NAME

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

SYNOPSIS

       package require Tcl  ?8.3?

       package require math::fuzzy  ?0.2?

       ::math::fuzzy::teq value1 value2

       ::math::fuzzy::tne value1 value2

       ::math::fuzzy::tge value1 value2

       ::math::fuzzy::tle value1 value2

       ::math::fuzzy::tlt value1 value2

       ::math::fuzzy::tgt value1 value2

       ::math::fuzzy::tfloor value

       ::math::fuzzy::tceil value

       ::math::fuzzy::tround value

       ::math::fuzzy::troundn value ndigits

_________________________________________________________________

DESCRIPTION

       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.

PROCEDURES

       Effectively the package provides the following procedures:

       ::math::fuzzy::teq value1 value2
              Compares  two  floating-point  numbers  and returns 1 if their values fall within a
              small range. Otherwise it returns 0.

       ::math::fuzzy::tne value1 value2
              Returns the negation, that is, if the difference is  larger  than  the  margin,  it
              returns 1.

       ::math::fuzzy::tge value1 value2
              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::tle value1 value2
              Returns  1  if  the  two  numbers  are  equal according to [teq] or if the first is
              smaller than the second.

       ::math::fuzzy::tlt value1 value2
              Returns the opposite of [tge].

       ::math::fuzzy::tgt value1 value2
              Returns the opposite of [tle].

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

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

       ::math::fuzzy::tround value
              Rounds the floating-point number off.

       ::math::fuzzy::troundn value ndigits
              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]

TEST CASES

       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.

REFERENCES

       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.

BUGS, IDEAS, FEEDBACK

       This  document,  and  the  package  it  describes, will undoubtedly contain bugs and other
       problems.  Please report such in the category math :: fuzzy  of  the  Tcllib  SF  Trackers
       [http://sourceforge.net/tracker/?group_id=12883].    Please  also  report  any  ideas  for
       enhancements you may have for either package and/or documentation.

KEYWORDS

       floating-point, math, rounding

CATEGORY

       Mathematics