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math::fuzzy(n)		       Tcl Math Library			math::fuzzy(n)

______________________________________________________________________________

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 val‐
	      ues 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 val‐
	      ues 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 ver‐
       sion.

REFERENCES
       Original implementation in Fortran by dr. H.D. Knoble (Penn State  Uni‐
       versity).

       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://source‐
       forge.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

math				      0.2			math::fuzzy(n)
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