math::bigfloat - Arbitrary precision floating-point numbers
TABLE OF CONTENTS
SYNOPSIS
DESCRIPTION
INTRODUCTION
ARITHMETICS
COMPARISONS
ANALYSIS
ROUNDING
PRECISION
WHAT ABOUT TCL 8.4 ?
NAMESPACES AND OTHER PACKAGES
EXAMPLES
BUGS, IDEAS, FEEDBACK
KEYWORDS
COPYRIGHT
package require Tcl 8.5
package require math::bigfloat ?2.0?
The bigfloat package provides arbitrary precision floating-point math capabilities to the Tcl language. It is designed to work with Tcl 8.5, but for Tcl 8.4 is provided an earlier version of this package. See WHAT ABOUT TCL 8.4 ? for more explanations. By convention, we will talk about the numbers treated in this library as :
# x and y are BigFloats : the first string contained a dot, and the second an e sign set x [fromstr -1.000000] set y [fromstr 2000e30] # let's see how we get integers set t 20000000000000 # the old way (package 1.2) is still supported for backwards compatibility : set m [fromstr 10000000000] # but we do not need fromstr for integers anymore set n -39 # t, m and n are integers |
set x [fromstr 1.0000000000] # the next line does the same, but smarter set y [fromstr 1. 10] |
puts [tostr [fromstr 0.99999]] ;# 1.0000 puts [tostr [fromstr 1.00001]] ;# 1.0000 puts [tostr [fromstr 0.002]] ;# 0.e-2 |
tostr [fromstr 1.111 4] # returns : 1.111000 (3 zeros) tostr [fromdouble 1.111 4] # returns : 1.111 |
set n 10 set x [int2float $n]; # like fromstr 10.0 puts [tostr $x]; # prints "10." set x [int2float $n 3]; # like fromstr 10.000 puts [tostr $x]; # prints "10.00" |
tostr [fromstr 0.001] ; # -> 0.e-2 tostr [fromstr 0.000000] ; # -> 0.e-5 tostr [fromstr -0.000001] ; # -> 0.e-5 tostr [fromstr 0.0] ; # -> 0. tostr [fromstr 0.002] ; # -> 0.e-2 set a [fromstr 0.002] ; # uncertainty interval : 0.001, 0.003 tostr $a ; # 0.e-2 iszero $a ; # false set a [fromstr 0.001] ; # uncertainty interval : 0.000, 0.002 tostr $a ; # 0.e-2 iszero $a ; # true |
How do conversions work with precision ?
Now you may ask this question : What precision am I going to get after calling add, sub, mul or div? First you set a number from the string representation and, by the way, its uncertainty is set:
set a [fromstr 1.230] # $a belongs to [1.229, 1.231] set a [fromstr 1.000] # $a belongs to [0.999, 1.001] # $a has a relative uncertainty of 0.1% : 0.001(the uncertainty)/1.000(the medium value) |
set a [fromstr 1.230] set b [fromstr 2.340] set sum [add $a $b]] # the result is : [3.568, 3.572] (the last digit is known with an uncertainty of 2) tostr $sum ; # 3.57 |
set a [fromstr 0.999999999] # now something dangerous set b [fromstr 2.000] # the result has only 3 digits tostr [add $a $b] # how to keep precision at its maximum puts [tostr [add $a 2]] |
For multiplication and division, the relative uncertainties of the product or the quotient, is the sum of the relative uncertainties of the operands. Take care of division by zero : check each divider with iszero.
set num [fromstr 4.00] set denom [fromstr 0.01] puts [iszero $denom];# true set quotient [div $num $denom];# error : divide by zero # opposites of our operands puts [compare $num [opp $num]]; # 1 puts [compare $denom [opp $denom]]; # 0 !!! # No suprise ! 0 and its opposite are the same... |
puts [tostr [cos [fromstr 0. 10]]]; # -> 1.000000000 puts [tostr [cos [fromstr 0. 5]]]; # -> 1.0000 puts [tostr [cos [fromstr 0e-10]]]; # -> 1.000000000 puts [tostr [cos [fromstr 1e-10]]]; # -> 1.000000000 |
For most analysis functions (cosine, square root, logarithm, etc.), determining the precision of the result is difficult. It seems however that in many cases, the loss of precision in the result is of one or two digits. There are some exceptions : for example,
tostr [exp [fromstr 100.0 10]] # returns : 2.688117142e+43 which has only 10 digits of precision, although the entry # has 14 digits of precision. |
If your setup do not provide Tcl 8.5 but supports 8.4, the package can still be loaded, switching back to math::bigfloat 1.2. Indeed, an important function introduced in Tcl 8.5 is required - the ability to handle bignums, that we can do with expr. Before 8.5, this ability was provided by several packages, including the pure-Tcl math::bignum package provided by tcllib. In this case, all you need to know, is that arguments to the commands explained here, are expected to be in their internal representation. So even with integers, you will need to call fromstr and tostr in order to convert them between string and internal representations.
# # with Tcl 8.5 # ============ set a [pi 20] # round returns an integer and 'everything is a string' applies to integers # whatever big they are puts [round [mul $a 10000000000]] # # the same with Tcl 8.4 # ===================== set a [pi 20] # bignums (arbitrary length integers) need a conversion hook set b [fromstr 10000000000] # round returns a bignum: # before printing it, we need to convert it with 'tostr' puts [tostr [round [mul $a $b]]] |
We have not yet discussed about namespaces because we assumed that you had imported public commands into the global namespace, like this:
namespace import ::math::bigfloat::* |
package require math::bigfloat # beware: namespace ensembles are not available in Tcl 8.4 namespace eval ::math::bigfloat {namespace ensemble create -command ::bigfloat} # from now, the bigfloat command takes as subcommands all original math::bigfloat::* commands set a [bigfloat sub [bigfloat fromstr 2.000] [bigfloat fromstr 0.530]] puts [bigfloat tostr $a] |
Guess what happens when you are doing some astronomy. Here is an example :
# convert acurrate angles with a millisecond-rated accuracy proc degree-angle {degrees minutes seconds milliseconds} { set result 0 set div 1 foreach factor {1 1000 60 60} var [list $milliseconds $seconds $minutes $degrees] { # we convert each entry var into milliseconds set div [expr {$div*$factor}] incr result [expr {$var*$div}] } return [div [int2float $result] $div] } # load the package package require math::bigfloat namespace import ::math::bigfloat::* # work with angles : a standard formula for navigation (taking bearings) set angle1 [deg2rad [degree-angle 20 30 40 0]] set angle2 [deg2rad [degree-angle 21 0 50 500]] set opposite3 [deg2rad [degree-angle 51 0 50 500]] set sinProduct [mul [sin $angle1] [sin $angle2]] set cosProduct [mul [cos $angle1] [cos $angle2]] set angle3 [asin [add [mul $sinProduct [cos $opposite3]] $cosProduct]] puts "angle3 : [tostr [rad2deg $angle3]]" |
This document, and the package it describes, will undoubtedly contain bugs and other problems. Please report such in the category math :: bignum :: float of the Tcllib SF Trackers. Please also report any ideas for enhancements you may have for either package and/or documentation.
computations, floating-point, interval, math, multiprecision, tcl
Copyright © 2004-2005, by Stephane Arnold <stephanearnold at yahoo dot fr>