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Martin Becker
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NAME

Math::Logic::Ternary::TFP_81 - 81-trit ternary floating point arithmetic

VERSION

This documentation refers to version 0.004 of Math::Logic::Ternary::TFP_81.

SYNOPSIS

  use Math::Logic::Ternary::TFP_81 qw(float81);

  $fp = float81($mantissa_72t, $exponent_9t);
  $fp = Math::Logic::Ternary::TFP_81->new($mantissa_72t, $exponent_9t);
  $fp = Math::Logic::Ternary::TFP_81->from_int($word_81t);
  $fp = Math::Logic::Ternary::TFP_81->from_ratio($numerator, $denominator);
  $fp = Math::Logic::Ternary::TFP_81->from_word($word_81t);

  $fp = Math::Logic::Ternary::TFP_81->plus_infinity;
  $fp = Math::Logic::Ternary::TFP_81->minus_infinity;
  $fp = Math::Logic::Ternary::TFP_81->plus_epsilon;
  $fp = Math::Logic::Ternary::TFP_81->minus_epsilon;
  $fp = Math::Logic::Ternary::TFP_81->not_a_number;
  $fp = Math::Logic::Ternary::TFP_81->undefined;

  $mantissa_72t = $fp->mantissa;
  $exponent_9t  = $fp->exponent;
  $trit         = $fp->sign;
  $word_81t     = $fp->as_word;
  $word_81t     = $fp->as_int;
  $bool         = $fp->is_number;
  $bool         = $fp->is_symbol;
  $bool         = $fp->is_well_formed;
  $bool         = $fp->is_zero;
  $bool         = $fp->is_p_inf;
  $bool         = $fp->is_m_inf;
  $bool         = $fp->is_nan;
  $bool         = $fp->is_p_eps;
  $bool         = $fp->is_m_eps;
  $bool         = $fp->is_undef;

  $fp2 = $fp1->Fneg;
  $fp3 = $fp1->Fadd($fp2);
  $fp3 = $fp1->Fsub($fp2);
  $fp2 = $fp1->Finv;
  $fp3 = $fp1->Fmul($fp2);
  $fp3 = $fp1->Fdiv($fp2);
  $fp3 = $fp1->Fpow($fp2);
  $fp2 = $fp1->Flog;
  $fp2 = $fp1->Fexp;
  $fp2 = $fp1->Ftrunc;
  $fp2 = $fp1->Ffrac;
  $fp2 = $fp1->Ffloor;
  $fp2 = $fp1->Fceil;
  $fp2 = $fp1->Fabs;
  $fp2 = $fp1->Fnormalize;

  $trit = $fp1->Fcmp($fp2);
  $trit = $fp1->Fasc($fp2);

DESCRIPTION

This module defines an 81-trit floating point format and emulates some basic operations on numbers of this format.

Number Format

TFP_81 numbers are coded as 81-trit words with a mantissa of 72 trits followed by an exponent of 9 trits. The mantissa starts with a +1 or -1 trit for all non-zero numbers. It is a balanced ternary signed integer number spanning the most significant 72 trits of the 81-trit word. The exponent is a balanced ternary signed integer number, spanning the least significant 9 trits of the 81-trit word.

Zero is represented as 81 zero trits (mantissa 0, exponent 0). One is represented as a +1 trit followed by 80 zero trits (3 ** 71, 0).

Plus infinity is represented as 72 zero trits followed by nine +1 trits (0, 9841). Minus infinity is represented as 72 zero trits followed by eight +1 trits and a -1 trit (0, 9839). Not-a-number is represented as 72 zero trits followed by eight +1 trits and a zero trit (0, 9840). Plus epsilon, a symbol that can be used to denote a positive number very close to zero, is represented as 72 zero trits followed by eight -1 trits and a +1 trit (0, -9839). Minus epsilon is represented as 72 zero trits followed by nine -1 trits (0, -9841). CtN (close to naught), a symbol that can be used to denote a number with unknown sign very close to zero or actually zero, is represented as 72 zero trits followed by eight -1 trits and a zero trit (0, -9840).

Multiplying a non-zero number by three increases the exponent by one while leaving the mantissa unchanged, unless the exponent was already at its maximum (9841). Results greater than the largest representable number are replaced by plus infinity. Results smaller than the smallest representable number are replaced by minus infinity. Results closer to zero than any representable number are replaced by plus or minus epsilon, respectively. Operations with plus or minus epsilon that yield another very small entity of unknown sign are replaced by the CtN symbol. Results of undefined operations, such as division by zero, are replaced by not-a-number.

The largest representable number is (3 ** 72 - 1) / (2 * 3 ** 71) * 3 ** 9841, which is about 3.36 * 10 ** 4695 or about 1.15 * 2 ** 15598. The smallest well-formed representable positive number is (3 ** 71 + 1) / (2 * 3 ** 71) * 3 ** -9841, which is about 2.23 * 10 ** -4696 or about 1.30 * 2 ** -15599.

Numbers with a non-zero mantissa starting with a zero trit are legal to use but not well-formed; such numbers will not occur as operation results.

Numbers with a mantissa of zero and an exponent other than those reserved for the six symbolic values are treated as zero and will not occur as operation results either.

Operation Tables

  +------+------+------+--------+--------+--------+---------+
  |   a  |  -a  |  1/a | sgn(a) | abs(a) |trunc(a)| frac(a) |
  +------+------+------+--------+--------+--------+---------+
  |   0  |   0  |  NaN |    0   |    0   |    0   |     0   |
  |   1  |  -1  |   1  |    1   |    1   |    1   |     0   |
  |  -1  |   1  |  -1  |   -1   |    1   |   -1   |     0   |
  | +Inf | -Inf | +Eps |    1   |  +Inf  |  +Inf  |    NaN  |
  | -Inf | +Inf | -Eps |   -1   |  +Inf  |  -Inf  |    NaN  |
  | +Eps | -Eps | +Inf |    1   |  +Eps  |    0   |   +Eps  |
  | -Eps | +Eps | -Inf |   -1   |  +Eps  |    0   |   -Eps  |
  |  CtN |  CtN |  NaN |   NaN  |  +Eps  |    0   |    CtN  |
  |  NaN |  NaN |  NaN |   NaN  |   NaN  |   NaN  |    NaN  |
  +------+------+------+--------+--------+--------+---------+

  +------+--------+---------+---------+--------+--------+
  |   a  |floor(a)| ceil(a) | sqrt(a) | log(a) | exp(a) |
  +------+--------+---------+---------+--------+--------+
  |   0  |    0   |     0   |    0    |  -Inf  |    1   |
  |   1  |    1   |     1   |    1    |    0   |    e   |
  |  -1  |   -1   |    -1   |   NaN   |   NaN  |   1/e  |
  | +Inf |  +Inf  |   +Inf  |  +Inf   |  +Inf  |  +Inf  |
  | -Inf |  -Inf  |   -Inf  |   NaN   |   NaN  |  +Eps  |
  | +Eps |    0   |     1   |  +Eps   |  -Inf  |    1   |
  | -Eps |   -1   |     0   |   NaN   |   NaN  |    1   |
  |  CtN |   NaN  |    NaN  |   NaN   |   NaN  |    1   |
  |  NaN |   NaN  |    NaN  |   NaN   |   NaN  |   NaN  |
  +------+--------+---------+---------+--------+--------+

  +------+------+------+------+------+------+------+
  |   a  |   b  |  a+b |  a-b |  a*b |  a/b | a**b |
  +------+------+------+------+------+------+------+
  |   0  | +Inf | +Inf | -Inf |  NaN |   0  |  NaN |
  |   0  | -Inf | -Inf | +Inf |  NaN |   0  |  NaN |
  |   0  | +Eps | +Eps | -Eps |   0  |  NaN |  NaN |
  |   0  | -Eps | -Eps | +Eps |   0  |  NaN |  NaN |
  |   0  |  NaN |  NaN |  NaN |  NaN |  NaN |  NaN |
  |   1  | +Inf | +Inf | -Inf | +Inf | +Eps |  NaN |
  |   1  | -Inf | -Inf | +Inf | -Inf | -Eps |  NaN |
  |   1  | +Eps |   1  |   1  | +Eps | +Inf |   1  |
  |   1  | -Eps |   1  |   1  | -Eps | -Inf |   1  |
  |   1  |  NaN |  NaN |  NaN |  NaN |  NaN |  NaN |
  |  -1  | +Inf | +Inf | -Inf | -Inf | -Eps |  NaN |
  |  -1  | -Inf | -Inf | +Inf | +Inf | +Eps |  NaN |
  |  -1  | +Eps |  -1  |  -1  | -Eps | -Inf |  NaN |
  |  -1  | -Eps |  -1  |  -1  | +Eps | +Inf |  NaN |
  |  -1  |  NaN |  NaN |  NaN |  NaN |  NaN |  NaN |
  | +Inf |   0  | +Inf | +Inf |  NaN |  NaN |  NaN |
  | +Inf |   1  | +Inf | +Inf | +Inf | +Inf | +Inf |
  | +Inf |  -1  | +Inf | +Inf | -Inf | -Inf | +Eps |
  | +Inf | +Inf | +Inf |  NaN | +Inf |  NaN | +Inf |
  | +Inf | -Inf |  NaN | +Inf | -Inf |  NaN | +Eps |
  | +Inf | +Eps | +Inf | +Inf |  NaN | +Inf |  NaN |
  | +Inf | -Eps | +Inf | +Inf |  NaN | -Inf |  NaN |
  | +Inf |  NaN |  NaN |  NaN |  NaN |  NaN |  NaN |
  | -Inf |   0  | -Inf | -Inf |  NaN |  NaN |  NaN |
  | -Inf |   1  | -Inf | -Inf | -Inf | -Inf |  NaN |
  | -Inf |  -1  | -Inf | -Inf | +Inf | +Inf |  NaN |
  | -Inf | +Inf |  NaN | -Inf | -Inf |  NaN |  NaN |
  | -Inf | -Inf | -Inf |  NaN | +Inf |  NaN |  NaN |
  | -Inf | +Eps | -Inf | -Inf |  NaN | -Inf |  NaN |
  | -Inf | -Eps | -Inf | -Inf |  NaN | +Inf |  NaN |
  | -Inf |  NaN |  NaN |  NaN |  NaN |  NaN |  NaN |
  +------+------+------+------+------+------+------+

  +------+------+------+------+------+------+------+
  |   a  |   b  |  a+b |  a-b |  a*b |  a/b | a**b |
  +------+------+------+------+------+------+------+
  | +Eps |   0  | +Eps | +Eps |   0  |  NaN |   1  |
  | +Eps |   1  |   1  |  -1  | +Eps | +Eps | +Eps |
  | +Eps |  -1  |  -1  |   1  | -Eps | -Eps | +Inf |
  | +Eps | +Inf | +Inf | -Inf |  NaN | +Eps |  NaN |
  | +Eps | -Inf | -Inf | +Inf |  NaN | -Eps | +Inf |
  | +Eps | +Eps | +Eps |   0  | +Eps |  NaN |  NaN |
  | +Eps | -Eps |   0  | +Eps | -Eps |  NaN |  NaN |
  | +Eps |  NaN |  NaN |  NaN |  NaN |  NaN |  NaN |
  | -Eps |   0  | -Eps | -Eps |   0  |  NaN |  NaN |
  | -Eps |   1  |   1  |  -1  | -Eps | -Eps | -Eps |
  | -Eps |  -1  |  -1  |   1  | +Eps | +Eps | -Inf |
  | -Eps | +Inf | +Inf | -Inf |  NaN | -Eps |  NaN |
  | -Eps | -Inf | -Inf | +Inf |  NaN | +Eps |  NaN |
  | -Eps | +Eps |  CtN | -Eps | -Eps |  NaN |  NaN |
  | -Eps | -Eps | -Eps |  CtN | +Eps |  NaN |  NaN |
  | -Eps |  NaN |  NaN |  NaN |  NaN |  NaN |  NaN |
  |  NaN |   0  |  NaN |  NaN |  NaN |  NaN |  NaN |
  |  NaN |   1  |  NaN |  NaN |  NaN |  NaN |  NaN |
  |  NaN |  -1  |  NaN |  NaN |  NaN |  NaN |  NaN |
  |  NaN | +Inf |  NaN |  NaN |  NaN |  NaN |  NaN |
  |  NaN | -Inf |  NaN |  NaN |  NaN |  NaN |  NaN |
  |  NaN | +Eps |  NaN |  NaN |  NaN |  NaN |  NaN |
  |  NaN | -Eps |  NaN |  NaN |  NaN |  NaN |  NaN |
  |  NaN |  NaN |  NaN |  NaN |  NaN |  NaN |  NaN |
  +------+------+------+------+------+------+------+

Exports

By default, nothing is exported into the caller's namespace. The constructor float81 can be imported explicitly, though.

DEPENDENCIES

This module depends on Math::BigFloat and Math::Logic::Ternary::Word. Installing Math::BigInt::GMP or Math::BigInt::Pari should make it faster.

BUGS AND LIMITATIONS

As of version 0.004, this module is not fully implemented. The documentation is intended as a preview of its eventual content.

However, the definition of the TFP_81 number format, as given here, should be taken as final, and can be used as a reference.

SEE ALSO

AUTHOR

Martin Becker <becker-cpan-mp@cozap.com>

COPYRIGHT AND LICENSE

Copyright (c) 2012-2017 by Martin Becker. All rights reserved.

This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself, either Perl version 5.8.0 or, at your option, any later version of Perl 5 you may have available.