math::fuzzy(n) Math math::fuzzy(n)

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

SYNOPSIS
       ::::mmaatthh::::ffuuzzzzyy::::tteeqq value1 value2

       ::::mmaatthh::::ffuuzzzzyy::::ttnnee value1 value2

       ::::mmaatthh::::ffuuzzzzyy::::ttggee value1 value2

       ::::mmaatthh::::ffuuzzzzyy::::ttllee value1 value2

       ::::mmaatthh::::ffuuzzzzyy::::ttlltt value1 value2

       ::::mmaatthh::::ffuuzzzzyy::::ttggtt value1 value2

       ::::mmaatthh::::ffuuzzzzyy::::ttfflloooorr value

       ::::mmaatthh::::ffuuzzzzyy::::ttcceeiill value

       ::::mmaatthh::::ffuuzzzzyy::::ttrroouunndd value

       ::::mmaatthh::::ffuuzzzzyy::::ttrroouunnddnn value ndigits



DESCRIPTION
       The package Fuzzy is meant to solve common problems with floating-point
       numbers in a systematic way:

       +o      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.

       +o      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.

PPRROOCCEEDDUURREESS
       Effectively the package provides the following procedures:

       ::::mmaatthh::::ffuuzzzzyy::::tteeqq value1 value2
              Compares  two floating-point numbers and returns 1 if their val-
              ues fall within a small range. Otherwise it returns 0.

       ::::mmaatthh::::ffuuzzzzyy::::ttnnee value1 value2
              Returns the negation, that is, if the difference is larger  than
              the margin, it returns 1.

       ::::mmaatthh::::ffuuzzzzyy::::ttggee 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.

       ::::mmaatthh::::ffuuzzzzyy::::ttllee value1 value2
              Returns  1 if the two numbers are equal according to [teq] or if
              the first is smaller than the second.

       ::::mmaatthh::::ffuuzzzzyy::::ttlltt value1 value2
              Returns the opposite of [tge].

       ::::mmaatthh::::ffuuzzzzyy::::ttggtt value1 value2
              Returns the opposite of [tle].

       ::::mmaatthh::::ffuuzzzzyy::::ttfflloooorr value
              Returns the integer number that is lower or equal to  the  given
              floating-point number, within a well-defined tolerance.

       ::::mmaatthh::::ffuuzzzzyy::::ttcceeiill value
              Returns the integer number that is greater or equal to the given
              floating-point number, within a well-defined tolerance.

       ::::mmaatthh::::ffuuzzzzyy::::ttrroouunndd value
              Rounds the floating-point number off.

       ::::mmaatthh::::ffuuzzzzyy::::ttrroouunnddnn 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]

TTEESSTT CCAASSEESS
       The problems that can occur with floating-point numbers are illustrated
       by the test cases in the file "fuzzy.test":

       +o      Several test case use the ordinary comparisons,  and  they  fail
              invariably to produce understandable results

       +o      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.

KKEEYYWWOORRDDSS
       floating-point, math, rounding



math                                  1.0                       math::fuzzy(n)