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math::fuzzy(n) 1.0 "Math"
math::fuzzy - Fuzzy comparison of floating-point numbers
TABLE OF
CONTENTS
SYNOPSIS
DESCRIPTION
PROCEDURES
TEST CASES
REFERENCES
KEYWORDS
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.
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:
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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]
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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.
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.
floating-point , math , rounding