Technical note: if you examine the code for the function routines, you will see
 1211 that most subroutines are simply doing some basic error-checking and then
 1212 passing the parameters on to the low-level FUNCTION_CMP(). Because this routine
 1213 is set up to handle multivariable functions, with single-variable functions as
 1214 a special case, it is possible to pass multivariable parameters to single-
 1215 variable functions. This usage is strongly discouraged as unnecessarily
 1216 confusing. Avoid it.
 1217 
 1218 Default Values (As of 7/24/2000) (Option -- Variable Name -- Value)
 1219 
 1220   Variable      --  $functVarDefault      --  'x'
 1221   Relative Tolerance    --  $functRelPercentTolDefault    --  .1
 1222   Absolute Tolerance    --  $functAbsTolDefault     --  .001
 1223   Lower Limit     --  $functLLimitDefault     --  .0000001
 1224   Upper Limit     --  $functULimitDefault     --  1
 1225   Number of Points    --  $functNumOfPoints     --  3
 1226   Zero Level      --  $functZeroLevelDefault      --  1E-14
 1227   Zero Level Tolerance    --  $functZeroLevelTolDefault   --  1E-12
 1228   Maximum Constant    --  $functMaxConstantOfIntegration    --  1E8
 1229     of Integration
 1230 
 1231 =cut
 1232 
 1233 
 1234 
 1235 =head3 fun_cmp()
 1236 
 1237 Compares a function or a list of functions, using a named hash of options to set
 1238 parameters. This can make for more readable code than using the function_cmp()
 1239 style, but some people find one or the other easier to remember.
 1240 
 1241 ANS( fun_cmp( answer or answer_array_ref, options_hash ) );
 1242 
 1243   1. a string containing the correct function, or a reference to an
 1244     array of correct functions
 1245   2. a hash containing the following items (all optional):
 1246     var           --  either the number of variables or a reference to an
 1247                       array of variable names (see below)
 1248     limits            --  reference to an array of arrays of limits (see below), or:
 1249     mode            --  'std' (default) (function must match exactly), or:
 1250                     'antider' (function must match up to a constant)
 1251     relTol            --  (default) a relative tolerance (as a percentage), or:
 1252     tol           --  an absolute tolerance for error
 1253     numPoints         --  the number of points to evaluate the function at
 1254     maxConstantOfIntegration      --  maximum size of the constant of integration
 1255     zeroLevel         --  if the correct answer is this close to zero, then
 1256                       zeroLevelTol applies
 1257     zeroLevelTol          --  absolute tolerance to allow when answer is close to zero
 1258     params                an array of "free" parameters which can be used to adapt
 1259                     the correct answer to the submitted answer. (e.g. ['c'] for
 1260                     a constant of integration in the answer x^3/3 + c.
 1261     debug           --  when set to 1 this provides extra information while checking the
 1262                         the answer.
 1263 
 1264   Returns an answer evaluator, or (if given a reference to an array
 1265   of answers), a list of answer evaluators
 1266 
 1267 ANSWER:
 1268 
 1269   The answer must be in the form of a string. The answer can contain
 1270   functions, pi, e, and arithmetic operations. However, the correct answer
 1271   string follows a slightly stricter syntax than student answers; specifically,
 1272   there is no implicit multiplication. So the correct answer must be "3*x" rather
 1273   than "3 x". Students can still enter "3 x".
 1274 
 1275 VARIABLES:
 1276 
 1277   The var parameter can contain either a number or a reference to an array of
 1278   variable names. If it contains a number, the variables are named automatically
 1279   as follows: 1 variable  --  x
 1280       2 variables --  x, y
 1281       3 variables --  x, y, z
 1282       4 or more --  x_1, x_2, x_3, etc.
 1283   If the var parameter contains a reference to an array of variable names, then
 1284   the number of variables is determined by the number of items in the array. A
 1285   reference to an array is created with brackets, e.g. "var => ['r', 's', 't']".
 1286   If only one variable is being used, you can write either "var => ['t']" for
 1287   consistency or "var => 't'" as a shortcut. The default is one variable, x.
 1288 
 1289 LIMITS:
 1290 
 1291   Limits are specified with the limits parameter. You may NOT use llimit/ulimit.
 1292   If you specify limits for one variable, you must specify them for all variables.
 1293   The limit parameter must be a reference to an array of arrays of the form
 1294   [lower_limit. upper_limit], each array corresponding to the lower and upper
 1295   endpoints of the (half-open) domain of one variable. For example,
 1296   "vars => 2, limits => [[0,2], [-3,8]]" would cause x to be evaluated in [0,2) and
 1297   y to be evaluated in [-3,8). If only one variable is being used, you can write
 1298   either "limits => [[0,3]]" for consistency or "limits => [0,3]" as a shortcut.
 1299 
 1300 EXAMPLES:
 1301 
 1302   fun_cmp( "3*x" )  --  standard compare, variable is x
 1303   fun_cmp( ["3*x", "4*x+3", "3*x**2"] ) --  standard compare, defaults used for all three functions
 1304   fun_cmp( "3*t", var => 't' )  --  standard compare, variable is t
 1305   fun_cmp( "5*x*y*z", var => 3 )  --  x, y and z are the variables
 1306   fun_cmp( "5*x", mode => 'antider' ) --  student answer must match up to constant (i.e., 5x+C)
 1307   fun_cmp( ["3*x*y", "4*x*y"], limits => [[0,2], [5,7]] ) --  x evaluated in [0,2)
 1308                                 y evaluated in [5,7)
 1309 
 1310 =cut
 1311 
 1312 sub fun_cmp {
 1313   my $correctAnswer = shift @_;
 1314   my %opt = @_;
 1315 
 1316     assign_option_aliases( \%opt,
 1317         'vars'    =>  'var',    # set the standard option 'var' to the one specified as vars
 1318           'domain'  =>  'limits', # set the standard option 'limits' to the one specified as domain
 1319           'reltol'    =>  'relTol',
 1320           'param'   =>  'params',
 1321     );
 1322 
 1323     set_default_options(  \%opt,
 1324         'var'         =>  $functVarDefault,
 1325             'params'        =>  [],
 1326         'limits'        =>  [[$functLLimitDefault, $functULimitDefault]],
 1327         'mode'          =>  'std',
 1328         'tolType'       =>    (defined($opt{tol}) ) ? 'absolute' : 'relative',
 1329         'tol'         =>  .01, # default mode should be relative, to obtain this tol must not be defined
 1330             'relTol'        =>  $functRelPercentTolDefault,
 1331         'numPoints'       =>  $functNumOfPoints,
 1332         'maxConstantOfIntegration'  =>  $functMaxConstantOfIntegration,
 1333         'zeroLevel'       =>  $functZeroLevelDefault,
 1334         'zeroLevelTol'      =>  $functZeroLevelTolDefault,
 1335             'debug'         =>  0,
 1336      );
 1337 
 1338     # allow var => 'x' as an abbreviation for var => ['x']
 1339   my %out_options = %opt;
 1340   unless ( ref($out_options{var}) eq 'ARRAY' ) {
 1341     $out_options{var} = [$out_options{var}];
 1342   }
 1343   # allow params => 'c' as an abbreviation for params => ['c']
 1344   unless ( ref($out_options{params}) eq 'ARRAY' ) {
 1345     $out_options{params} = [$out_options{params}];
 1346   }
 1347   my ($tolType, $tol);
 1348     if ($out_options{tolType} eq 'absolute') {
 1349     $tolType = 'absolute';
 1350     $tol = $out_options{'tol'};
 1351     delete($out_options{'relTol'}) if exists( $out_options{'relTol'} );
 1352   } else {
 1353     $tolType = 'relative';
 1354     $tol = $out_options{'relTol'};
 1355     delete($out_options{'tol'}) if exists( $out_options{'tol'} );
 1356   }
 1357 
 1358   my @output_list = ();
 1359   # thread over lists
 1360   my @ans_list = ();
 1361 
 1362   if ( ref($correctAnswer) eq 'ARRAY' ) {
 1363     @ans_list = @{$correctAnswer};
 1364   }
 1365   else {
 1366     push( @ans_list, $correctAnswer );
 1367   }
 1368 
 1369   # produce answer evaluators
 1370   foreach my $ans (@ans_list) {
 1371     push(@output_list,
 1372       FUNCTION_CMP(
 1373           'correctEqn'    =>  $ans,
 1374           'var'       =>  $out_options{'var'},
 1375           'limits'      =>  $out_options{'limits'},
 1376           'tolerance'     =>  $tol,
 1377           'tolType'     =>  $tolType,
 1378           'numPoints'     =>  $out_options{'numPoints'},
 1379           'mode'        =>  $out_options{'mode'},
 1380           'maxConstantOfIntegration'  =>  $out_options{'maxConstantOfIntegration'},
 1381           'zeroLevel'     =>  $out_options{'zeroLevel'},
 1382           'zeroLevelTol'    =>  $out_options{'zeroLevelTol'},
 1383           'params'      =>  $out_options{'params'},
 1384           'debug'       =>  $out_options{'debug'},
 1385       ),
 1386     );
 1387   }
 1388 
 1389   return (wantarray) ? @output_list : $output_list[0];
 1390 }
 1391 
 1392 =head3 Single-variable Function Comparisons
 1393 
 1394 There are four single-variable function answer evaluators: "normal," absolute
 1395 tolerance, antiderivative, and antiderivative with absolute tolerance. All
 1396 parameters (other than the correct equation) are optional.
 1397 
 1398  function_cmp( $correctEqn ) OR
 1399  function_cmp( $correctEqn, $var ) OR
 1400  function_cmp( $correctEqn, $var, $llimit, $ulimit ) OR
 1401  function_cmp( $correctEqn, $var, $llimit, $ulimit, $relPercentTol ) OR
 1402  function_cmp( $correctEqn, $var, $llimit, $ulimit,
 1403         $relPercentTol, $numPoints ) OR
 1404  function_cmp( $correctEqn, $var, $llimit, $ulimit,
 1405         $relPercentTol, $numPoints, $zeroLevel ) OR
 1406  function_cmp( $correctEqn, $var, $llimit, $ulimit, $relPercentTol, $numPoints,
 1407         $zeroLevel,$zeroLevelTol )
 1408 
 1409   $correctEqn   --  the correct equation, as a string
 1410   $var      --  the string representing the variable (optional)
 1411   $llimit     --  the lower limit of the interval to evaluate the
 1412               variable in (optional)
 1413   $ulimit     --  the upper limit of the interval to evaluate the
 1414               variable in (optional)
 1415   $relPercentTol  --  the error tolerance as a percentage (optional)
 1416   $numPoints    --  the number of points at which to evaluate the
 1417               variable (optional)
 1418   $zeroLevel    --  if the correct answer is this close to zero, then
 1419               zeroLevelTol applies (optional)
 1420   $zeroLevelTol --  absolute tolerance to allow when answer is close to zero
 1421 
 1422   function_cmp() uses standard comparison and relative tolerance. It takes a
 1423   string representing a single-variable function and compares the student
 1424   answer to that function numerically.
 1425 
 1426  function_cmp_up_to_constant( $correctEqn ) OR
 1427  function_cmp_up_to_constant( $correctEqn, $var ) OR
 1428  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit ) OR
 1429  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1430                 $relpercentTol ) OR
 1431  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1432                 $relpercentTol, $numOfPoints ) OR
 1433  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1434                 $relpercentTol, $numOfPoints,
 1435                 $maxConstantOfIntegration ) OR
 1436  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1437                 $relpercentTol, $numOfPoints,
 1438                 $maxConstantOfIntegration, $zeroLevel)  OR
 1439  function_cmp_up_to_constant( $correctEqn, $var, $llimit, $ulimit,
 1440                 $relpercentTol, $numOfPoints,
 1441                 $maxConstantOfIntegration,
 1442                 $zeroLevel, $zeroLevelTol )
 1443 
 1444   $maxConstantOfIntegration --  the maximum size of the constant of
 1445                   integration
 1446 
 1447   function_cmp_up_to_constant() uses antiderivative compare and relative
 1448   tolerance. All options work exactly like function_cmp(), except of course
 1449   $maxConstantOfIntegration. It will accept as correct any function which
 1450   differs from $correctEqn by at most a constant; that is, if
 1451     $studentEqn = $correctEqn + C
 1452   the answer is correct.
 1453 
 1454  function_cmp_abs( $correctFunction ) OR
 1455  function_cmp_abs( $correctFunction, $var ) OR
 1456  function_cmp_abs( $correctFunction, $var, $llimit, $ulimit ) OR
 1457  function_cmp_abs( $correctFunction, $var, $llimit, $ulimit, $absTol ) OR
 1458  function_cmp_abs( $correctFunction, $var, $llimit, $ulimit, $absTol,
 1459           $numOfPoints )
 1460 
 1461   $absTol --  the tolerance as an absolute value
 1462 
 1463   function_cmp_abs() uses standard compare and absolute tolerance. All
 1464   other options work exactly as for function_cmp().
 1465 
 1466  function_cmp_up_to_constant_abs( $correctFunction ) OR
 1467  function_cmp_up_to_constant_abs( $correctFunction, $var ) OR
 1468  function_cmp_up_to_constant_abs( $correctFunction, $var, $llimit, $ulimit ) OR
 1469  function_cmp_up_to_constant_abs( $correctFunction, $var, $llimit, $ulimit,
 1470                   $absTol ) OR
 1471  function_cmp_up_to_constant_abs( $correctFunction, $var, $llimit, $ulimit,
 1472                   $absTol, $numOfPoints ) OR
 1473  function_cmp_up_to_constant_abs( $correctFunction, $var, $llimit, $ulimit,
 1474                   $absTol, $numOfPoints,
 1475                   $maxConstantOfIntegration )
 1476 
 1477   function_cmp_up_to_constant_abs() uses antiderivative compare
 1478   and absolute tolerance. All other options work exactly as with
 1479   function_cmp_up_to_constant().
 1480 
 1481 Examples:
 1482 
 1483   ANS( function_cmp( "cos(x)" ) ) --  Accepts cos(x), sin(x+pi/2),
 1484     sin(x)^2 + cos(x) + cos(x)^2 -1, etc. This assumes
 1485     $functVarDefault has been set to "x".
 1486   ANS( function_cmp( $answer, "t" ) ) --  Assuming $answer is "cos(t)",
 1487     accepts cos(t), etc.
 1488   ANS( function_cmp_up_to_constant( "cos(x)" ) )  --  Accepts any
 1489     antiderivative of sin(x), e.g. cos(x) + 5.
 1490   ANS( function_cmp_up_to_constant( "cos(z)", "z" ) ) --  Accepts any
 1491     antiderivative of sin(z), e.g. sin(z+pi/2) + 5.
 1492 
 1493 =cut
 1494 
 1495 sub adaptive_function_cmp {
 1496   my $correctEqn = shift;
 1497   my %options = @_;
 1498   set_default_options(  \%options,
 1499       'vars'      =>  [qw( x y )],
 1500                   'params'    =>  [],
 1501                   'limits'    =>  [ [0,1], [0,1]],
 1502                   'reltol'    =>  $main::functRelPercentTolDefault,
 1503                   'numPoints'   =>  $main::functNumOfPoints,
 1504                   'zeroLevel'   =>  $main::functZeroLevelDefault,
 1505                   'zeroLevelTol'    =>  $main::functZeroLevelTolDefault,
 1506                   'debug'     =>  0,
 1507   );
 1508 
 1509     my $var_ref = $options{'vars'};
 1510     my $ra_params = $options{ 'params'};
 1511     my $limit_ref = $options{'limits'};
 1512     my $relPercentTol= $options{'reltol'};
 1513     my $numPoints = $options{'numPoints'};
 1514     my $zeroLevel = $options{'zeroLevel'};
 1515     my $zeroLevelTol = $options{'zeroLevelTol'};
 1516 
 1517   FUNCTION_CMP( 'correctEqn'          =>  $correctEqn,
 1518       'var'           =>  $var_ref,
 1519       'limits'          =>  $limit_ref,
 1520       'tolerance'         =>  $relPercentTol,
 1521       'tolType'         =>  'relative',
 1522       'numPoints'         =>  $numPoints,
 1523       'mode'            =>  'std',
 1524       'maxConstantOfIntegration'      =>  10**100,
 1525       'zeroLevel'         =>  $zeroLevel,
 1526       'zeroLevelTol'          =>  $zeroLevelTol,
 1527       'scale_norm'                      =>    1,
 1528       'params'                          =>    $ra_params,
 1529       'debug'               =>  $options{debug} ,
 1530   );
 1531 }
 1532 
 1533 sub function_cmp {
 1534   my ($correctEqn,$var,$llimit,$ulimit,$relPercentTol,$numPoints,$zeroLevel,$zeroLevelTol) = @_;
 1535 
 1536   if ( (scalar(@_) == 3) or (scalar(@_) > 8) or (scalar(@_) == 0) ) {
 1537     function_invalid_params( $correctEqn );
 1538   }
 1539   else {
 1540     FUNCTION_CMP( 'correctEqn'          =>  $correctEqn,
 1541         'var'           =>  $var,
 1542         'limits'          =>  [$llimit, $ulimit],
 1543         'tolerance'         =>  $relPercentTol,
 1544         'tolType'         =>  'relative',
 1545         'numPoints'         =>  $numPoints,
 1546         'mode'            =>  'std',
 1547         'maxConstantOfIntegration'      =>  0,
 1548         'zeroLevel'         =>  $zeroLevel,
 1549         'zeroLevelTol'          =>  $zeroLevelTol
 1550           );
 1551   }
 1552 }
 1553 
 1554 sub function_cmp_up_to_constant { ## for antiderivative problems
 1555   my ($correctEqn,$var,$llimit,$ulimit,$relPercentTol,$numPoints,$maxConstantOfIntegration,$zeroLevel,$zeroLevelTol) = @_;
 1556 
 1557   if ( (scalar(@_) == 3) or (scalar(@_) > 9) or (scalar(@_) == 0) ) {
 1558     function_invalid_params( $correctEqn );
 1559   }
 1560   else {
 1561     FUNCTION_CMP( 'correctEqn'          =>  $correctEqn,
 1562         'var'           =>  $var,
 1563         'limits'          =>  [$llimit, $ulimit],
 1564         'tolerance'         =>  $relPercentTol,
 1565         'tolType'         =>  'relative',
 1566         'numPoints'         =>  $numPoints,
 1567         'mode'            =>  'antider',
 1568         'maxConstantOfIntegration'      =>  $maxConstantOfIntegration,
 1569         'zeroLevel'         =>  $zeroLevel,
 1570         'zeroLevelTol'          =>  $zeroLevelTol
 1571           );
 1572   }
 1573 }
 1574 
 1575 sub function_cmp_abs {      ## similar to function_cmp but uses absolute tolerance
 1576   my ($correctEqn,$var,$llimit,$ulimit,$absTol,$numPoints) = @_;
 1577 
 1578   if ( (scalar(@_) == 3) or (scalar(@_) > 6) or (scalar(@_) == 0) ) {
 1579     function_invalid_params( $correctEqn );
 1580   }
 1581   else {
 1582     FUNCTION_CMP( 'correctEqn'      =>  $correctEqn,
 1583         'var'       =>  $var,
 1584         'limits'      =>  [$llimit, $ulimit],
 1585         'tolerance'     =>  $absTol,
 1586         'tolType'     =>  'absolute',
 1587         'numPoints'     =>  $numPoints,
 1588         'mode'        =>  'std',
 1589         'maxConstantOfIntegration'  =>  0,
 1590         'zeroLevel'     =>  0,
 1591         'zeroLevelTol'      =>  0
 1592     );
 1593   }
 1594 }
 1595 
 1596 
 1597 sub function_cmp_up_to_constant_abs  {  ## for antiderivative problems
 1598                     ## similar to function_cmp_up_to_constant
 1599                     ## but uses absolute tolerance
 1600   my ($correctEqn,$var,$llimit,$ulimit,$absTol,$numPoints,$maxConstantOfIntegration) = @_;
 1601 
 1602   if ( (scalar(@_) == 3) or (scalar(@_) > 7) or (scalar(@_) == 0) ) {
 1603     function_invalid_params( $correctEqn );
 1604   }
 1605 
 1606   else {
 1607     FUNCTION_CMP( 'correctEqn'          =>  $correctEqn,
 1608         'var'           =>  $var,
 1609         'limits'          =>  [$llimit, $ulimit],
 1610         'tolerance'         =>  $absTol,
 1611         'tolType'         =>  'absolute',
 1612         'numPoints'         =>  $numPoints,
 1613         'mode'            =>  'antider',
 1614         'maxConstantOfIntegration'      =>  $maxConstantOfIntegration,
 1615         'zeroLevel'         =>  0,
 1616         'zeroLevelTol'          =>  0
 1617     );
 1618   }
 1619 }
 1620 
 1621 ## The following answer evaluator for comparing multivarable functions was
 1622 ## contributed by Professor William K. Ziemer
 1623 ## (Note: most of the multivariable functionality provided by Professor Ziemer
 1624 ## has now been integrated into fun_cmp and FUNCTION_CMP)
 1625 ############################
 1626 # W.K. Ziemer, Sep. 1999
 1627 # Math Dept. CSULB
 1628 # email: wziemer@csulb.edu
 1629 ############################
 1630 
 1631 =head3 multivar_function_cmp
 1632 
 1633 NOTE: this function is maintained for compatibility. fun_cmp() is
 1634     slightly preferred.
 1635 
 1636 usage:
 1637 
 1638   multivar_function_cmp( $answer, $var_reference, options)
 1639     $answer       --  string, represents function of several variables
 1640     $var_reference    --  number (of variables), or list reference (e.g. ["var1","var2"] )
 1641   options:
 1642     $limit_reference  --  reference to list of lists (e.g. [[1,2],[3,4]])
 1643     $relPercentTol    --  relative percent tolerance in answer
 1644     $numPoints      --  number of points to sample in for each variable
 1645     $zeroLevel      --  if the correct answer is this close to zero, then zeroLevelTol applies
 1646     $zeroLevelTol   --  absolute tolerance to allow when answer is close to zero
 1647 
 1648 =cut
 1649 
 1650 sub multivar_function_cmp {
 1651   my ($correctEqn,$var_ref,$limit_ref,$relPercentTol,$numPoints,$zeroLevel,$zeroLevelTol) = @_;
 1652 
 1653   if ( (scalar(@_) > 7) or (scalar(@_) < 2) ) {
 1654     function_invalid_params( $correctEqn );
 1655   }
 1656 
 1657   FUNCTION_CMP( 'correctEqn'      =>  $correctEqn,
 1658       'var'       =>  $var_ref,
 1659       'limits'      =>  $limit_ref,
 1660       'tolerance'     =>  $relPercentTol,
 1661       'tolType'     =>  'relative',
 1662       'numPoints'     =>  $numPoints,
 1663       'mode'        =>  'std',
 1664       'maxConstantOfIntegration'  =>  0,
 1665       'zeroLevel'     =>  $zeroLevel,
 1666       'zeroLevelTol'      =>  $zeroLevelTol
 1667   );
 1668 }
 1669 
 1670 ## LOW-LEVEL ROUTINE -- NOT NORMALLY FOR END USERS -- USE WITH CAUTION
 1671 ## NOTE: PG_answer_eval is used instead of PG_restricted_eval in order to insure that the answer
 1672 ## evaluated within the context of the package the problem was originally defined in.
 1673 ## Includes multivariable modifications contributed by Professor William K. Ziemer
 1674 ##
 1675 ## IN:  a hash consisting of the following keys (error checking to be added later?)
 1676 ##      correctEqn      --  the correct equation as a string
 1677 ##      var       --  the variable name as a string,
 1678 ##                or a reference to an array of variables
 1679 ##      limits        --  reference to an array of arrays of type [lower,upper]
 1680 ##      tolerance     --  the allowable margin of error
 1681 ##      tolType       --  'relative' or 'absolute'
 1682 ##      numPoints     --  the number of points to evaluate the function at
 1683 ##      mode        --  'std' or 'antider'
 1684 ##      maxConstantOfIntegration  --  maximum size of the constant of integration
 1685 ##      zeroLevel     --  if the correct answer is this close to zero,
 1686 ##                        then zeroLevelTol applies
 1687 ##      zeroLevelTol      --  absolute tolerance to allow when answer is close to zero
 1688 
 1689 
 1690 sub FUNCTION_CMP {
 1691   my %func_params = @_;
 1692 
 1693   my $correctEqn          = $func_params{'correctEqn'};
 1694   my $var           = $func_params{'var'};
 1695   my $ra_limits         = $func_params{'limits'};
 1696   my $tol           = $func_params{'tolerance'};
 1697   my $tolType         = $func_params{'tolType'};
 1698   my $numPoints         = $func_params{'numPoints'};
 1699   my $mode          = $func_params{'mode'};
 1700   my $maxConstantOfIntegration      = $func_params{'maxConstantOfIntegration'};
 1701   my $zeroLevel         = $func_params{'zeroLevel'};
 1702   my $zeroLevelTol        = $func_params{'zeroLevelTol'};
 1703 
 1704 
 1705     # Check that everything is defined:
 1706     $func_params{debug} = 0 unless defined($func_params{debug});
 1707     $mode = 'std' unless defined($mode);
 1708     my @VARS = get_var_array( $var );
 1709   my @limits = get_limits_array( $ra_limits );
 1710   my @PARAMS = ();
 1711   @PARAMS = @{$func_params{'params'}} if defined($func_params{'params'});
 1712 
 1713   if ($mode eq 'antider' ) {
 1714     # doctor the equation to allow addition of a constant
 1715     my $CONSTANT_PARAM = 'Q';  # unfortunately parameters must be single letters.
 1716                    # There is the possibility of conflict here.
 1717                    #  'Q' seemed less dangerous than  'C'.
 1718     $correctEqn = "( $correctEqn ) + $CONSTANT_PARAM";
 1719     push(@PARAMS, $CONSTANT_PARAM);
 1720   }
 1721     my $dim_of_param_space = @PARAMS;      # dimension of equivalence space
 1722 
 1723   if( $tolType eq 'relative' ) {
 1724     $tol = $functRelPercentTolDefault     unless defined $tol;
 1725     $tol *= .01;
 1726   }
 1727   else {
 1728     $tol = $functAbsTolDefault        unless defined $tol;
 1729   }
 1730 
 1731   #loop ensures that number of limits matches number of variables
 1732   for( my $i = 0; $i < scalar(@VARS); $i++ ) {
 1733     $limits[$i][0] = $functLLimitDefault      unless defined $limits[$i][0];
 1734     $limits[$i][1] = $functULimitDefault      unless defined $limits[$i][1];
 1735   }
 1736   $numPoints = $functNumOfPoints          unless defined $numPoints;
 1737   $maxConstantOfIntegration = $functMaxConstantOfIntegration  unless defined $maxConstantOfIntegration;
 1738   $zeroLevel = $functZeroLevelDefault       unless defined $zeroLevel;
 1739   $zeroLevelTol = $functZeroLevelTolDefault     unless defined $zeroLevelTol;
 1740 
 1741   $func_params{'var'}       = $var;
 1742       $func_params{'limits'}        = \@limits;
 1743       $func_params{'tolerance'}     = $tol;
 1744       $func_params{'tolType'}       = $tolType;
 1745       $func_params{'numPoints'}     = $numPoints;
 1746       $func_params{'mode'}        = $mode;
 1747       $func_params{'maxConstantOfIntegration'}  = $maxConstantOfIntegration;
 1748       $func_params{'zeroLevel'}     = $zeroLevel;
 1749       $func_params{'zeroLevelTol'}        =   $zeroLevelTol;
 1750 
 1751 ########################################################
 1752 #   End of cleanup of calling parameters
 1753 ########################################################
 1754   my $i;            #for use with loops
 1755   my $PGanswerMessage = "";
 1756   my $originalCorrEqn = $correctEqn;
 1757 
 1758 #prepare the correct answer and check it's syntax
 1759       my $rh_correct_ans = new AnswerHash;
 1760   $rh_correct_ans->input($correctEqn);
 1761   $rh_correct_ans = check_syntax($rh_correct_ans);
 1762   warn  $rh_correct_ans->{error_message} if $rh_correct_ans->{error_flag};
 1763   $rh_correct_ans->clear_error();
 1764   $rh_correct_ans = function_from_string2($rh_correct_ans, ra_vars => [ @VARS, @PARAMS ],
 1765                                                            store_in =>'rf_correct_ans',
 1766                                                            debug =>  $func_params{debug});
 1767   my $correct_eqn_sub = $rh_correct_ans->{rf_correct_ans};
 1768   warn $rh_correct_ans->{error_message} if $rh_correct_ans->{error_flag};
 1769 
 1770 #create the evaluation points
 1771   my $random_for_answers = new PGrandom($main::PG_original_problemSeed);
 1772       my $NUMBER_OF_STEPS_IN_RANDOM = 1000;    # determines the granularity of the random_for_answers number generator
 1773   my (@evaluation_points);
 1774   for( my $count = 0; $count < @PARAMS+1+$numPoints; $count++ ) {
 1775       my (@vars,$iteration_limit);
 1776     for( my $i = 0; $i < @VARS; $i++ ) {
 1777       my $iteration_limit = 10;
 1778       while (  0 < --$iteration_limit ) {  # make sure that the endpoints of the interval are not included
 1779           $vars[$i] = $random_for_answers->random($limits[$i][0], $limits[$i][1], abs($limits[$i][1] - $limits[$i][0])/$NUMBER_OF_STEPS_IN_RANDOM );
 1780           last if $vars[$i]!=$limits[$i][0] and $vars[$i]!=$limits[$i][1];
 1781         }
 1782         warn "Unable to properly choose  evaluation points for this function in the interval ( $limits[$i][0] , $limits[$i][1] )"
 1783           if $iteration_limit == 0;
 1784     };
 1785 
 1786     push(@evaluation_points,\@vars);
 1787   }
 1788   my $evaluation_points = Matrix->new_from_array_ref(\@evaluation_points);
 1789 
 1790   #my $COEFFS = determine_param_coeffs($correct_eqn_sub,$evaluation_points[0],$numOfParameters);
 1791       #warn "coeff", join(" | ", @{$COEFFS});
 1792 
 1793 #construct the answer evaluator
 1794     my $answer_evaluator = new AnswerEvaluator;
 1795     $answer_evaluator->{debug} = $func_params{debug};
 1796     $answer_evaluator->ans_hash(  correct_ans     =>  $originalCorrEqn,
 1797           rf_correct_ans    =>  $rh_correct_ans->{rf_correct_ans},
 1798           evaluation_points =>  \@evaluation_points,
 1799           ra_param_vars     =>  \@PARAMS,
 1800           ra_vars     =>  \@VARS,
 1801           type      =>  'function',
 1802     );
 1803 
 1804     $answer_evaluator->install_pre_filter(\&check_syntax);
 1805     $answer_evaluator->install_pre_filter(\&function_from_string2, ra_vars => \@VARS,debug=>$func_params{debug},); # @VARS has been guaranteed to be an array, $var might be a single string.
 1806     $answer_evaluator->install_pre_filter(\&best_approx_parameters, %func_params, param_vars => \@PARAMS);
 1807     $answer_evaluator->install_evaluator(\&calculate_difference_vector, %func_params);
 1808     $answer_evaluator->install_evaluator(\&is_zero_array, tolerance => $tol );
 1809     $answer_evaluator->install_post_filter(sub {my $rh_ans = shift; $rh_ans->clear_error('SYNTAX'); $rh_ans;} );
 1810     $answer_evaluator->install_post_filter(
 1811             sub {my $rh_ans = shift;
 1812             if ($rh_ans->catch_error('EVAL') ) {
 1813               $rh_ans->{ans_message} = $rh_ans->{error_message};
 1814               $rh_ans->clear_error('EVAL');
 1815             }
 1816             $rh_ans;
 1817           }
 1818   );
 1819     $answer_evaluator;
 1820 }
 1821 
 1822 
 1823 ## LOW-LEVEL ROUTINE -- NOT NORMALLY FOR END USERS -- USE WITH CAUTION
 1824 ##
 1825 ## IN:  a hash containing the following items (error-checking to be added later?):
 1826 ##      correctAnswer --  the correct answer
 1827 ##      tolerance   --  the allowable margin of error
 1828 ##      tolType     --  'relative' or 'absolute'
 1829 ##      format      --  the display format of the answer
 1830 ##      mode      --  one of 'std', 'strict', 'arith', or 'frac';
 1831 ##                  determines allowable formats for the input
 1832 ##      zeroLevel   --  if the correct answer is this close to zero, then zeroLevelTol applies
 1833 ##      zeroLevelTol  --  absolute tolerance to allow when answer is close to zero
 1834 
 1835 
 1836 ##########################################################################
 1837 ##########################################################################
 1838 ## String answer evaluators
 1839 
 1840 =head2 String Answer Evaluators
 1841 
 1842 String answer evaluators compare a student string to the correct string.
 1843 Different filters can be applied to allow various degrees of variation.
 1844 Both the student and correct answers are subject to the same filters, to
 1845 ensure that there are no unexpected matches or rejections.
 1846 
 1847 String Filters
 1848 
 1849   remove_whitespace --  Removes all whitespace from the string.
 1850             It applies the following substitution
 1851             to the string:
 1852               $filteredAnswer =~ s/\s+//g;
 1853 
 1854   compress_whitespace --  Removes leading and trailing whitespace, and
 1855             replaces all other blocks of whitespace by a
 1856             single space. Applies the following substitutions:
 1857               $filteredAnswer =~ s/^\s*//;
 1858               $filteredAnswer =~ s/\s*$//;
 1859               $filteredAnswer =~ s/\s+/ /g;
 1860 
 1861   trim_whitespace   --  Removes leading and trailing whitespace.
 1862             Applies the following substitutions:
 1863               $filteredAnswer =~ s/^\s*//;
 1864               $filteredAnswer =~ s/\s*$//;
 1865 
 1866   ignore_case     --  Ignores the case of the string. More accurately,
 1867             it converts the string to uppercase (by convention).
 1868             Applies the following function:
 1869               $filteredAnswer = uc $filteredAnswer;
 1870 
 1871   ignore_order    --  Ignores the order of the letters in the string.
 1872             This is used for problems of the form "Choose all
 1873             that apply." Specifically, it removes all
 1874             whitespace and lexically sorts the letters in
 1875             ascending alphabetical order. Applies the following
 1876             functions:
 1877               $filteredAnswer = join( "", lex_sort(
 1878                 split( /\s*/, $filteredAnswer ) ) );
 1879 
 1880 =cut
 1881 
 1882 ################################
 1883 ## STRING ANSWER FILTERS
 1884 
 1885 ## IN:  --the string to be filtered
 1886 ##    --a list of the filters to use
 1887 ##
 1888 ## OUT: --the modified string
 1889 ##
 1890 ## Use this subroutine instead of the
 1891 ## individual filters below it
 1892 
 1893 sub str_filters {
 1894   my $stringToFilter = shift @_;
 1895   my @filters_to_use = @_;
 1896   my %known_filters = ( 'remove_whitespace'   =>  undef,
 1897         'compress_whitespace'   =>  undef,
 1898         'trim_whitespace'   =>  undef,
 1899         'ignore_case'     =>  undef,
 1900         'ignore_order'      =>  undef
 1901   );
 1902 
 1903   #test for unknown filters
 1904   my $filter;
 1905   foreach $filter (@filters_to_use) {
 1906     die "Unknown string filter $filter (try checking the parameters to str_cmp() )"
 1907                 unless exists $known_filters{$filter};
 1908   }
 1909 
 1910   if( grep( /remove_whitespace/i, @filters_to_use ) ) {
 1911     $stringToFilter = remove_whitespace( $stringToFilter );
 1912   }
 1913   if( grep( /compress_whitespace/i, @filters_to_use ) ) {
 1914     $stringToFilter = compress_whitespace( $stringToFilter );
 1915   }
 1916   if( grep( /trim_whitespace/i, @filters_to_use ) ) {
 1917     $stringToFilter = trim_whitespace( $stringToFilter );
 1918   }
 1919   if( grep( /ignore_case/i, @filters_to_use ) ) {
 1920     $stringToFilter = ignore_case( $stringToFilter );
 1921   }
 1922   if( grep( /ignore_order/i, @filters_to_use ) ) {
 1923     $stringToFilter = ignore_order( $stringToFilter );
 1924   }
 1925 
 1926   return $stringToFilter;
 1927 }
 1928 
 1929 sub remove_whitespace {
 1930   my $filteredAnswer = shift;
 1931 
 1932   $filteredAnswer =~ s/\s+//g;    # remove all whitespace
 1933 
 1934   return $filteredAnswer;
 1935 }
 1936 
 1937 sub compress_whitespace {
 1938   my $filteredAnswer = shift;
 1939 
 1940   $filteredAnswer =~ s/^\s*//;    # remove initial whitespace
 1941   $filteredAnswer =~ s/\s*$//;    # remove trailing whitespace
 1942   $filteredAnswer =~ s/\s+/ /g;   # replace spaces by single space
 1943 
 1944   return $filteredAnswer;
 1945 }
 1946 
 1947 sub trim_whitespace {
 1948   my $filteredAnswer = shift;
 1949 
 1950   $filteredAnswer =~ s/^\s*//;    # remove initial whitespace
 1951   $filteredAnswer =~ s/\s*$//;    # remove trailing whitespace
 1952 
 1953   return $filteredAnswer;
 1954 }
 1955 
 1956 sub ignore_case {
 1957   my $filteredAnswer = shift;
 1958 
 1959   $filteredAnswer = uc $filteredAnswer;
 1960 
 1961   return $filteredAnswer;
 1962 }
 1963 
 1964 sub ignore_order {
 1965   my $filteredAnswer = shift;
 1966 
 1967   $filteredAnswer = join( "", lex_sort( split( /\s*/, $filteredAnswer ) ) );
 1968 
 1969   return $filteredAnswer;
 1970 }
 1971 ################################
 1972 ## END STRING ANSWER FILTERS
 1973 
 1974 
 1975 =head3 str_cmp()
 1976 
 1977 Compares a string or a list of strings, using a named hash of options to set
 1978 parameters. This can make for more readable code than using the "mode"_str_cmp()
 1979 style, but some people find one or the other easier to remember.
 1980 
 1981 ANS( str_cmp( answer or answer_array_ref, options_hash ) );
 1982 
 1983   1. the correct answer or a reference to an array of answers
 1984   2. either a list of filters, or:
 1985      a hash consisting of
 1986     filters - a reference to an array of filters
 1987 
 1988   Returns an answer evaluator, or (if given a reference to an array of answers),
 1989   a list of answer evaluators
 1990 
 1991 FILTERS:
 1992 
 1993   remove_whitespace --  removes all whitespace
 1994   compress_whitespace --  removes whitespace from the beginning and end of the string,
 1995               and treats one or more whitespace characters in a row as a
 1996               single space (true by default)
 1997   trim_whitespace   --  removes whitespace from the beginning and end of the string
 1998   ignore_case   --  ignores the case of the letters (true by default)
 1999   ignore_order    --  ignores the order in which letters are entered
 2000 
 2001 EXAMPLES:
 2002 
 2003   str_cmp( "Hello" )  --  matches "Hello", "  hello" (same as std_str_cmp() )
 2004   str_cmp( ["Hello", "Goodbye"] ) --  same as std_str_cmp_list()
 2005   str_cmp( " hello ", trim_whitespace ) --  matches "hello", " hello  "
 2006   str_cmp( "ABC", filters => 'ignore_order' ) --  matches "ACB", "A B C", but not "abc"
 2007   str_cmp( "D E F", remove_whitespace, ignore_case )  --  matches "def" and "d e f" but not "fed"
 2008 
 2009 
 2010 =cut
 2011 
 2012 sub str_cmp {
 2013   my $correctAnswer = shift @_;
 2014   $correctAnswer = '' unless defined($correctAnswer);
 2015   my @options = @_;
 2016   my $ra_filters;
 2017 
 2018   # error-checking for filters occurs in the filters() subroutine
 2019   if( not defined( $options[0] ) ) {    # used with no filters as alias for std_str_cmp()
 2020     @options = ( 'compress_whitespace', 'ignore_case' );
 2021   }
 2022 
 2023   if( $options[0] eq 'filters' ) {    # using filters => [f1, f2, ...] notation
 2024     $ra_filters = $options[1];
 2025   }
 2026   else {            # using a list of filters
 2027     $ra_filters = \@options;
 2028   }
 2029 
 2030   # thread over lists
 2031   my @ans_list = ();
 2032 
 2033   if ( ref($correctAnswer) eq 'ARRAY' ) {
 2034     @ans_list = @{$correctAnswer};
 2035   }
 2036   else {
 2037     push( @ans_list, $correctAnswer );
 2038   }
 2039 
 2040   # final_answer;
 2041   my @output_list = ();
 2042 
 2043   foreach my $ans (@ans_list) {
 2044     push(@output_list, STR_CMP( 'correctAnswer' =>  $ans,
 2045             'filters' =>  $ra_filters,
 2046             'type'    =>  'str_cmp'
 2047          )
 2048     );
 2049   }
 2050 
 2051   return (wantarray) ? @output_list : $output_list[0] ;
 2052 }
 2053 
 2054 =head3 "mode"_str_cmp functions
 2055 
 2056 The functions of the the form "mode"_str_cmp() use different functions to
 2057 specify which filters to apply. They take no options except the correct
 2058 string. There are also versions which accept a list of strings.
 2059 
 2060  std_str_cmp( $correctString )
 2061  std_str_cmp_list( @correctStringList )
 2062   Filters: compress_whitespace, ignore_case
 2063 
 2064  std_cs_str_cmp( $correctString )
 2065  std_cs_str_cmp_list( @correctStringList )
 2066   Filters: compress_whitespace
 2067 
 2068  strict_str_cmp( $correctString )
 2069  strict_str_cmp_list( @correctStringList )
 2070   Filters: trim_whitespace
 2071 
 2072  unordered_str_cmp( $correctString )
 2073  unordered_str_cmp_list( @correctStringList )
 2074   Filters: ignore_order, ignore_case
 2075 
 2076  unordered_cs_str_cmp( $correctString )
 2077  unordered_cs_str_cmp_list( @correctStringList )
 2078   Filters: ignore_order
 2079 
 2080  ordered_str_cmp( $correctString )
 2081  ordered_str_cmp_list( @correctStringList )
 2082   Filters: remove_whitespace, ignore_case
 2083 
 2084  ordered_cs_str_cmp( $correctString )
 2085  ordered_cs_str_cmp_list( @correctStringList )
 2086   Filters: remove_whitespace
 2087 
 2088 Examples
 2089 
 2090   ANS( std_str_cmp( "W. Mozart" ) ) --  Accepts "W. Mozart", "W. MOZarT",
 2091     and so forth. Case insensitive. All internal spaces treated
 2092     as single spaces.
 2093   ANS( std_cs_str_cmp( "Mozart" ) ) --  Rejects "mozart". Same as
 2094     std_str_cmp() but case sensitive.
 2095   ANS( strict_str_cmp( "W. Mozart" ) )  --  Accepts only the exact string.
 2096   ANS( unordered_str_cmp( "ABC" ) ) --  Accepts "a c B", "CBA" and so forth.
 2097     Unordered, case insensitive, spaces ignored.
 2098   ANS( unordered_cs_str_cmp( "ABC" ) )  --  Rejects "abc". Same as
 2099     unordered_str_cmp() but case sensitive.
 2100   ANS( ordered_str_cmp( "ABC" ) ) --  Accepts "a b C", "A B C" and so forth.
 2101     Ordered, case insensitive, spaces ignored.
 2102   ANS( ordered_cs_str_cmp( "ABC" ) )  --  Rejects "abc", accepts "A BC" and
 2103     so forth. Same as ordered_str_cmp() but case sensitive.
 2104 
 2105 =cut
 2106 
 2107 sub std_str_cmp {         # compare strings
 2108   my $correctAnswer = shift @_;
 2109   my @filters = ( 'compress_whitespace', 'ignore_case' );
 2110   my $type = 'std_str_cmp';
 2111   STR_CMP(  'correctAnswer' =>  $correctAnswer,
 2112       'filters' =>  \@filters,
 2113       'type'    =>  $type
 2114   );
 2115 }
 2116 
 2117 sub std_str_cmp_list {        # alias for std_str_cmp
 2118   my @answerList = @_;
 2119   my @output;
 2120   while (@answerList) {
 2121     push( @output, std_str_cmp(shift @answerList) );
 2122   }
 2123   @output;
 2124 }
 2125 
 2126 sub std_cs_str_cmp {        # compare strings case sensitive
 2127   my $correctAnswer = shift @_;
 2128   my @filters = ( 'compress_whitespace' );
 2129   my $type = 'std_cs_str_cmp';
 2130   STR_CMP(  'correctAnswer' =>  $correctAnswer,
 2131       'filters' =>  \@filters,
 2132       'type'    =>  $type
 2133   );
 2134 }
 2135 
 2136 sub std_cs_str_cmp_list {     # alias for std_cs_str_cmp
 2137   my @answerList = @_;
 2138   my @output;
 2139   while (@answerList) {
 2140     push( @output, std_cs_str_cmp(shift @answerList) );
 2141   }
 2142   @output;
 2143 }
 2144 
 2145 sub strict_str_cmp {        # strict string compare
 2146   my $correctAnswer = shift @_;
 2147   my @filters = ( 'trim_whitespace' );
 2148   my $type = 'strict_str_cmp';
 2149   STR_CMP(  'correctAnswer' =>  $correctAnswer,
 2150       'filters' =>  \@filters,
 2151       'type'    =>  $type
 2152   );
 2153 }
 2154 
 2155 sub strict_str_cmp_list {     # alias for strict_str_cmp
 2156   my @answerList = @_;
 2157   my @output;
 2158   while (@answerList) {
 2159     push( @output, strict_str_cmp(shift @answerList) );
 2160   }
 2161   @output;
 2162 }
 2163 
 2164 sub unordered_str_cmp {       # unordered, case insensitive, spaces ignored
 2165   my $correctAnswer = shift @_;
 2166   my @filters = ( 'ignore_order', 'ignore_case' );
 2167   my $type = 'unordered_str_cmp';
 2168   STR_CMP(  'correctAnswer'   =>  $correctAnswer,
 2169       'filters'   =>  \@filters,
 2170       'type'      =>  $type
 2171   );
 2172 }
 2173 
 2174 sub unordered_str_cmp_list {    # alias for unordered_str_cmp
 2175   my @answerList = @_;
 2176   my @output;
 2177   while (@answerList) {
 2178     push( @output, unordered_str_cmp(shift @answerList) );
 2179   }
 2180   @output;
 2181 }
 2182 
 2183 sub unordered_cs_str_cmp {      # unordered, case sensitive, spaces ignored
 2184   my $correctAnswer = shift @_;
 2185   my @filters = ( 'ignore_order' );
 2186   my $type = 'unordered_cs_str_cmp';
 2187   STR_CMP(  'correctAnswer'   =>  $correctAnswer,
 2188       'filters'   =>  \@filters,
 2189       'type'      =>  $type
 2190   );
 2191 }
 2192 
 2193 sub unordered_cs_str_cmp_list {   # alias for unordered_cs_str_cmp
 2194   my @answerList = @_;
 2195   my @output;
 2196   while (@answerList) {
 2197     push( @output, unordered_cs_str_cmp(shift @answerList) );
 2198   }
 2199   @output;
 2200 }
 2201 
 2202 sub ordered_str_cmp {       # ordered, case insensitive, spaces ignored
 2203   my $correctAnswer = shift @_;
 2204   my @filters = ( 'remove_whitespace', 'ignore_case' );
 2205   my $type = 'ordered_str_cmp';
 2206   STR_CMP(  'correctAnswer' =>  $correctAnswer,
 2207       'filters' =>  \@filters,
 2208       'type'    =>  $type
 2209   );
 2210 }
 2211 
 2212 sub ordered_str_cmp_list {      # alias for ordered_str_cmp
 2213   my @answerList = @_;
 2214   my @output;
 2215   while (@answerList) {
 2216     push( @output, ordered_str_cmp(shift @answerList) );
 2217   }
 2218   @output;
 2219 }
 2220 
 2221 sub ordered_cs_str_cmp {      # ordered,  case sensitive, spaces ignored
 2222   my $correctAnswer = shift @_;
 2223   my @filters = ( 'remove_whitespace' );
 2224   my $type = 'ordered_cs_str_cmp';
 2225   STR_CMP(  'correctAnswer' =>  $correctAnswer,
 2226       'filters' =>  \@filters,
 2227       'type'    =>  $type
 2228   );
 2229 }
 2230 
 2231 sub ordered_cs_str_cmp_list {   # alias for ordered_cs_str_cmp
 2232   my @answerList = @_;
 2233   my @output;
 2234   while (@answerList) {
 2235     push( @output, ordered_cs_str_cmp(shift @answerList) );
 2236   }
 2237   @output;
 2238 }
 2239 
 2240 
 2241 ## LOW-LEVEL ROUTINE -- NOT NORMALLY FOR END USERS -- USE WITH CAUTION
 2242 ##
 2243 ## IN:  a hashtable with the following entries (error-checking to be added later?):
 2244 ##      correctAnswer --  the correct answer, before filtering
 2245 ##      filters     --  reference to an array containing the filters to be applied
 2246 ##      type      --  a string containing the type of answer evaluator in use
 2247 ## OUT: a reference to an answer evaluator subroutine
 2248 
 2249 sub STR_CMP {
 2250   my %str_params = @_;
 2251   $str_params{'correctAnswer'} = str_filters( $str_params{'correctAnswer'}, @{$str_params{'filters'}} );
 2252   my $answer_evaluator = sub {
 2253     my $in = shift @_;
 2254     $in = '' unless defined $in;
 2255     my $original_student_ans = $in;
 2256     $in = str_filters( $in, @{$str_params{'filters'}} );
 2257     my $correctQ = ( $in eq $str_params{'correctAnswer'} ) ? 1: 0;
 2258     my $ans_hash = new AnswerHash(    'score'       =>  $correctQ,
 2259               'correct_ans'     =>  $str_params{'correctAnswer'},
 2260               'student_ans'     =>  $in,
 2261               'ans_message'     =>  '',
 2262               'type'        =>  $str_params{'type'},
 2263               'preview_text_string'   =>  $in,
 2264               'preview_latex_string'    =>  $in,
 2265               'original_student_ans'    =>  $original_student_ans
 2266     );
 2267     return $ans_hash;
 2268   };
 2269   return $answer_evaluator;
 2270 }
 2271 
 2272 ##########################################################################
 2273 ##########################################################################
 2274 ## Miscellaneous answer evaluators
 2275 
 2276 =head2 Miscellaneous Answer Evaluators (Checkboxes and Radio Buttons)
 2277 
 2278 These evaluators do not fit any of the other categories.
 2279 
 2280 checkbox_cmp( $correctAnswer )
 2281 
 2282   $correctAnswer  --  a string containing the names of the correct boxes,
 2283             e.g. "ACD". Note that this means that individual
 2284             checkbox names can only be one character. Internally,
 2285             this is largely the same as unordered_cs_str_cmp().
 2286 
 2287 radio_cmp( $correctAnswer )
 2288 
 2289   $correctAnswer  --  a string containing the name of the correct radio
 2290             button, e.g. "Choice1". This is case sensitive and
 2291             whitespace sensitive, so the correct answer must match
 2292             the name of the radio button exactly.
 2293 
 2294 =cut
 2295 
 2296 # added 6/14/2000 by David Etlinger
 2297 # because of the conversion of the answer
 2298 # string to an array, I thought it better not
 2299 # to force STR_CMP() to work with this
 2300 sub checkbox_cmp {
 2301   my  $correctAnswer = shift @_;
 2302   $correctAnswer = str_filters( $correctAnswer, 'ignore_order' );
 2303 
 2304   my  $answer_evaluator = sub {
 2305     my $in = shift @_;
 2306     $in = '' unless defined $in;      #in case no boxes checked
 2307 
 2308     my @temp = split( "\0", $in );      #convert "\0"-delimited string to array...
 2309     $in = join( "", @temp );        #and then to a single no-delimiter string
 2310 
 2311     my $original_student_ans = $in;     #well, almost original
 2312     $in = str_filters( $in, 'ignore_order' );
 2313 
 2314     my $correctQ = ($in eq $correctAnswer) ? 1: 0;
 2315 
 2316     my $ans_hash = new AnswerHash(
 2317               'score'     =>  $correctQ,
 2318               'correct_ans'   =>  $correctAnswer,
 2319               'student_ans'   =>  $in,
 2320               'ans_message'   =>  "",
 2321               'type'      =>  "checkbox_cmp",
 2322               'preview_text_string' =>  $in,
 2323               'original_student_ans'  =>  $original_student_ans
 2324     );
 2325     return $ans_hash;
 2326 
 2327   };
 2328   return $answer_evaluator;
 2329 }
 2330 
 2331 #added 6/28/2000 by David Etlinger
 2332 #exactly the same as strict_str_cmp,
 2333 #but more intuitive to the user
 2334 sub radio_cmp {
 2335   strict_str_cmp( @_ );
 2336 }
 2337 
 2338 ##########################################################################
 2339 ##########################################################################
 2340 ## Text and e-mail routines
 2341 
 2342 sub store_ans_at {
 2343   my $answerStringRef = shift;
 2344   my %options = @_;
 2345   my $ans_eval= '';
 2346   if ( ref($answerStringRef) eq 'SCALAR' ) {
 2347     $ans_eval= sub {
 2348       my $text = shift;
 2349       $text = '' unless defined($text);
 2350       $$answerStringRef = $$answerStringRef  . $text;
 2351       my $ans_hash = new AnswerHash(
 2352                'score'      =>  1,
 2353                'correct_ans'      =>  '',
 2354                'student_ans'      =>  $text,
 2355                'ans_message'      =>  '',
 2356                'type'       =>  'store_ans_at',
 2357                'original_student_ans'   =>  $text,
 2358                'preview_text_string'    =>  ''
 2359       );
 2360 
 2361     return $ans_hash;
 2362     };
 2363   }
 2364   else {
 2365     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";
 2366   }
 2367 
 2368   return $ans_eval;
 2369 }
 2370 
 2371 #### subroutines used in producing a questionnaire
 2372 #### these are at least good models for other answers of this type
 2373 
 2374 my $QUESTIONNAIRE_ANSWERS=''; #  stores the answers until it is time to send them
 2375        #  this must be initialized before the answer evaluators are run
 2376        #  but that happens long after all of the text in the problem is
 2377        #  evaluated.
 2378 # this is a utility script for cleaning up the answer output for display in
 2379 #the answers.
 2380 
 2381 sub DUMMY_ANSWER {
 2382   my $num = shift;
 2383   qq{<INPUT TYPE="HIDDEN" NAME="answer$num" VALUE="">}
 2384 }
 2385 
 2386 sub escapeHTML {
 2387   my $string = shift;
 2388   $string =~ s/\n/$BR/ge;
 2389   $string;
 2390 }
 2391 
 2392 # these next three subroutines show how to modify the "store_ans_at()" answer
 2393 # evaluator to add extra information before storing the info
 2394 # They provide a good model for how to tweak answer evaluators in special cases.
 2395 
 2396 sub anstext {
 2397   my $num = shift;
 2398   my $ans_eval_template = store_ans_at(\$QUESTIONNAIRE_ANSWERS);
 2399   my $ans_eval = sub {
 2400          my $text = shift;
 2401          $text = '' unless defined($text);
 2402          my $new_text = "\n$main::psvnNumber-Problem-$main::probNum-Question-$num:\n $text "; # modify entered text
 2403          my $out = &$ans_eval_template($new_text);       # standard evaluator
 2404          #warn "$QUESTIONNAIRE_ANSWERS";
 2405          $out->{student_ans} = escapeHTML($text);  #  restore original entered text
 2406          $out->{correct_ans} = "Question  $num answered";
 2407          $out->{original_student_ans} = escapeHTML($text);
 2408          $out;
 2409     };
 2410    $ans_eval;
 2411 }
 2412 
 2413 sub ansradio {
 2414   my $num = shift;
 2415   my $ans_eval_template = store_ans_at(\$QUESTIONNAIRE_ANSWERS);
 2416   my $ans_eval = sub {
 2417          my $text = shift;
 2418          $text = '' unless defined($text);
 2419          my $new_text = "\n$main::psvnNumber-Problem-$main::probNum-RADIO-$num:\n $text ";       # modify entered text
 2420          my $out = $ans_eval_template->($new_text);       # standard evaluator
 2421          $out->{student_ans} =escapeHTML($text);  # restore original entered text
 2422          $out->{original_student_ans} = escapeHTML($text);
 2423          $out;
 2424    };
 2425 
 2426    $ans_eval;
 2427 }
 2428 
 2429 sub anstext_non_anonymous {
 2430   ## this emails identifying information
 2431   my $num = shift;
 2432   my $ans_eval_template = store_ans_at(\$QUESTIONNAIRE_ANSWERS);
 2433   my $ans_eval = sub {
 2434          my $text = shift;
 2435          $text = '' unless defined($text);
 2436          my $new_text = "\n$main::psvnNumber-Problem-$main::probNum-Question-$num:\n$main::studentLogin $main::studentID $main::studentName\n$text "; # modify entered text
 2437          my $out = &$ans_eval_template($new_text);       # standard evaluator
 2438          #warn "$QUESTIONNAIRE_ANSWERS";
 2439          $out->{student_ans} = escapeHTML($text);  #  restore original entered text
 2440          $out->{correct_ans} = "Question  $num answered";
 2441          $out->{original_student_ans} = escapeHTML($text);
 2442          $out;
 2443     };
 2444    $ans_eval;
 2445 }
 2446 
 2447 
 2448 #  This is another example of how to modify an  answer evaluator to obtain
 2449 #  the desired behavior in a special case.  Here the object is to have
 2450 #  have the last answer trigger the send_mail_to subroutine which mails
 2451 #  all of the answers to the designated address.
 2452 #  (This address must be listed in PG_environment{'ALLOW_MAIL_TO'} or an error occurs.)
 2453 
 2454 sub mail_answers_to {  #accepts the last answer and mails off the result
 2455   my $user_address = shift;
 2456   my $ans_eval = sub {
 2457 
 2458     # then mail out all of the answers, including this last one.
 2459 
 2460     send_mail_to( $user_address,
 2461           'subject'   =>  "$main::courseName WeBWorK questionnaire",
 2462           'body'      =>  $QUESTIONNAIRE_ANSWERS,
 2463           'ALLOW_MAIL_TO'   =>  $envir{ALLOW_MAIL_TO}
 2464     );
 2465 
 2466     my $ans_hash = new AnswerHash(  'score'   =>  1,
 2467             'correct_ans' =>  '',
 2468             'student_ans' =>  'Answer recorded',
 2469             'ans_message' =>  '',
 2470             'type'    =>  'send_mail_to',
 2471     );
 2472 
 2473     return $ans_hash;
 2474   };
 2475 
 2476   return $ans_eval;
 2477 }
 2478 sub mail_answers_to2 {  #accepts the last answer and mails off the result
 2479   my $user_address = shift;
 2480   my $subject = shift;
 2481   $subject = "$main::courseName WeBWorK questionnaire" unless defined $subject;
 2482 
 2483   send_mail_to($user_address,
 2484       'subject'     => $subject,
 2485       'body'        => $QUESTIONNAIRE_ANSWERS,
 2486       'ALLOW_MAIL_TO'   => $envir{ALLOW_MAIL_TO}
 2487   );
 2488 }
 2489 
 2490 ##########################################################################
 2491 ##########################################################################
 2492 
 2493 
 2494 ###########################################################################
 2495 ### THE FOLLOWING ARE LOCAL SUBROUTINES THAT ARE MEANT TO BE CALLED ONLY FROM THIS SCRIPT.
 2496 
 2497 ## Internal routine that converts variables into the standard array format
 2498 ##
 2499 ## IN:  one of the following:
 2500 ##      an undefined value (i.e., no variable was specified)
 2501 ##      a reference to an array of variable names -- [var1, var2]
 2502 ##      a number (the number of variables desired) -- 3
 2503 ##      one or more variable names -- (var1, var2)
 2504 ## OUT: an array of variable names
 2505 
 2506 sub get_var_array {
 2507   my $in = shift @_;
 2508   my @out;
 2509 
 2510   if( not defined($in) ) {      #if nothing defined, build default array and return
 2511     @out = ( $functVarDefault );
 2512     return @out;
 2513   }
 2514   elsif( ref( $in ) eq 'ARRAY' ) {  #if given an array ref, dereference and return
 2515     return @{$in};
 2516   }
 2517   elsif( $in =~ /^\d+/ ) {      #if given a number, set up the array and return
 2518     if( $in == 1 ) {
 2519       $out[0] = 'x';
 2520     }
 2521     elsif( $in == 2 ) {
 2522       $out[0] = 'x';
 2523       $out[1] = 'y';
 2524     }
 2525     elsif( $in == 3 ) {
 2526       $out[0] = 'x';
 2527       $out[1] = 'y';
 2528       $out[2] = 'z';
 2529     }
 2530     else {  #default to the x_1, x_2, ... convention
 2531       my ($i, $tag);
 2532       for( $i=0; $i < $in; $i++ ) {
 2533               ## akp the above seems to be off by one 1/4/00
 2534         $tag = $i + 1;                            ## akp 1/4/00
 2535         $out[$i] = "${functVarDefault}_" . $tag;              ## akp 1/4/00
 2536       }
 2537     }
 2538     return @out;
 2539   }
 2540   else {            #if given one or more names, return as an array
 2541     unshift( @_, $in );
 2542     return @_;
 2543   }
 2544 }
 2545 
 2546 ## Internal routine that converts limits into the standard array of arrays format
 2547 ##  Some of the cases are probably unneccessary, but better safe than sorry
 2548 ##
 2549 ## IN:  one of the following:
 2550 ##      an undefined value (i.e., no limits were specified)
 2551 ##      a reference to an array of arrays of limits -- [[llim,ulim], [llim,ulim]]
 2552 ##      a reference to an array of limits -- [llim, ulim]
 2553 ##      an array of array references -- ([llim,ulim], [llim,ulim])
 2554 ##      an array of limits -- (llim,ulim)
 2555 ## OUT: an array of array references -- ([llim,ulim], [llim,ulim]) or ([llim,ulim])
 2556 
 2557 sub get_limits_array {
 2558   my $in = shift @_;
 2559   my @out;
 2560 
 2561   if( not defined($in) ) {        #if nothing defined, build default array and return
 2562     @out = ( [$functLLimitDefault, $functULimitDefault] );
 2563     return @out;
 2564   }
 2565   elsif( ref($in) eq 'ARRAY' ) {        #$in is either ref to array, or ref to array of refs
 2566     my @deref = @{$in};
 2567 
 2568     if( ref( $in->[0] ) eq 'ARRAY' ) {    #$in is a ref to an array of array refs
 2569       return @deref;
 2570     }
 2571     else {            #$in was just a ref to an array of numbers
 2572       @out = ( $in );
 2573       return @out;
 2574     }
 2575   }
 2576   else {              #$in was an array of references or numbers
 2577     unshift( @_, $in );
 2578 
 2579     if( ref($_[0]) eq 'ARRAY' ) {     #$in was an array of references, so just return it
 2580       return @_;
 2581     }
 2582     else {            #$in was an array of numbers
 2583       @out = ( \@_ );
 2584       return @out;
 2585     }
 2586   }
 2587 }
 2588 
 2589 #sub check_option_list {
 2590 # my $size = scalar(@_);
 2591 # if( ( $size % 2 ) != 0 ) {
 2592 #   warn "ERROR in answer evaluator generator:\n" .
 2593 #     "Usage: <CODE>str_cmp([\$ans1,  \$ans2],%options)</CODE>
 2594 #     or <CODE> num_cmp([\$num1, \$num2], %options)</CODE><BR>
 2595 #     A list of inputs must be inclosed in square brackets <CODE>[\$ans1, \$ans2]</CODE>";
 2596 # }
 2597 #}
 2598 
 2599 # simple subroutine to display an error message when
 2600 # function compares are called with invalid parameters
 2601 sub function_invalid_params {
 2602   my $correctEqn = shift @_;
 2603   my $error_response = sub {
 2604     my $PGanswerMessage = "Tell your professor that there is an error with the parameters " .
 2605             "to the function answer evaluator";
 2606     return ( 0, $correctEqn, "", $PGanswerMessage );
 2607   };
 2608   return $error_response;
 2609 }
 2610 
 2611 sub clean_up_error_msg {
 2612   my $msg = $_[0];
 2613   $msg =~ s/^\[[^\]]*\][^:]*://;
 2614   $msg =~ s/Unquoted string//g;
 2615   $msg =~ s/may\s+clash.*/does not make sense here/;
 2616   $msg =~ s/\sat.*line [\d]*//g;
 2617   $msg = 'error: '. $msg;
 2618 
 2619   return $msg;
 2620 }
 2621 
 2622 #formats the student and correct answer as specified
 2623 #format must be of a form suitable for sprintf (e.g. '%0.5g'),
 2624 #with the exception that a '#' at the end of the string
 2625 #will cause trailing zeros in the decimal part to be removed
 2626 sub prfmt {
 2627   my($number,$format) = @_;  # attention, the order of format and number are reversed
 2628   my $out;
 2629   if ($format) {
 2630     warn "Incorrect format used: $format. <BR> Format should look something like %4.5g<BR>"
 2631                 unless $format =~ /^\s*%\d*\.?\d*\w#?\s*$/;
 2632 
 2633     if( $format =~ s/#\s*$// ) {  # remove trailing zeros in the decimal
 2634       $out = sprintf( $format, $number );
 2635       $out =~ s/(\.\d*?)0+$/$1/;
 2636       $out =~ s/\.$//;      # in case all decimal digits were zero, remove the decimal
 2637       $out =~ s/e/E/g;        # only use capital E's for exponents. Little e is for 2.71828...
 2638     } elsif (is_a_number($number) ){
 2639       $out = sprintf( $format, $number );
 2640       $out =~ s/e/E/g;        # only use capital E's for exponents. Little e is for 2.71828...
 2641     } else { # number is probably a string representing an arithmetic expression
 2642       $out = $number;
 2643     }
 2644 
 2645   } else {
 2646     if (is_a_number($number)) {# only use capital E's for exponents. Little e is for 2.71828...
 2647       $out = $number;
 2648       $out =~ s/e/E/g;
 2649     } else { # number is probably a string representing an arithmetic expression
 2650       $out = $number;
 2651     }
 2652   }
 2653   return $out;
 2654 }
 2655 #########################################################################
 2656 # Filters for answer evaluators
 2657 #########################################################################
 2658 
 2659 =head2 Filters
 2660 
 2661 =pod
 2662 
 2663 A filter is a short subroutine with the following structure.  It accepts an
 2664 AnswerHash, followed by a hash of options.  It returns an AnswerHash
 2665 
 2666   $ans_hash = filter($ans_hash, %options);
 2667 
 2668 See the AnswerHash.pm file for a list of entries which can be expected to be found
 2669 in an AnswerHash, such as 'student_ans', 'score' and so forth.  Other entries
 2670 may be present for specialized answer evaluators.
 2671 
 2672 The hope is that a well designed set of filters can easily be combined to form
 2673 a new answer_evaluator and that this method will produce answer evaluators which are
 2674 are more robust than the method of copying existing answer evaluators and modifying them.
 2675 
 2676 Here is an outline of how a filter is constructed:
 2677 
 2678   sub filter{
 2679     my $rh_ans = shift;
 2680     my %options = @_;
 2681     assign_option_aliases(\%options,
 2682         'alias1'  => 'option5'
 2683         'alias2'  => 'option7'
 2684     );
 2685     set_default_options(\%options,
 2686         '_filter_name'  =>  'filter',
 2687         'option5'   =>  .0001,
 2688         'option7'   =>  'ascii',
 2689         'allow_unknown_options  =>  0,
 2690     }
 2691     .... body code of filter .......
 2692       if ($error) {
 2693         $rh_ans->throw_error("FILTER_ERROR", "Something went wrong");
 2694         # see AnswerHash.pm for details on using the throw_error method.
 2695 
 2696     $rh_ans;  #reference to an AnswerHash object is returned.
 2697   }
 2698 
 2699 =cut
 2700 
 2701 =head4 compare_numbers
 2702 
 2703 
 2704 =cut
 2705 
 2706 
 2707 sub compare_numbers {
 2708   my ($rh_ans, %options) = @_;
 2709   my ($inVal,$PG_eval_errors,$PG_full_error_report) = PG_answer_eval($rh_ans->{student_ans});
 2710   if ($PG_eval_errors) {
 2711     $rh_ans->throw_error('EVAL','There is a syntax error in your answer');
 2712     $rh_ans->{ans_message} = clean_up_error_msg($PG_eval_errors);
 2713     # return $rh_ans;
 2714   } else {
 2715     $rh_ans->{student_ans} = prfmt($inVal,$options{format});
 2716   }
 2717 
 2718   my $permitted_error;
 2719 
 2720   if ($rh_ans->{tolType} eq 'absolute') {
 2721     $permitted_error = $rh_ans->{tolerance};
 2722   }
 2723   elsif ( abs($rh_ans->{correct_ans}) <= $options{zeroLevel}) {
 2724       $permitted_error = $options{zeroLevelTol};  ## want $tol to be non zero
 2725   }
 2726   else {
 2727     $permitted_error = abs($rh_ans->{tolerance}*$rh_ans->{correct_ans});
 2728   }
 2729 
 2730   my $is_a_number = is_a_number($inVal);
 2731   $rh_ans->{score} = 1 if ( ($is_a_number) and
 2732       (abs( $inVal - $rh_ans->{correct_ans} ) <= $permitted_error) );
 2733   if (not $is_a_number) {
 2734     $rh_ans->{error_message} = "$rh_ans->{error_message}". 'Your answer does not evaluate to a number ';
 2735   }
 2736 
 2737   $rh_ans;
 2738 }
 2739 
 2740 =head4 std_num_filter
 2741 
 2742   std_num_filter($rh_ans, %options)
 2743   returns $rh_ans
 2744 
 2745 Replaces some constants using math_constants, then evaluates a perl expression.
 2746 
 2747 
 2748 =cut
 2749 
 2750 sub std_num_filter {
 2751   my $rh_ans = shift;
 2752   my %options = @_;
 2753   my $in = $rh_ans->input();
 2754   $in = math_constants($in);
 2755   $rh_ans->{type} = 'std_number';
 2756   my ($inVal,$PG_eval_errors,$PG_full_error_report);
 2757   if ($in =~ /\S/) {
 2758     ($inVal,$PG_eval_errors,$PG_full_error_report) = PG_answer_eval($in);
 2759   } else {
 2760     $PG_eval_errors = '';
 2761   }
 2762 
 2763   if ($PG_eval_errors) {        ##error message from eval or above
 2764     $rh_ans->{ans_message} = 'There is a syntax error in your answer';
 2765     $rh_ans->{student_ans} = clean_up_error_msg($PG_eval_errors);
 2766   } else {
 2767     $rh_ans->{student_ans} = $inVal;
 2768   }
 2769   $rh_ans;
 2770 }
 2771 
 2772 =head std_num_array_filter
 2773 
 2774   std_num_array_filter($rh_ans, %options)
 2775   returns $rh_ans
 2776 
 2777 Assumes the {student_ans} field is a numerical  array, and applies BOTH check_syntax and std_num_filter
 2778 to each element of the array.  Does it's best to generate sensible error messages for syntax errors.
 2779 A typical error message displayed in {studnet_ans} might be ( 56, error message, -4).
 2780 
 2781 =cut
 2782 
 2783 sub std_num_array_filter {
 2784   my $rh_ans= shift;
 2785   my %options = @_;
 2786   set_default_options(  \%options,
 2787         '_filter_name'  =>  'std_num_array_filter',
 2788     );
 2789   my @in = @{$rh_ans->{student_ans}};
 2790   my $temp_hash = new AnswerHash;
 2791   my @out=();
 2792   my $PGanswerMessage = '';
 2793   foreach my $item (@in)   {  # evaluate each number in the vector
 2794     $temp_hash->input($item);
 2795     $temp_hash = check_syntax($temp_hash);
 2796     if (defined($temp_hash->{error_flag}) and $temp_hash->{error_flag} eq 'SYNTAX') {
 2797       $PGanswerMessage .= $temp_hash->{ans_message};
 2798       $temp_hash->{ans_message} = undef;
 2799     } else {
 2800       #continue processing
 2801       $temp_hash = std_num_filter($temp_hash);
 2802       if (defined($temp_hash->{ans_message}) and $temp_hash->{ans_message} ) {
 2803         $PGanswerMessage .= $temp_hash->{ans_message};
 2804         $temp_hash->{ans_message} = undef;
 2805       }
 2806     }
 2807     push(@out, $temp_hash->input());
 2808 
 2809   }
 2810   if ($PGanswerMessage) {
 2811     $rh_ans->input( "( " . join(", ", @out ) . " )" );
 2812         $rh_ans->throw_error('SYNTAX', 'There is a syntax error in your answer.');
 2813   } else {
 2814     $rh_ans->input( [@out] );
 2815   }
 2816   $rh_ans;
 2817 }
 2818 
 2819 =head4 function_from_string2
 2820 
 2821 
 2822 
 2823 =cut
 2824 
 2825 sub function_from_string2 {
 2826     my $rh_ans = shift;
 2827     my %options = @_;
 2828   my $eqn = $rh_ans->{student_ans};
 2829   assign_option_aliases(\%options,
 2830         'vars'      => 'ra_vars',
 2831         'var'           => 'ra_vars',
 2832   );
 2833   set_default_options(  \%options,
 2834               'store_in'    =>      'rf_student_ans',
 2835           'ra_vars'   =>  [qw( x y )],
 2836           'debug'     =>  0,
 2837           '_filter_name'  =>  'function_from_string2',
 2838     );
 2839     $rh_ans->{_filter_name} = $options{_filter_name};
 2840     my @VARS = @{ $options{ 'ra_vars'}};
 2841     #warn "VARS = ", join("<>", @VARS) if defined($options{debug}) and $options{debug} ==1;
 2842     my $originalEqn = $eqn;
 2843     $eqn = &math_constants($eqn);
 2844     for( my $i = 0; $i < @VARS; $i++ ) {
 2845         #  This next line is a hack required for 5.6.0 -- it doesn't appear to be needed in 5.6.1
 2846         my ($temp,$er1,$er2) = PG_restricted_eval('"'. $VARS[$i] . '"');
 2847   #$eqn =~ s/\b$VARS[$i]\b/\$VARS[$i]/g;
 2848         $eqn  =~ s/\b$temp\b/\$VARS[$i]/g;
 2849 
 2850   }
 2851   #warn "equation evaluated = $eqn",$rh_ans->pretty_print(), "<br>\noptions<br>\n",
 2852   #     pretty_print(\%options)
 2853   #     if defined($options{debug}) and $options{debug} ==1;
 2854     my ($function_sub,$PG_eval_errors, $PG_full_errors) = PG_answer_eval( q!
 2855       sub {
 2856         my @VARS = @_;
 2857         my $input_str = '';
 2858         for( my $i=0; $i<@VARS; $i++ ) {
 2859           $input_str .= "\$VARS[$i] = $VARS[$i]; ";
 2860         }
 2861         my $PGanswerMessage;
 2862         $input_str .= '! . $eqn . q!';  # need the single quotes to keep the contents of $eqn from being
 2863                                         # evaluated when it is assigned to $input_str;
 2864         my ($out, $PG_eval_errors, $PG_full_errors) = PG_answer_eval($input_str); #Finally evaluated
 2865 
 2866         if ( defined($PG_eval_errors) and $PG_eval_errors =~ /\S/ ) {
 2867             $PGanswerMessage  = clean_up_error_msg($PG_eval_errors);
 2868 # This message seemed too verbose, but it does give extra information, we'll see if it is needed.
 2869 #                    "<br> There was an error in evaluating your function <br>
 2870 #           !. $originalEqn . q! <br>
 2871 #           at ( " . join(', ', @VARS) . " ) <br>
 2872 #            $PG_eval_errors
 2873 #           ";   # this message appears in the answer section which is not process by Latex2HTML so it must
 2874 #                # be in HTML.  That is why $BR is NOT used.
 2875 
 2876       }
 2877       (wantarray) ? ($out, $PGanswerMessage): $out;   # PGanswerMessage may be undefined.
 2878       };
 2879   !);
 2880 
 2881   if (defined($PG_eval_errors) and $PG_eval_errors =~/\S/ ) {
 2882         $PG_eval_errors = clean_up_error_msg($PG_eval_errors);
 2883 
 2884     my $PGanswerMessage = "There was an error in converting the expression
 2885       $main::BR $originalEqn $main::BR into a function.
 2886       $main::BR $PG_eval_errors.";
 2887     $rh_ans->{rf_student_ans} = $function_sub;
 2888     $rh_ans->{ans_message} = $PGanswerMessage;
 2889     $rh_ans->{error_message} = $PGanswerMessage;
 2890     $rh_ans->{error_flag} = 1;
 2891      # we couldn't compile the equation, we'll return an error message.
 2892   } else {
 2893 #     if (defined($options{store_in} )) {
 2894 #       $rh_ans ->{$options{store_in}} = $function_sub;
 2895 #     } else {
 2896 #         $rh_ans->{rf_student_ans} = $function_sub;
 2897 #       }
 2898       $rh_ans ->{$options{store_in}} = $function_sub;
 2899   }
 2900 
 2901     $rh_ans;
 2902 }
 2903 
 2904 =head4 is_zero_array
 2905 
 2906 
 2907 =cut
 2908 
 2909 
 2910 sub is_zero_array {
 2911     my $rh_ans = shift;
 2912     my %options = @_;
 2913     set_default_options(  \%options,
 2914         '_filter_name'  =>  'is_zero_array',
 2915         'tolerance' =>  0.000001,
 2916     );
 2917     my $array = $rh_ans -> {ra_differences};
 2918   my $num = @$array;
 2919   my $i;
 2920   my $max = 0; my $mm;
 2921   for ($i=0; $i< $num; $i++) {
 2922     $mm = $array->[$i] ;
 2923     if  (not is_a_number($mm) ) {
 2924       $max = $mm;  # break out if one of the elements is not a number
 2925       last;
 2926     }
 2927     $max = abs($mm) if abs($mm) > $max;
 2928   }
 2929   if (not is_a_number($max)) {
 2930     $rh_ans->{score} = 0;
 2931       my $error = "WeBWorK was unable evaluate your function. Please check that your
 2932                 expression doesn't take roots of negative numbers, or divide by zero.";
 2933     $rh_ans->throw_error('EVAL',$error);
 2934   } else {
 2935       $rh_ans->{score} = ($max < $options{tolerance} ) ? 1: 0;       # 1 if the array is close to 0;
 2936   }
 2937   $rh_ans;
 2938 }
 2939 
 2940 =head4 best_approx_parameters
 2941 
 2942   best_approx_parameters($rh_ans,%options);   #requires the following fields in $rh_ans
 2943                         {rf_student_ans}      # reference to the test answer
 2944                         {rf_correct_ans}      # reference to the comparison answer
 2945                         {evaluation_points},  # an array of row vectors indicating the points
 2946                                       # to evaluate when comparing the functions
 2947 
 2948                          %options       # debug => 1   gives more error answers
 2949                                     # param_vars => ['']  additional parameters used to adapt to function
 2950                          )
 2951 
 2952 
 2953 The parameters for the comparison function which best approximates the test_function are stored
 2954 in the field {ra_parameters}.
 2955 
 2956 
 2957 The last $dim_of_parms_space variables are assumed to be parameters, and it is also
 2958 assumed that the function \&comparison_fun
 2959 depends linearly on these variables.  This function finds the  values for these parameters which minimizes the
 2960 Euclidean distance (L2 distance) between the test function and the comparison function and the test points specified
 2961 by the array reference  \@rows_of_test_points.  This is assumed to be an array of arrays, with the inner arrays
 2962 determining a test point.
 2963 
 2964 The comparison function should have $dim_of_params_space more input variables than the test function.
 2965 
 2966 
 2967 
 2968 
 2969 
 2970 =cut
 2971 
 2972 # Used internally:
 2973 #
 2974 #   &$determine_param_coeff( $rf_comparison_function # a reference to the correct answer function
 2975 #                    $ra_variables                   # an array of the active input variables to the functions
 2976 #                    $dim_of_params_space            # indicates the number of parameters upon which the
 2977 #                                                    # the comparison function depends linearly.  These are assumed to
 2978 #                                                    # be the last group of inputs to the comparison function.
 2979 #
 2980 #                    %options                        # $options{debug} gives more error messages
 2981 #
 2982 #                                                    # A typical function might look like
 2983 #                                                    # f(x,y,z,a,b) = x^2+a*cos(xz) + b*sin(x) with a parameter
 2984 #                                                    # space of dimension 2 and a variable space of dimension 3.
 2985 #                   )
 2986 #         # returns a list of coefficients
 2987 
 2988 sub best_approx_parameters {
 2989     my $rh_ans = shift;
 2990     my %options = @_;
 2991     set_default_options(\%options,
 2992         '_filter_name'      =>  'best_approx_paramters',
 2993         'allow_unknown_options' =>  1,
 2994     );
 2995     my $errors = undef;
 2996     # This subroutine for the determining the coefficents of the parameters at a given point
 2997     # is pretty specialized, so it is included here as a sub-subroutine.
 2998     my $determine_param_coeffs  = sub {
 2999     my ($rf_fun, $ra_variables, $dim_of_params_space, %options) =@_;
 3000     my @zero_params=();
 3001     for(my $i=1;$i<=$dim_of_params_space;$i++){push(@zero_params,0); }
 3002     my @vars = @$ra_variables;
 3003     my @coeff = ();
 3004     my @inputs = (@vars,@zero_params);
 3005     my ($f0, $f1, $err);
 3006     ($f0, $err) = &{$rf_fun}(@inputs);
 3007     if (defined($err) ) {
 3008       $errors .= "$err ";
 3009     } else {
 3010       for (my $i=@vars;$i<@inputs;$i++) {
 3011         $inputs[$i]=1;  # set one parameter to 1;
 3012         my($f1,$err) = &$rf_fun(@inputs);
 3013         if (defined($err) ) {
 3014           $errors .= " $err ";
 3015         } else {
 3016           push(@coeff, $f1-$f0);
 3017         }
 3018         $inputs[$i]=0;  # set it back
 3019       }
 3020     }
 3021     (\@coeff, $errors);
 3022   };
 3023     my $rf_fun = $rh_ans->{rf_student_ans};
 3024     my $rf_correct_fun = $rh_ans->{rf_correct_ans};
 3025     my $ra_vars_matrix = $rh_ans->{evaluation_points};
 3026     my $dim_of_param_space = @{$options{param_vars}};
 3027     # Short cut.  Bail if there are no param_vars
 3028     unless ($dim_of_param_space >0) {
 3029     $rh_ans ->{ra_parameters} = [];
 3030     return $rh_ans;
 3031     }
 3032     # inputs are row arrays in this case.
 3033     my @zero_params=();
 3034 
 3035     for(my $i=1;$i<=$dim_of_param_space;$i++){push(@zero_params,0); }
 3036     my @rows_of_vars = @$ra_vars_matrix;
 3037     warn "input rows ", pretty_print(\@rows_of_vars) if defined($options{debug}) and $options{debug};
 3038     my $rows = @rows_of_vars;
 3039     my $matrix =new Matrix($rows,$dim_of_param_space);
 3040     my $rhs_vec = new Matrix($rows, 1);
 3041     my $row_num = 1;
 3042     my ($ra_coeff,$val2, $val1, $err1,$err2,@inputs,@vars);
 3043     my $number_of_data_points = $dim_of_param_space +2;
 3044     while (@rows_of_vars and $row_num <= $number_of_data_points) {
 3045      # get one set of data points from the test function;
 3046       @vars = @{ shift(@rows_of_vars) };
 3047       ($val2, $err1) = &{$rf_fun}(@vars);
 3048       $errors .= " $err1 "  if defined($err1);
 3049       @inputs = (@vars,@zero_params);
 3050       ($val1, $err2) = &{$rf_correct_fun}(@inputs);
 3051       $errors .= " $err2 " if defined($err2);
 3052 
 3053       unless (defined($err1) or defined($err2) ) {
 3054           $rhs_vec->assign($row_num,1, $val2-$val1 );
 3055 
 3056     # warn "rhs data  val1=$val1, val2=$val2, val2 - val1 = ", $val2 - $val1 if $options{debug};
 3057     # warn "vars ", join(" | ", @vars) if $options{debug};
 3058 
 3059       ($ra_coeff, $err1) = &{$determine_param_coeffs}($rf_correct_fun,\@vars,$dim_of_param_space,%options);
 3060       if (defined($err1) ) {
 3061         $errors .= " $err1 ";
 3062       } else {
 3063         my @coeff = @$ra_coeff;
 3064         my $col_num=1;
 3065           while(@coeff) {
 3066             $matrix->assign($row_num,$col_num, shift(@coeff) );
 3067             $col_num++;
 3068           }
 3069         }
 3070       }
 3071       $row_num++;
 3072       last if $errors;  # break if there are any errors.
 3073                       # This cuts down on the size of error messages.
 3074                       # However it impossible to check for equivalence at 95% of points
 3075             # which might be useful for functions that are not defined at some points.
 3076   }
 3077     warn "<br> best_approx_parameters: matrix1 <br>  ", " $matrix " if $options{debug};
 3078     warn "<br> best_approx_parameters: vector <br>  ", " $rhs_vec " if $options{debug};
 3079 
 3080    # we have   Matrix * parameter = data_vec + perpendicular vector
 3081    # where the matrix has column vectors defining the span of the parameter space
 3082    # multiply both sides by Matrix_transpose and solve for the parameters
 3083    # This is exactly what the method proj_coeff method does.
 3084    my @array;
 3085    if (defined($errors) ) {
 3086     @array = ();   #     new Matrix($dim_of_param_space,1);
 3087    } else {
 3088     @array = $matrix->proj_coeff($rhs_vec)->list();
 3089    }
 3090   # check size (hack)
 3091   my $max = 0;
 3092   foreach my $val (@array ) {
 3093     $max = abs($val) if  $max < abs($val);
 3094     if (not is_a_number($val) ) {
 3095       $max = "NaN: $val";
 3096       last;
 3097     }
 3098   }
 3099   if ($max =~/NaN/) {
 3100     $errors .= "WeBWorK was unable evaluate your function. Please check that your
 3101                 expression doesn't take roots of negative numbers, or divide by zero.";
 3102   } elsif ($max > $options{maxConstantOfIntegration} ) {
 3103     $errors .= "At least one of the adapting parameters
 3104              (perhaps the constant of integration) is too large: $max,
 3105              ( the maximum allowed is $options{maxConstantOfIntegration} )";
 3106   }
 3107 
 3108     $rh_ans->{ra_parameters} = \@array;
 3109     $rh_ans->throw_error('EVAL', $errors) if defined($errors);
 3110     $rh_ans;
 3111 }
 3112 
 3113 =head4 calculate_difference_vector
 3114 
 3115   calculate_difference_vector( $ans_hash, %options);
 3116 
 3117                 {rf_student_ans},     # a reference to the test function
 3118                                {rf_correct_ans},      # a reference to the correct answer function
 3119                                {evaluation_points},   # an array of row vectors indicating the points
 3120                                           # to evaluate when comparing the functions
 3121                                {ra_parameters}        # these are the (optional) additional inputs to
 3122                                                       # the comparison function which adapt it properly
 3123                                                       # to the problem at hand.
 3124 
 3125                                %options               # mode => 'rel'  specifies that each element in the
 3126                                                       # difference matrix is divided by the correct answer.
 3127                                                       # unless the correct answer is nearly 0.
 3128                               )
 3129 
 3130 =cut
 3131 
 3132 sub calculate_difference_vector {
 3133   my $rh_ans = shift;
 3134   my %options = @_;
 3135   # initialize
 3136   my $rf_fun = $rh_ans -> {rf_student_ans};
 3137   my $rf_correct_fun = $rh_ans -> {rf_correct_ans};
 3138   my $ra_parameters = $rh_ans ->{ra_parameters};
 3139   my @evaluation_points = @{$rh_ans->{evaluation_points} };
 3140   my @parameters = ();
 3141   @parameters = @$ra_parameters if defined($ra_parameters) and ref($ra_parameters) eq 'ARRAY';
 3142   my $errors = undef;
 3143   my @zero_params=();
 3144   for(my $i=1;$i<=@{$ra_parameters};$i++){push(@zero_params,0); }
 3145   my @differences = ();
 3146   my @student_values;
 3147   my @adjusted_student_values;
 3148   my @instructorVals;
 3149   my ($diff,$instructorVal);
 3150   # calculate the vector of differences between the test function and the comparison function.
 3151   while (@evaluation_points) {
 3152     my ($err1, $err2,$err3);
 3153     my @vars = @{ shift(@evaluation_points) };
 3154     my @inputs = (@vars, @parameters);
 3155     my ($inVal,  $correctVal);
 3156     ($inVal, $err1) = &{$rf_fun}(@vars);
 3157     $errors .= " $err1 "  if defined($err1);
 3158     $errors .= " Error detected evaluating student input at (".join(' , ',@vars) ." ) " if  defined($options{debug}) and $options{debug}=1 and defined($err1);
 3159     ($correctVal, $err2) =&{$rf_correct_fun}(@inputs);
 3160     $errors .= " There is an error in WeBWorK's answer to this problem, please alert your instructor.<br> $err2 " if defined($err2);
 3161     $errors .= " Error detected evaluating correct adapted answer  at (".join(' , ',@inputs) ." ) " if defined($options{debug}) and $options{debug}=1 and defined($err2);
 3162     ($instructorVal,$err3)= &$rf_correct_fun(@vars, @zero_params);
 3163     $errors .= " There is an error in WeBWorK's answer to this problem, please alert your instructor.<br> $err3 " if defined($err3);
 3164     $errors .= " Error detected evaluating instructor answer  at (".join(' , ',@vars, @zero_params) ." ) " if defined($options{debug}) and $options{debug}=1 and defined($err3);
 3165     unless (defined($err1) or defined($err2) or defined($err3) ) {
 3166       $diff = ( $inVal - ($correctVal -$instructorVal ) ) - $instructorVal;  #prevents entering too high a number?
 3167       #warn "taking the difference of ", $inVal, " and ", $correctVal, " is ", $diff;
 3168       if (defined($options{tolType}) and $options{tolType} eq 'relative' ) {  #relative tolerance
 3169         #warn "diff = $diff";
 3170         #$diff = ( $inVal - ($correctVal-$instructorVal ) )/abs($instructorVal) -1    if abs($instructorVal) > $options{zeroLevel};
 3171         $diff = ( $inVal - ($correctVal-$instructorVal ) )/$instructorVal -1    if abs($instructorVal) > $options{zeroLevel};
 3172         #$diff = ( $inVal - ($correctVal-$instructorVal- $instructorVal ) )/abs($instructorVal)    if abs($instructorVal) > $options{zeroLevel};
 3173         #warn "diff = $diff,   ", abs( &$rf_correct_fun(@inputs) ) , "-- $correctVal";
 3174       }
 3175     }
 3176     last if $errors;  # break if there are any errors.
 3177                   # This cuts down on the size of error messages.
 3178                   # However it impossible to check for equivalence at 95% of points
 3179                   # which might be useful for functions that are not defined at some points.
 3180         push(@student_values,$inVal);
 3181         push(@adjusted_student_values,(  $inVal - ($correctVal -$instructorVal) ) );
 3182     push(@differences, $diff);
 3183     push(@instructorVals,$instructorVal);
 3184   }
 3185   $rh_ans ->{ra_differences} = \@differences;
 3186   $rh_ans ->{ra_student_values} = \@student_values;
 3187   $rh_ans ->{ra_adjusted_student_values} = \@adjusted_student_values;
 3188   $rh_ans->{ra_instructor_values}=\@instructorVals;
 3189   $rh_ans->throw_error('EVAL', $errors) if defined($errors);
 3190   $rh_ans;
 3191 }
 3192 
 3193 =head4 fix_answer_for_display
 3194 
 3195 =cut
 3196 
 3197 sub fix_answers_for_display {
 3198   my ($rh_ans, %options) = @_;
 3199   if ( $rh_ans->{answerIsString} ==1) {
 3200     $rh_ans = evaluatesToNumber ($rh_ans, %options);
 3201   }
 3202   if (defined ($rh_ans->{student_units})) {
 3203     $rh_ans->{student_ans} = $rh_ans->{student_ans}. ' '. $rh_ans->{student_units};
 3204   }
 3205   $rh_ans->{correct_ans} = $rh_ans->{original_correct_ans};
 3206 
 3207   $rh_ans;
 3208 }
 3209 
 3210 =head4 evaluatesToNumber
 3211 
 3212 =cut
 3213 
 3214 sub evaluatesToNumber {
 3215   my ($rh_ans, %options) = @_;
 3216   if (is_a_numeric_expression($rh_ans->{student_ans})) {
 3217     my ($inVal,$PG_eval_errors,$PG_full_error_report) = PG_answer_eval($rh_ans->{student_ans});
 3218     if ($PG_eval_errors) { # this if statement should never be run
 3219       # change nothing
 3220     } else {
 3221       # change this
 3222       $rh_ans->{student_ans} = prfmt($inVal,$options{format});
 3223     }
 3224   }
 3225   $rh_ans;
 3226 }
 3227 
 3228 =head4 is_numeric_expression
 3229 
 3230 =cut
 3231 
 3232 sub is_a_numeric_expression {
 3233   my $testString = shift;
 3234   my $is_a_numeric_expression = 0;
 3235   my ($inVal,$PG_eval_errors,$PG_full_error_report) = PG_answer_eval($testString);
 3236   if ($PG_eval_errors) {
 3237     $is_a_numeric_expression = 0;
 3238   } else {
 3239     $is_a_numeric_expression = 1;
 3240   }
 3241   $is_a_numeric_expression;
 3242 }
 3243 
 3244 =head4 is_a_number
 3245 
 3246 =cut
 3247 
 3248 sub is_a_number {
 3249   my ($num,%options) =  @_;
 3250   my $process_ans_hash = ( ref( $num ) eq 'AnswerHash' ) ? 1 : 0 ;
 3251   my ($rh_ans);
 3252   if ($process_ans_hash) {
 3253     $rh_ans = $num;
 3254     $num = $rh_ans->{student_ans};
 3255   }
 3256 
 3257   my $is_a_number = 0;
 3258   return $is_a_number unless defined($num);
 3259   $num =~ s/^\s*//; ## remove initial spaces
 3260   $num =~ s/\s*$//; ## remove trailing spaces
 3261 
 3262   ## the following is copied from the online perl manual
 3263   if ($num =~ /^([+-]?)(?=\d|\.\d)\d*(\.\d*)?([Ee]([+-]?\d+))?$/){
 3264     $is_a_number = 1;
 3265   }
 3266 
 3267   if ($process_ans_hash)   {
 3268         if ($is_a_number == 1 ) {
 3269           $rh_ans->{student_ans}=$num;
 3270           return $rh_ans;
 3271         } else {
 3272           $rh_ans->{student_ans} = "Incorrect number format:  You must enter a number, e.g. -6, 5.3, or 6.12E-3";
 3273           $rh_ans->throw_error('NUMBER', 'You must enter a number, e.g. -6, 5.3, or 6.12E-3');
 3274           return $rh_ans;
 3275         }
 3276   } else {
 3277     return $is_a_number;
 3278   }
 3279 }
 3280 
 3281 =head4 is_a_fraction
 3282 
 3283 =cut
 3284 
 3285 sub is_a_fraction {
 3286   my ($num,%options) =  @_;
 3287   my $process_ans_hash = ( ref( $num ) eq 'AnswerHash' ) ? 1 : 0 ;
 3288   my ($rh_ans);
 3289   if ($process_ans_hash) {
 3290     $rh_ans = $num;
 3291     $num = $rh_ans->{student_ans};
 3292   }
 3293 
 3294   my $is_a_fraction = 0;
 3295   return $is_a_fraction unless defined($num);
 3296   $num =~ s/^\s*//; ## remove initial spaces
 3297   $num =~ s/\s*$//; ## remove trailing spaces
 3298 
 3299   if ($num =~ /^\s*\-?\s*[\/\d\.Ee\s]*$/) {
 3300     $is_a_fraction = 1;
 3301   }
 3302 
 3303     if ($process_ans_hash)   {
 3304       if ($is_a_fraction == 1 ) {
 3305         $rh_ans->{student_ans}=$num;
 3306         return $rh_ans;
 3307       } else {
 3308         $rh_ans->{student_ans} = "Not a number of fraction: You must enter a number or fraction, e.g. -6 or 7/13";
 3309         $rh_ans->throw_error('NUMBER', 'You must enter a number, e.g. -6, 5.3, or 6.12E-3');
 3310         return $rh_ans;
 3311       }
 3312 
 3313       } else {
 3314     return $is_a_fraction;
 3315   }
 3316 }
 3317 
 3318 =head4 phase_pi
 3319   I often discovered that the answers I was getting, when using the arctan function would be off by phases of
 3320   pi, which for the tangent function, were equivalent values. This method allows for this.
 3321 =cut
 3322 
 3323 sub phase_pi {
 3324   my ($num,%options) =  @_;
 3325   my $process_ans_hash = ( ref( $num ) eq 'AnswerHash' ) ? 1 : 0 ;
 3326   my ($rh_ans);
 3327   if ($process_ans_hash) {
 3328     $rh_ans = $num;
 3329     $num = $rh_ans->{correct_ans};
 3330   }
 3331   while( ($rh_ans->{correct_ans}) >  3.14159265358979/2 ){
 3332     $rh_ans->{correct_ans} -= 3.14159265358979;
 3333   }
 3334   while( ($rh_ans->{correct_ans}) <= -3.14159265358979/2 ){
 3335     $rh_ans->{correct_ans} += 3.14159265358979;
 3336   }
 3337   $rh_ans;
 3338 }
 3339 
 3340 =head4 is_an_arithemetic_expression
 3341 
 3342 =cut
 3343 
 3344 sub is_an_arithmetic_expression {
 3345   my ($num,%options) =  @_;
 3346   my $process_ans_hash = ( ref( $num ) eq 'AnswerHash' ) ? 1 : 0 ;
 3347   my ($rh_ans);
 3348   if ($process_ans_hash) {
 3349     $rh_ans = $num;
 3350     $num = $rh_ans->{student_ans};
 3351   }
 3352 
 3353   my $is_an_arithmetic_expression = 0;
 3354   return $is_an_arithmetic_expression unless defined($num);
 3355   $num =~ s/^\s*//; ## remove initial spaces
 3356   $num =~ s/\s*$//; ## remove trailing spaces
 3357 
 3358   if ($num =~ /^[+\-*\/\^\(\)\[\]\{\}\s\d\.Ee]*$/) {
 3359     $is_an_arithmetic_expression =  1;
 3360   }
 3361 
 3362     if ($process_ans_hash)   {
 3363       if ($is_an_arithmetic_expression == 1 ) {
 3364         $rh_ans->{student_ans}=$num;
 3365         return $rh_ans;
 3366       } else {
 3367 
 3368     $rh_ans->{student_ans} = "Not an arithmetic expression: You must enter an arithmetic expression, e.g. -6 or (2.3*4+5/3)^2";
 3369         $rh_ans->throw_error('NUMBER', 'You must enter an arithmetic expression, e.g. -6 or (2.3*4+5/3)^2');
 3370         return $rh_ans;
 3371       }
 3372 
 3373       } else {
 3374     return $is_an_arithmetic_expression;
 3375   }
 3376 }
 3377 
 3378 #
 3379 
 3380 =head4 math_constants
 3381 
 3382 replaces pi, e, and ^ with their Perl equivalents
 3383 if useBaseTenLog is non-zero, convert log to logten
 3384 
 3385 =cut
 3386 
 3387 sub math_constants {
 3388   my($in,%options) = @_;
 3389   my $rh_ans;
 3390   my $process_ans_hash = ( ref( $in ) eq 'AnswerHash' ) ? 1 : 0 ;
 3391   if ($process_ans_hash) {
 3392     $rh_ans = $in;
 3393     $in = $rh_ans->{student_ans};
 3394   }
 3395   # The code fragment above allows this filter to be used when the input is simply a string
 3396   # as well as when the input is an AnswerHash, and options.
 3397   $in =~s/\bpi\b/(4*atan2(1,1))/ge;
 3398   $in =~s/\be\b/(exp(1))/ge;
 3399   $in =~s/\^/**/g;
 3400   if($main::useBaseTenLog) {
 3401     $in =~ s/\blog\b/logten/g;
 3402   }
 3403 
 3404   if ($process_ans_hash)   {
 3405       $rh_ans->{student_ans}=$in;
 3406       return $rh_ans;
 3407     } else {
 3408     return $in;
 3409   }
 3410 }
 3411 
 3412 
 3413 
 3414 =head4 is_array
 3415 
 3416   is_array($rh_ans)
 3417     returns: $rh_ans.   Throws error "NOTARRAY" if this is not an array
 3418 
 3419 =cut
 3420 
 3421 sub is_array  {
 3422   my $rh_ans = shift;
 3423     # return if the result is an array
 3424   return($rh_ans) if  ref($rh_ans->{student_ans}) eq 'ARRAY' ;
 3425   $rh_ans->throw_error("NOTARRAY","The answer is not an array");
 3426   $rh_ans;
 3427 }
 3428 
 3429 =head4 check_syntax
 3430 
 3431   check_syntax( $rh_ans, %options)
 3432     returns an answer hash.
 3433 
 3434 latex2html preview code are installed in the answer hash.
 3435 The input has been transformed, changing 7pi to 7*pi  or 7x to 7*x.
 3436 Syntax error messages may be generated and stored in student_ans
 3437 Additional syntax error messages are stored in {ans_message} and duplicated in {error_message}
 3438 
 3439 
 3440 =cut
 3441 
 3442 sub check_syntax {
 3443         my $rh_ans = shift;
 3444         my %options = @_;
 3445         unless ( defined( $rh_ans->{student_ans} ) ) {
 3446           warn "Check_syntax requires an equation in the field {student_ans} or input";
 3447           $rh_ans->throw_error("1","{student_ans} field not defined");
 3448           return $rh_ans;
 3449         }
 3450         my $in = $rh_ans->{student_ans};
 3451     my $parser = new AlgParserWithImplicitExpand;
 3452     my $ret = $parser -> parse($in);      #for use with loops
 3453 
 3454     if ( ref($ret) )  {   ## parsed successfully
 3455       $parser -> tostring();
 3456       $parser -> normalize();
 3457       $rh_ans->input( $parser -> tostring() );
 3458       $rh_ans->{preview_text_string} = $in;
 3459       $rh_ans->{preview_latex_string} = $parser -> tolatex();
 3460 
 3461     } else {          ## error in parsing
 3462 
 3463       $rh_ans->{'student_ans'}      = 'syntax error:'. $parser->{htmlerror},
 3464       $rh_ans->{'ans_message'}      = $parser -> {error_msg},
 3465       $rh_ans->{'preview_text_string'}  = '',
 3466       $rh_ans->{'preview_latex_string'} = '',
 3467       $rh_ans->throw_error('SYNTAX',  'syntax error in answer:'. $parser->{htmlerror} . "$BR" .$parser -> {error_msg});
 3468     }
 3469 $rh_ans;
 3470 
 3471 }
 3472 
 3473 =head4 check_strings
 3474 
 3475   check_strings ($rh_ans, %options)
 3476     returns $rh_ans
 3477 
 3478 =cut
 3479 
 3480 sub check_strings {
 3481   my ($rh_ans, %options) = @_;
 3482 
 3483   # if the student's answer is a number, simply return the answer hash (unchanged).
 3484 
 3485   #  we allow constructions like -INF to be treated as a string. Thus we ignore an initial
 3486   # - in deciding whether the student's answer is a number or string
 3487 
 3488   my $temp_ans = $rh_ans->{student_ans};
 3489   $temp_ans =~ s/^\s*\-//;   # remove an initial -
 3490 
 3491   if  ( $temp_ans =~ m/[\d+\-*\/^(){}\[\]]|^\s*e\s*$|^\s*pi\s*$/)   {
 3492   # if ( $rh_ans->{answerIsString} == 1) {
 3493   #     #$rh_ans->throw_error('STRING','Incorrect Answer'); # student's answer is a number
 3494   # }
 3495     return $rh_ans;
 3496   }
 3497   # the student's answer is recognized as a string
 3498   my $ans = $rh_ans->{student_ans};
 3499 
 3500 # OVERVIEW of reminder of function:
 3501 # if answer is correct, return correct.  (adjust score to 1)
 3502 # if answer is incorect:
 3503 # 1) determine if the answer is sensible.  if it is, return incorrect.
 3504 # 2) if the answer is not sensible (and incorrect), then return an error message indicating so.
 3505 # no matter what:  throw a 'STRING' error to skip numerical evaluations.  (error flag skips remainder of pre_filters and evaluators)
 3506 # last: 'STRING' post_filter will clear the error (avoiding pink screen.)
 3507 
 3508   my $sensibleAnswer = 0;
 3509   $ans = str_filters( $ans, 'compress_whitespace' );  # remove trailing, leading, and double spaces.
 3510   my ($ans_eval) = str_cmp($rh_ans->{correct_ans});
 3511   my $temp_ans_hash = &$ans_eval($ans);
 3512   $rh_ans->{test} = $temp_ans_hash;
 3513   if ($temp_ans_hash->{score} ==1 ) {     # students answer matches the correct answer.
 3514     $rh_ans->{score} = 1;
 3515     $sensibleAnswer = 1;
 3516   } else {            # students answer does not match the correct answer.
 3517     my $legalString = '';       # find out if string makes sense
 3518     my @legalStrings = @{$options{strings}};
 3519     foreach $legalString (@legalStrings) {
 3520       if ( uc($ans) eq uc($legalString) ) {
 3521         $sensibleAnswer = 1;
 3522         last;
 3523         }
 3524       }
 3525     $sensibleAnswer = 1 unless $ans =~ /\S/;  ## empty answers are sensible
 3526     $rh_ans->throw_error('EVAL', "Your answer is not a recognized answer") unless ($sensibleAnswer);
 3527     # $temp_ans_hash -> setKeys( 'ans_message' => 'Your answer is not a recognized answer' ) unless ($sensibleAnswer);
 3528     # $temp_ans_hash -> setKeys( 'student_ans' => uc($ans) );
 3529   }
 3530   $rh_ans->{student_ans} = $ans;
 3531   if ($sensibleAnswer) {
 3532     $rh_ans->throw_error('STRING', "The student's answer $rh_ans->{student_ans} is interpreted as a string.");
 3533   }
 3534   # warn ("\$rh_ans->{answerIsString} = $rh_ans->{answerIsString}");
 3535   $rh_ans;
 3536 }
 3537 
 3538 =head4 check_units
 3539 
 3540   check_strings ($rh_ans, %options)
 3541     returns $rh_ans
 3542 
 3543 
 3544 =cut
 3545 
 3546 sub check_units {
 3547   my ($rh_ans, %options) = @_;
 3548   my %correct_units = %{$rh_ans-> {rh_correct_units}};
 3549   my $ans = $rh_ans->{student_ans};
 3550   # $ans = '' unless defined ($ans);
 3551   $ans = str_filters ($ans, 'trim_whitespace');
 3552   my $original_student_ans = $ans;
 3553   $rh_ans->{original_student_ans} = $original_student_ans;
 3554 
 3555   # it surprises me that the match below works since the first .* is greedy.
 3556   my ($num_answer, $units) = $ans =~ /^(.*)\s+([^\s]*)$/;
 3557 
 3558   unless ( defined($num_answer) && $units ) {
 3559     # there is an error reading the input
 3560     if ( $ans =~ /\S/ )  {  # the answer is not blank
 3561       $rh_ans -> setKeys( 'ans_message' => "The answer \"$ans\" could not be interpreted " .
 3562         "as a number or an arithmetic expression followed by a unit specification. " .
 3563         "Your answer must contain units." );
 3564       $rh_ans->throw_error('UNITS', "The answer \"$ans\" could not be interpreted " .
 3565         "as a number or an arithmetic expression followed by a unit specification. " .
 3566         "Your answer must contain units." );
 3567     }
 3568     return $rh_ans;
 3569   }
 3570 
 3571   # we have been able to parse the answer into a numerical part and a unit part
 3572 
 3573   # $num_answer = $1;   #$1 and $2 from the regular expression above
 3574   # $units    = $2;
 3575 
 3576   my %units = Units::evaluate_units($units);
 3577   if ( defined( $units{'ERROR'} ) ) {
 3578      # handle error condition
 3579           $units{'ERROR'} = clean_up_error_msg($units{'ERROR'});
 3580     $rh_ans -> setKeys( 'ans_message' => "$units{'ERROR'}" );
 3581     $rh_ans -> throw_error('UNITS', "$units{'ERROR'}");
 3582     return $rh_ans;
 3583   }
 3584 
 3585   my $units_match = 1;
 3586   my $fund_unit;
 3587   foreach $fund_unit (keys %correct_units) {
 3588     next if $fund_unit eq 'factor';
 3589     $units_match = 0 unless $correct_units{$fund_unit} == $units{$fund_unit};
 3590   }
 3591 
 3592   if ( $units_match ) {
 3593         # units are ok.  Evaluate the numerical part of the answer
 3594     $rh_ans->{'tolerance'} = $rh_ans->{'tolerance'}* $correct_units{'factor'}/$units{'factor'}  if
 3595           $rh_ans->{'tolType'} eq 'absolute'; # the tolerance is in the units specified by the instructor.
 3596     $rh_ans->{correct_ans} =  prfmt($rh_ans->{correct_ans}*$correct_units{'factor'}/$units{'factor'});
 3597     $rh_ans->{student_units} = $units;
 3598     $rh_ans->{student_ans} = $num_answer;
 3599 
 3600   } else {
 3601         $rh_ans -> setKeys( ans_message => 'There is an error in the units for this answer.' );
 3602         $rh_ans -> throw_error ( 'UNITS', 'There is an error in the units for this answer.' );
 3603   }
 3604 
 3605   return $rh_ans;
 3606 }
 3607 
 3608 
 3609 
 3610 =head2 Filter utilities
 3611 
 3612 These two subroutines can be used in filters to set default options.  They
 3613 help make filters perform in uniform, predictable ways, and also make it
 3614 easy to recognize from the code which options a given filter expects.
 3615 
 3616 
 3617 =head4 assign_option_aliases
 3618 
 3619 Use this to assign aliases for the standard options.  It must come before set_default_options
 3620 within the subroutine.
 3621 
 3622     assign_option_aliases(\%options,
 3623         'alias1'  => 'option5'
 3624         'alias2'  => 'option7'
 3625     );
 3626 
 3627 
 3628 If the subroutine is called with an option  " alias1 => 23 " it will behave as if it had been
 3629 called with the option " option5 => 23 "
 3630 
 3631 =cut
 3632 
 3633 
 3634 
 3635 sub assign_option_aliases {
 3636   my $rh_options = shift;
 3637   warn "The first entry to set_default_options must be a reference to the option hash" unless ref($rh_options) eq 'HASH';
 3638   my @option_aliases = @_;
 3639   while (@option_aliases) {
 3640     my $alias = shift @option_aliases;
 3641     my $option_key = shift @option_aliases;
 3642 
 3643     if (defined($rh_options->{$alias} )) {                       # if the alias appears in the option list
 3644       if (not defined($rh_options->{$option_key}) ) {          # and the option itself is not defined,
 3645         $rh_options->{$option_key} = $rh_options->{$alias};  # insert the value defined by the alias into the option value
 3646                                                              # the FIRST alias for a given option takes precedence
 3647                                                              # (after the option itself)
 3648       } else {
 3649         warn "option $option_key is already defined as", $rh_options->{$option_key}, "<br>\n",
 3650              "The attempt to override this option with the alias $alias with value ", $rh_options->{$alias},
 3651              " was ignored.";
 3652       }
 3653     }
 3654     delete($rh_options->{$alias});                               # remove the alias from the initial list
 3655   }
 3656 
 3657 }
 3658 
 3659 =head4 set_default_options
 3660 
 3661     set_default_options(\%options,
 3662         '_filter_name'  =>  'filter',
 3663         'option5'   =>  .0001,
 3664         'option7'   =>  'ascii',
 3665         'allow_unknown_options  =>  0,
 3666     }
 3667 
 3668 Note that the first entry is a reference to the options with which the filter was called.
 3669 
 3670 The option5 is set to .0001 unless the option is explicitly set when the subroutine is called.
 3671 
 3672 The B<'_filter_name'> option should always be set, although there is no error if it is missing.
 3673 It is used mainly for debugging answer evaluators and allows
 3674 you to keep track of which filter is currently processing the answer.
 3675 
 3676 If B<'allow_unknown_options'> is set to 0 then if the filter is called with options which do NOT appear in the
 3677 set_default_options list an error will be signaled and a warning message will be printed out.  This provides
 3678 error checking against misspelling an option and is generally what is desired for most filters.
 3679 
 3680 Occasionally one wants to write a filter which accepts a long list of options, not all of which are known in advance,
 3681 but only uses a subset of the options
 3682 provided.  In this case, setting 'allow_unkown_options' to 1 prevents the error from being signaled.
 3683 
 3684 =cut
 3685 
 3686 sub set_default_options {
 3687   my $rh_options = shift;
 3688   warn "The first entry to set_default_options must be a reference to the option hash" unless ref($rh_options) eq 'HASH';
 3689   my %default_options = @_;
 3690   unless ( defined($default_options{allow_unknown_options}) and $default_options{allow_unknown_options} == 1 ) {
 3691     foreach  my $key1 (keys %$rh_options) {
 3692       warn "This option |$key1| is not recognized in this subroutine<br> ", pretty_print($rh_options) unless exists($default_options{$key1});
 3693     }
 3694   }
 3695   foreach my $key (keys %default_options) {
 3696     if  ( not defined($rh_options->{$key} ) and defined( $default_options{$key} )  ) {
 3697       $rh_options->{$key} = $default_options{$key};  #this allows     tol   => undef to allow the tol option, but doesn't define
 3698                                                      # this key unless tol is explicitly defined.
 3699     }
 3700   }
 3701 }
 3702 
 3703 =head2 Problem Grader Subroutines
 3704 
 3705 =cut
 3706 
 3707 ## Problem Grader Subroutines
 3708 
 3709 #####################################
 3710 # This is a model for plug-in problem graders
 3711 #####################################
 3712 sub install_problem_grader {
 3713   my $rf_problem_grader = shift;
 3714   $main::PG_FLAGS{PROBLEM_GRADER_TO_USE} = $rf_problem_grader;
 3715 }
 3716 
 3717 =head4 std_problem_grader
 3718 
 3719 This is an all-or-nothing grader.  A student must get all parts of the problem write
 3720 before receiving credit.  You should make sure to use this grader on multiple choice
 3721 and true-false questions, otherwise students will be able to deduce how many
 3722 answers are correct by the grade reported by webwork.
 3723 
 3724 
 3725   install_problem_grader(~~&std_problem_grader);
 3726 
 3727 =cut
 3728 
 3729 sub std_problem_grader {
 3730   my $rh_evaluated_answers = shift;
 3731   my $rh_problem_state = shift;
 3732   my %form_options = @_;
 3733   my %evaluated_answers = %{$rh_evaluated_answers};
 3734   #  The hash $rh_evaluated_answers typically contains:
 3735   #    'answer1' => 34, 'answer2'=> 'Mozart', etc.
 3736 
 3737   # By default the  old problem state is simply passed back out again.
 3738   my %problem_state = %$rh_problem_state;
 3739 
 3740   # %form_options might include
 3741   # The user login name
 3742   # The permission level of the user
 3743   # The studentLogin name for this psvn.
 3744   # Whether the form is asking for a refresh or is submitting a new answer.
 3745 
 3746   # initial setup of the answer
 3747   my %problem_result = ( score    => 0,
 3748                errors   => '',
 3749              type   => 'std_problem_grader',
 3750              msg    => '',
 3751   );
 3752   # Checks
 3753 
 3754   my $ansCount = keys %evaluated_answers;  # get the number of answers
 3755 
 3756   unless ($ansCount > 0 ) {
 3757 
 3758     $problem_result{msg} = "This problem did not ask any questions.";
 3759     return(\%problem_result,\%problem_state);
 3760   }
 3761 
 3762   if ($ansCount > 1 ) {
 3763     $problem_result{msg} = 'In order to get credit for this problem all answers must be correct.' ;
 3764   }
 3765 
 3766   unless ($form_options{answers_submitted} == 1) {
 3767     return(\%problem_result,\%problem_state);
 3768   }
 3769 
 3770   my $allAnswersCorrectQ=1;
 3771   foreach my $ans_name (keys %evaluated_answers) {
 3772   # I'm not sure if this check is really useful.
 3773     if ( ( ref($evaluated_answers{$ans_name} ) eq 'HASH' ) or ( ref($evaluated_answers{$ans_name}) eq 'AnswerHash' ) )  {
 3774       $allAnswersCorrectQ = 0 unless( 1 == $evaluated_answers{$ans_name}->{score} );
 3775     }
 3776     else {
 3777       die "Error at file ",__FILE__,"line ", __LINE__,":  Answer |$ans_name| is not a hash reference\n".
 3778          $evaluated_answers{$ans_name} .
 3779          "This probably means that the answer evaluator for this answer\n" .
 3780          "is not working correctly.";
 3781       $problem_result{error} = "Error: Answer $ans_name is not a hash: $evaluated_answers{$ans_name}";
 3782     }
 3783   }
 3784   # report the results
 3785   $problem_result{score} = $allAnswersCorrectQ;
 3786 
 3787   # I don't like to put in this bit of code.
 3788   # It makes it hard to construct error free problem graders
 3789   # I would prefer to know that the problem score was numeric.
 3790   unless (defined($problem_state{recorded_score}) and $problem_state{recorded_score} =~ /^([+-]?)(?=\d|\.\d)\d*(\.\d*)?([Ee]([+-]?\d+))?$/ ) {
 3791     $problem_state{recorded_score} = 0;  # This gets rid of non-numeric scores
 3792   }
 3793   #
 3794   if ($allAnswersCorrectQ == 1 or $problem_state{recorded_score} == 1) {
 3795     $problem_state{recorded_score} = 1;
 3796   }
 3797   else {
 3798     $problem_state{recorded_score} = 0;
 3799   }
 3800 
 3801   $problem_state{num_of_correct_ans}++ if $allAnswersCorrectQ == 1;
 3802   $problem_state{num_of_incorrect_ans}++ if $allAnswersCorrectQ == 0;
 3803   (\%problem_result, \%problem_state);
 3804 }
 3805 
 3806 =head4 std_problem_grader2
 3807 
 3808 This is an all-or-nothing grader.  A student must get all parts of the problem write
 3809 before receiving credit.  You should make sure to use this grader on multiple choice
 3810 and true-false questions, otherwise students will be able to deduce how many
 3811 answers are correct by the grade reported by webwork.
 3812 
 3813 
 3814   install_problem_grader(~~&std_problem_grader2);
 3815 
 3816 The only difference between the two versions
 3817 is at the end of the subroutine, where std_problem_grader2
 3818 records the attempt only if there have been no syntax errors,
 3819 whereas std_problem_grader records it regardless.
 3820 
 3821 =cut
 3822 
 3823 
 3824 
 3825 sub std_problem_grader2 {
 3826   my $rh_evaluated_answers = shift;
 3827   my $rh_problem_state = shift;
 3828   my %form_options = @_;
 3829   my %evaluated_answers = %{$rh_evaluated_answers};
 3830   #  The hash $rh_evaluated_answers typically contains:
 3831   #    'answer1' => 34, 'answer2'=> 'Mozart', etc.
 3832 
 3833   # By default the  old problem state is simply passed back out again.
 3834   my %problem_state = %$rh_problem_state;
 3835 
 3836   # %form_options might include
 3837   # The user login name
 3838   # The permission level of the user
 3839   # The studentLogin name for this psvn.
 3840   # Whether the form is asking for a refresh or is submitting a new answer.
 3841 
 3842   # initial setup of the answer
 3843   my %problem_result = ( score        => 0,
 3844              errors       => '',
 3845              type       => 'std_problem_grader',
 3846              msg        => '',
 3847   );
 3848 
 3849   # syntax errors are not counted.
 3850   my $record_problem_attempt = 1;
 3851   # Checks
 3852 
 3853   my $ansCount = keys %evaluated_answers;  # get the number of answers
 3854   unless ($ansCount > 0 ) {
 3855     $problem_result{msg} = "This problem did not ask any questions.";
 3856     return(\%problem_result,\%problem_state);
 3857   }
 3858 
 3859   if ($ansCount > 1 ) {
 3860     $problem_result{msg} = 'In order to get credit for this problem all answers must be correct.' ;
 3861   }
 3862 
 3863   unless ($form_options{answers_submitted} == 1) {
 3864     return(\%problem_result,\%problem_state);
 3865   }
 3866 
 3867   my  $allAnswersCorrectQ=1;
 3868   foreach my $ans_name (keys %evaluated_answers) {
 3869   # I'm not sure if this check is really useful.
 3870     if ( ( ref($evaluated_answers{$ans_name} ) eq 'HASH' ) or ( ref($evaluated_answers{$ans_name}) eq 'AnswerHash' ) )  {
 3871       $allAnswersCorrectQ = 0 unless( 1 == $evaluated_answers{$ans_name}->{score} );
 3872     }
 3873     else {
 3874       die "Error at file ",__FILE__,"line ", __LINE__,":  Answer |$ans_name| is not a hash reference\n".
 3875          $evaluated_answers{$ans_name} .
 3876          "This probably means that the answer evaluator for this answer\n" .
 3877          "is not working correctly.";
 3878       $problem_result{error} = "Error: Answer $ans_name is not a hash: $evaluated_answers{$ans_name}";
 3879     }
 3880   }
 3881   # report the results
 3882   $problem_result{score} = $allAnswersCorrectQ;
 3883 
 3884   # I don't like to put in this bit of code.
 3885   # It makes it hard to construct error free problem graders
 3886   # I would prefer to know that the problem score was numeric.
 3887   unless ($problem_state{recorded_score} =~ /^([+-]?)(?=\d|\.\d)\d*(\.\d*)?([Ee]([+-]?\d+))?$/ ) {
 3888     $problem_state{recorded_score} = 0;  # This gets rid of non-numeric scores
 3889   }
 3890   #
 3891   if ($allAnswersCorrectQ == 1 or $problem_state{recorded_score} == 1) {
 3892     $problem_state{recorded_score} = 1;
 3893   }
 3894   else {
 3895     $problem_state{recorded_score} = 0;
 3896   }
 3897   # record attempt only if there have been no syntax errors.
 3898 
 3899   if ($record_problem_attempt == 1) {
 3900     $problem_state{num_of_correct_ans}++ if $allAnswersCorrectQ == 1;
 3901     $problem_state{num_of_incorrect_ans}++ if $allAnswersCorrectQ == 0;
 3902   }
 3903   else {
 3904     $problem_result{show_partial_correct_answers} = 0 ;  # prevent partial correct answers from being shown for syntax errors.
 3905   }
 3906   (\%problem_result, \%problem_state);
 3907 }
 3908 
 3909 =head4 avg_problem_grader
 3910 
 3911 This grader gives a grade depending on how many questions from the problem are correct.  (The highest
 3912 grade is the one that is kept.  One can never lower the recorded grade on a problem by repeating it.)
 3913 Many professors (and almost all students :-)  ) prefer this grader.
 3914 
 3915 
 3916   install_problem_grader(~~&avg_problem_grader);
 3917 
 3918 =cut
 3919 
 3920 
 3921 sub avg_problem_grader {
 3922     my $rh_evaluated_answers = shift;
 3923   my $rh_problem_state = shift;
 3924   my %form_options = @_;
 3925   my %evaluated_answers = %{$rh_evaluated_answers};
 3926   #  The hash $rh_evaluated_answers typically contains:
 3927   #    'answer1' => 34, 'answer2'=> 'Mozart', etc.
 3928 
 3929   # By default the  old problem state is simply passed back out again.
 3930   my %problem_state = %$rh_problem_state;
 3931 
 3932 
 3933   # %form_options might include
 3934   # The user login name
 3935   # The permission level of the user
 3936   # The studentLogin name for this psvn.
 3937   # Whether the form is asking for a refresh or is submitting a new answer.
 3938 
 3939   # initial setup of the answer
 3940   my  $total=0;
 3941   my %problem_result = ( score        => 0,
 3942              errors       => '',
 3943              type       => 'avg_problem_grader',
 3944              msg        => '',
 3945   );
 3946   my $count = keys %evaluated_answers;
 3947   $problem_result{msg} = 'You can earn partial credit on this problem.' if $count >1;
 3948   # Return unless answers have been submitted
 3949   unless ($form_options{answers_submitted} == 1) {
 3950     return(\%problem_result,\%problem_state);
 3951   }
 3952 
 3953   # Answers have been submitted -- process them.
 3954   foreach my $ans_name (keys %evaluated_answers) {
 3955     # I'm not sure if this check is really useful.
 3956     if ( ( ref($evaluated_answers{$ans_name} ) eq 'HASH' ) or ( ref($evaluated_answers{$ans_name}) eq 'AnswerHash' ) )  {
 3957       $total += $evaluated_answers{$ans_name}->{score};
 3958     }
 3959     else {
 3960       die "Error: Answer |$ans_name| is not a hash reference\n".
 3961          $evaluated_answers{$ans_name} .
 3962          "This probably means that the answer evaluator for this answer\n" .
 3963          "is not working correctly.";
 3964       $problem_result{error} = "Error: Answer $ans_name is not a hash: $evaluated_answers{$ans_name}";
 3965     }
 3966   }
 3967   # Calculate score rounded to three places to avoid roundoff problems
 3968   $problem_result{score} = $total/$count if $count;
 3969   # increase recorded score if the current score is greater.
 3970   $problem_state{recorded_score} = $problem_result{score} if $problem_result{score} > $problem_state{recorded_score};
 3971 
 3972 
 3973   $problem_state{num_of_correct_ans}++ if $total == $count;
 3974   $problem_state{num_of_incorrect_ans}++ if $total < $count ;
 3975   warn "Error in grading this problem the total $total is larger than $count" if $total > $count;
 3976   (\%problem_result, \%problem_state);
 3977 }
 3978 
 3979 =head2 Utility subroutines
 3980 
 3981 =head4
 3982 
 3983   warn pretty_print( $rh_hash_input)
 3984 
 3985 This can be very useful for printing out messages about objects while debugging
 3986 
 3987 =cut
 3988 
 3989 sub pretty_print {
 3990     my $r_input = shift;
 3991     my $out = '';
 3992     if ( not ref($r_input) ) {
 3993       $out = $r_input;    # not a reference
 3994     } elsif ("$r_input" =~/hash/i) {  # this will pick up objects whose '$self' is hash and so works better than ref($r_iput).
 3995       local($^W) = 0;
 3996     $out .= "$r_input " ."<TABLE border = \"2\" cellpadding = \"3\" BGCOLOR = \"#FFFFFF\">";
 3997     foreach my $key (lex_sort( keys %$r_input )) {
 3998       $out .= "<tr><TD> $key</TD><TD>=&gt;</td><td>&nbsp;".pretty_print($r_input->{$key}) . "</td></tr>";
 3999     }
 4000     $out .="</table>";
 4001   } elsif (ref($r_input) eq 'ARRAY' ) {
 4002     my @array = @$r_input;
 4003     $out .= "( " ;
 4004     while (@array) {
 4005       $out .= pretty_print(shift @array) . " , ";
 4006     }
 4007     $out .= " )";
 4008   } elsif (ref($r_input) eq 'CODE') {
 4009     $out = "$r_input";
 4010   } else {
 4011     $out = $r_input;
 4012   }
 4013     $out;
 4014 }
 4015 
 4016 1;

aubreyja at gmail dot com
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