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 Trackers [
http://core.tcl.tk/tcllib/reportlist]. Please also
report any ideas for enhancements you may have for either package and/or
documentation.
KEYWORDS¶
floating-point, math, rounding
CATEGORY¶
Mathematics