math::fuzzy - Fuzzy comparison of floating-point numbers

       This  document,  and  the package it describes, will undoubtedly contain bugs and other problems.  Please
       report such in the category math::fuzzy of the TcllibTrackers  [http://core.tcl.tk/tcllib/reportlist].
       Please also report any ideas for enhancements you may have for either package and/or documentation.

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       Mathematics

tcllib                                                 0.2                                     math::fuzzy(3tcl)

       The package Fuzzy is meant to solve common problems with floating-point numbers in a systematic way:

       •      Comparing  two  numbers  that  are  "supposed"  to be identical, like 1.0 and 2.1/(1.2+0.9) is not
              guaranteed to give the intuitive result.

       •      Rounding  a  number  that  is  halfway  two  integer  numbers  can  cause  strange  errors,   like
              int(100.0*2.8) != 28 but 27

       The  Fuzzy  package  is  meant  to  help sorting out this type of problems by defining "fuzzy" comparison
       procedures for floating-point numbers.  It does so by allowing for a  small  margin  that  is  determined
       automatically  -  the  margin is three times the "epsilon" value, that is three times the smallest number
       eps such that 1.0 and 1.0+$eps canbe distinguished. In Tcl, which uses  double  precision  floating-point
       numbers, this is typically 1.1e-16.

       floating-point, math, rounding

       math::fuzzy - Fuzzy comparison of floating-point numbers

       Effectively the package provides the following procedures:

       ::math::fuzzy::teqvalue1value2
              Compares  two  floating-point  numbers  and  returns  1 if their values fall within a small range.
              Otherwise it returns 0.

       ::math::fuzzy::tnevalue1value2
              Returns the negation, that is, if the difference is larger than the margin, it returns 1.

       ::math::fuzzy::tgevalue1value2
              Compares two floating-point numbers and returns 1 if their values either fall within a small range
              or if the first number is larger than the second. Otherwise it returns 0.

       ::math::fuzzy::tlevalue1value2
              Returns 1 if the two numbers are equal according to [teq] or if the  first  is  smaller  than  the
              second.

       ::math::fuzzy::tltvalue1value2
              Returns the opposite of [tge].

       ::math::fuzzy::tgtvalue1value2
              Returns the opposite of [tle].

       ::math::fuzzy::tfloorvalue
              Returns  the  integer  number  that is lower or equal to the given floating-point number, within a
              well-defined tolerance.

       ::math::fuzzy::tceilvalue
              Returns the integer number that is greater or equal to the given floating-point number,  within  a
              well-defined tolerance.

       ::math::fuzzy::troundvalue
              Rounds the floating-point number off.

       ::math::fuzzy::troundnvaluendigits
              Rounds the floating-point number off to the specified number of decimals (Pro memorie).

       Usage:

              if { [teq $x $y] } { puts "x == y" }
              if { [tne $x $y] } { puts "x != y" }
              if { [tge $x $y] } { puts "x >= y" }
              if { [tgt $x $y] } { puts "x > y" }
              if { [tlt $x $y] } { puts "x < y" }
              if { [tle $x $y] } { puts "x <= y" }

              set fx      [tfloor $x]
              set fc      [tceil  $x]
              set rounded [tround $x]
              set roundn  [troundn $x $nodigits]

       Original implementation in Fortran by dr. H.D. Knoble (Penn State University).

       P. E. Hagerty, "More on Fuzzy Floor and Ceiling,"  APL  QUOTE  QUAD  8(4):20-24,  June  1978.  Note  that
       TFLOOR=FL5 took five years of refereed evolution (publication).

       L. M. Breed, "Definitions for Fuzzy Floor and Ceiling", APL QUOTE QUAD 8(3):16-23, March 1978.

       D. Knuth, Art of Computer Programming, Vol. 1, Problem 1.2.4-5.

       package require Tcl?8.3?

       package require math::fuzzy?0.2?::math::fuzzy::teqvalue1value2::math::fuzzy::tnevalue1value2::math::fuzzy::tgevalue1value2::math::fuzzy::tlevalue1value2::math::fuzzy::tltvalue1value2::math::fuzzy::tgtvalue1value2::math::fuzzy::tfloorvalue::math::fuzzy::tceilvalue::math::fuzzy::troundvalue::math::fuzzy::troundnvaluendigits

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       The  problems  that  can  occur with floating-point numbers are illustrated by the test cases in the file
       "fuzzy.test":

       •      Several test case use the ordinary comparisons, and they fail invariably to produce understandable
              results

       •      One test case uses [expr] without braces ({ and }). It too fails.

       The conclusion from this is that any expression should be surrounded by braces,  because  otherwise  very
       awkward  things  can  happen  if  you need accuracy. Furthermore, accuracy and understandable results are
       enhanced by using these "tolerant" or fuzzy comparisons.

       Note that besides the Tcl-only package, there is also a C-based version.