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Math::Utils - Useful mathematical functions not in Perl.


    use Math::Utils qw(:utility);    # Useful functions

    # Base 10 and base 2 logarithms.
    $scale = log10($pagewidth);
    $bits = log2(1/$probability);

    # Two uses of sign().
    $d = sign($z - $w);

    @ternaries = sign(@coefficients);

    # Using copysign(), $dist will be doubled negative or
    # positive $offest, depending upon whether ($from - $to)
    # is positive or negative.
    my $dist = copysign(2 * $offset, $from - $to);

    # Change increment direction if goal is negative.
    $incr = flipsign($incr, $goal);

    # floor() and ceil() functions.
    $point = floor($goal);
    $limit = ceil($goal);

    # gcd() and lcm() functions.
    $divisor = gcd(@multipliers);
    $numerator = lcm(@multipliers);

    # Safer summation.
    $tot = fsum(@inputs);

    # The remainders of n after successive divisions of b, or
    # remainders after a set of divisions.
    @rems = moduli($n, $b);


    use Math::Utils qw(:compare);    # Make comparison functions with tolerance.

    # Floating point comparison function.
    my $fltcmp = generate_fltmcp(1.0e-7);

    if (&$fltcmp($x0, $x1) < 0)

    # Or we can create single-operation comparison functions.
    # Here we are only interested in the greater than and less than
    # comparison functions.
    my(undef, undef,
        $approx_gt, undef, $approx_lt) = generate_relational(1.5e-5);


    use Math::Utils qw(:polynomial);    # Basic polynomial ops

    # Coefficient lists run from 0th degree upward, left to right.
    my @c1 = (1, 3, 5, 7, 11, 13, 17, 19);
    my @c2 = (1, 3, 1, 7);
    my @c3 = (1, -1, 1)

    my $c_ref = pl_mult(\@c1, \@c2);
    $c_ref = pl_add($c_ref, \@c3);


All functions can be exported by name, or by using the tag that they're grouped under.

utility tag

Useful, general-purpose functions, including those that originated in FORTRAN and were implemented in Perl in the module Math::Fortran, by J. A. R. Williams.

There is a name change -- copysign() was known as sign() in Math::Fortran.


    $xlog10 = log10($x);
    @xlog10 = log10(@x);

Return the log base ten of the argument. A list form of the function is also provided.


    $xlog2 = log2($x);
    @xlog2 = log2(@x);

Return the log base two of the argument. A list form of the function is also provided.


    $s = sign($x);
    @valsigns = sign(@values);

Returns -1 if the argument is negative, 0 if the argument is zero, and 1 if the argument is positive.

In list form it applies the same operation to each member of the list.


    $ms = copysign($m, $n);
    $s = copysign($x);

Take the sign of the second argument and apply it to the first. Zero is considered part of the positive signs.

    copysign(-5, 0);  # Returns 5.
    copysign(-5, 7);  # Returns 5.
    copysign(-5, -7); # Returns -5.
    copysign(5, -7);  # Returns -5.

If there is only one argument, return -1 if the argument is negative, otherwise return 1. For example, copysign(1, -4) and copysign(-4) both return -1.


    $ms = flipsign($m, $n);

Multiply the signs of the arguments and apply it to the first. As with copysign(), zero is considered part of the positive signs.

Effectively this means change the sign of the first argument if the second argument is negative.

    flipsign(-5, 0);  # Returns -5.
    flipsign(-5, 7);  # Returns -5.
    flipsign(-5, -7); # Returns 5.
    flipsign(5, -7);  # Returns -5.

If for some reason flipsign() is called with a single argument, that argument is returned unchanged.


    $b = floor($a/2);

    @ilist = floor(@numbers);

Returns the greatest integer less than or equal to its argument. A list form of the function also exists.

    floor(1.5, 1.87, 1);        # Returns (1, 1, 1)
    floor(-1.5, -1.87, -1);     # Returns (-2, -2, -1)


    $b = ceil($a/2);

    @ilist = ceil(@numbers);

Returns the lowest integer greater than or equal to its argument. A list form of the function also exists.

    ceil(1.5, 1.87, 1);        # Returns (2, 2, 1)
    ceil(-1.5, -1.87, -1);     # Returns (-1, -1, -1)


Return a sum of the values in the list, done in a manner to avoid rounding and cancellation errors. Currently this is done via Kahan's summation algorithm.


Return a list of values as probabilities.

The function takes the list, and creates a new list by raising e to each value. The function then returns each value divided by the sum of the list. Each value in the new list is now a set of probabilities that sum to 1.0.

The summation is performed using fsum() above.

See Softmax function at Wikipedia.



Uniformly, or linearly, scale a number either from one range to another range (uniform_scaling()), or to a default range of [0 .. 1] (uniform_01scaling()).

    @v = uniform_scaling(\@original_range, \@new_range, @oldvalues);

For example, these two lines are equivalent, and both return 0:

    $y = uniform_scaling([50, 100], [0, 1], 50);

    $y = uniform_01scaling([50, 100], 50);

They may also be called with a list or array of numbers:

    @cm_measures = uniform_scaling([0, 10000], [0, 25400], @in_measures);

    @melt_centigrade = uniform_scaling([0, 2000], [-273.15, 1726.85], \@melting_points);

A number that is outside the original bounds will be proportionally changed to be outside of the new bounds, but then again having a number outside the original bounds is probably an error that should be checked before calling this function.



Return the greatest common divisor (also known as the highest common factor) of a list of integers. These are simply synomyms:

    $factor = gcd(@numbers);
    $factor = hcf(@numbers);


Return the least common multiple of a list of integers.

    $factor = lcm(@values);


Return the moduli of an integer after repeated divisions. The remainders are returned in a list from left to right.

    @digits = moduli(1899, 10);   # Returns (9, 9, 8, 1)
    @rems = moduli(29, 3);        # Returns (2, 0, 0, 1)

compare tag

Create comparison functions for floating point (non-integer) numbers.

Since exact comparisons of floating point numbers tend to be iffy, the comparison functions use a tolerance chosen by you. You may then use those functions from then on confident that comparisons will be consistent.

If you do not provide a tolerance, a default tolerance of 1.49012e-8 (approximately the square root of an Intel Pentium's machine epsilon) will be used.


Returns a comparison function that will compare values using a tolerance that you supply. The generated function will return -1 if the first argument compares as less than the second, 0 if the two arguments compare as equal, and 1 if the first argument compares as greater than the second.

    my $fltcmp = generate_fltcmp(1.5e-7);

    my(@xpos) = grep {&$fltcmp($_, 0) == 1} @xvals;


Returns a list of comparison functions that will compare values using a tolerance that you supply. The generated functions will be the equivalent of the equal, not equal, greater than, greater than or equal, less than, and less than or equal operators.

    my($eq, $ne, $gt, $ge, $lt, $le) = generate_relational(1.5e-7);

    my(@approx_5) = grep {&$eq($_, 5)} @xvals;

Of course, if you were only interested in not equal, you could use:

    my(undef, $ne) = generate_relational(1.5e-7);

    my(@not_around5) = grep {&$ne($_, 5)} @xvals;

polynomial tag

Perform some polynomial operations on plain lists of coefficients.

    # The coefficient lists are presumed to go from low order to high:
    @coefficients = (1, 2, 4, 8);    # 1 + 2x + 4x**2 + 8x**3

In all functions the coeffcient list is passed by reference to the function, and the functions that return coefficients all return references to a coefficient list.

It is assumed that any leading zeros in the coefficient lists have already been removed before calling these functions, and that any leading zeros found in the returned lists will be handled by the caller. This caveat is particularly important to note in the case of pl_div().

Although these functions are convenient for simple polynomial operations, for more advanced polynonial operations Math::Polynomial is recommended.


Returns either a y-value for a corresponding x-value, or a list of y-values on the polynomial for a corresponding list of x-values, using Horner's method.

    $y = pl_evaluate(\@coefficients, $x);
    @yvalues = pl_evaluate(\@coefficients, @xvalues);

    @ctemperatures = pl_evaluate([-160/9, 5/9], @ftemperatures);

The list of X values may also include X array references:

    @yvalues = pl_evaluate(\@coefficients, @xvalues, \@primes, $x, [-1, -10, -100]);


    ($y, $dy, $ddy) = pl_dxevaluate(\@coefficients, $x);

Returns p(x), p'(x), and p"(x) of the polynomial for an x-value, using Horner's method. Note that unlike pl_evaluate() above, the function can only use one x-value.

If the polynomial is a linear equation, the second derivative value will be zero. Similarly, if the polynomial is a simple constant, the first derivative value will be zero.


    $x = [8, 3, 1];
    $y = [3, 1];

    # Translating C<x**2 + 3*x + 8> by C<x + 3> returns [26, 9, 1]
    $z = pl_translate($x, $y);

Returns a polynomial transformed by substituting a polynomial variable with another polynomial. For example, a simple linear translation by 1 to the polynomial x**3 + x**2 + 4*x + 4 would be accomplished by setting x = (y - 1); resulting in x**3 - 2*x**2 + 5*x.

    $x = [4, 4, 1, 1];
    $y = [-1, 1];
    $z = pl_translate($x, $y);         # Becomes [0, 5, -2, 1]


    $polyn_ref = pl_add(\@m, \@n);

Add two lists of numbers as though they were polynomial coefficients.


    $polyn_ref = pl_sub(\@m, \@n);

Subtract the second list of numbers from the first as though they were polynomial coefficients.


    ($q_ref, $r_ref) = pl_div(\@numerator, \@divisor);

Synthetic division for polynomials. Divides the first list of coefficients by the second list.

Returns references to the quotient and the remainder.

Remember to check for leading zeros (which are rightmost in the list) in the returned values. For example,

    my @n = (4, 12, 9, 3);
    my @d = (1, 3, 3, 1);

    my($q_ref, $r_ref) = pl_div(\@n, \@d);

After division you will have returned (3) as the quotient, and (1, 3, 0) as the remainder. In general, you will want to remove the leading zero, or for that matter values within epsilon of zero, in the remainder.

    my($q_ref, $r_ref) = pl_div($f1, $f2);

    # Remove any leading zeros (i.e., numbers smaller in
    # magnitude than machine epsilon) in the remainder.
    my @remd = @{$r_ref};
    pop @remd while (@remd and abs($remd[$#remd]) < $epsilon);

    $f1 = $f2;
    $f2 = [@remd];

If $f1 and $f2 were to go through that bit of code again, not removing the leading zeros would lead to a divide-by-zero error.

If either list of coefficients is empty, pl_div() returns undefs for both quotient and remainder.


    $m_ref = pl_mult(\@coefficients1, \@coefficients2);

Returns the reference to the product of the two multiplicands.


    $poly_ref = pl_derivative(\@coefficients);

Returns the derivative of a polynomial.


    $poly_ref = pl_antiderivative(\@coefficients);

Returns the antiderivative of a polynomial. The constant value is always set to zero and will need to be changed by the caller if a different constant is needed.

  my @coefficients = (1, 2, -3, 2);
  my $integral = pl_antiderivative(\@coefficients);

  # Integral needs to be 0 at x = 1.
  my @coeff1 = @{$integral};
  $coeff1[0] = - pl_evaluate($integral, 1);


John M. Gamble, <jgamble at>


Math::Polynomial for a complete set of polynomial operations, with the added convenience that objects bring.

Among its other functions, List::Util has the mathematically useful functions max(), min(), product(), sum(), and sum0().

List::MoreUtils has the function minmax().

Math::Prime::Util has gcd() and lcm() functions, as well as vecsum(), vecprod(), vecmin(), and vecmax(), which are like the List::Util functions but which can force integer use, and when appropriate use Math::BigInt.

Math::VecStat Likewise has min(), max(), sum() (which can take as arguments array references as well as arrays), plus maxabs(), minabs(), sumbyelement(), convolute(), and other functions.


Please report any bugs or feature requests to bug-math-util at, or through the web interface at I will be notified, and then you'll automatically be notified of progress on your bug as I make changes.


This module is on Github at

You can also look for information at:


To J. A. R. Williams who got the ball rolling with Math::Fortran.


Copyright (c) 2017 John M. Gamble. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.