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math::fuzzy(n) 0.2 "Tcl Math Library"
Name
math::fuzzy  Fuzzy comparison of floatingpoint 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 floatingpoint 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 floatingpoint 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 floatingpoint numbers, this is typically 1.1e16.
PROCEDURES
Effectively the package provides the following procedures:
 ::math::fuzzy::teq value1 value2

Compares two floatingpoint 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 floatingpoint 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 floatingpoint number, within a welldefined tolerance.
 ::math::fuzzy::tceil value

Returns the integer number that is greater or equal to the given floatingpoint number, within a welldefined tolerance.
 ::math::fuzzy::tround value

Rounds the floatingpoint number off.
 ::math::fuzzy::troundn value ndigits

Rounds the floatingpoint 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 floatingpoint 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 Tclonly package, there is also a Cbased 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):2024, 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):1623, March 1978.
D. Knuth, Art of Computer Programming, Vol. 1, Problem 1.2.45.
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 Trackers. Please also report any ideas for enhancements you may have for either package and/or documentation.
Category
Mathematics