math::bigfloat(n) Tcl Math Library math::bigfloat(n)______________________________________________________________________________NAME
math::bigfloat - Arbitrary precision floating-point numbers
SYNOPSIS
package require Tcl 8.5
package require math::bigfloat ?2.0.1?
fromstr number ?trailingZeros?
tostr ?-nosci? number
fromdouble double ?decimals?
todouble number
isInt number
isFloat number
int2float integer ?decimals?
add x y
sub x y
mul x y
div x y
mod x y
abs x
opp x
pow x n
iszero x
equal x y
compare x y
sqrt x
log x
exp x
cos x
sin x
tan x
cotan x
acos x
asin x
atan x
cosh x
sinh x
tanh x
pi n
rad2deg radians
deg2rad degrees
round x
ceil x
floor x
_________________________________________________________________DESCRIPTION
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 :
· BigFloat for floating-point numbers of arbitrary length.
· integers for arbitrary length signed integers, just as basic
integers since Tcl 8.5.
Each BigFloat is an interval, namely [m-d, m+d], where m is the man‐
tissa and d the uncertainty, representing the limitation of that num‐
ber's precision. This is why we call such mathematics interval compu‐
tations. Just take an example in physics : when you measure a tempera‐
ture, not all digits you read are significant. Sometimes you just can‐
not trust all digits - not to mention if doubles (f.p. numbers) can
handle all these digits. BigFloat can handle this problem - trusting
the digits you get - plus the ability to store numbers with an arbi‐
trary precision. BigFloats are internally represented at Tcl lists:
this package provides a set of procedures operating against the inter‐
nal representation in order to :
· perform math operations on BigFloats and (optionnaly) with inte‐
gers.
· convert BigFloats from their internal representations to
strings, and vice versa.
INTRODUCTION
fromstr number ?trailingZeros?
Converts number into a BigFloat. Its precision is at least the
number of digits provided by number. If the number contains
only digits and eventually a minus sign, it is considered as an
integer. Subsequently, no conversion is done at all.
trailingZeros - the number of zeros to append at the end of the
floating-point number to get more precision. It cannot be
applied to an integer.
# 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
The number's last digit is considered by the procedure to be true at
+/-1, For example, 1.00 is the interval [0.99, 1.01], and 0.43 the
interval [0.42, 0.44]. The Pi constant may be approximated by the num‐
ber "3.1415". This string could be considered as the interval [3.1414
, 3.1416] by fromstr. So, when you mean 1.0 as a double, you may have
to write 1.000000 to get enough precision. To learn more about this
subject, see PRECISION.
For example :
set x [fromstr 1.0000000000]
# the next line does the same, but smarter
set y [fromstr 1. 10]
tostr ?-nosci? number
Returns a string form of a BigFloat, in which all digits are
exacts. All exact digits means a rounding may occur, for exam‐
ple to zero, if the uncertainty interval does not clearly show
the true digits. number may be an integer, causing the command
to return exactly the input argument. With the -nosci option,
the number returned is never shown in scientific notation, i.e.
not like '3.4523e+5' but like '345230.'.
puts [tostr [fromstr 0.99999]] ;# 1.0000
puts [tostr [fromstr 1.00001]] ;# 1.0000
puts [tostr [fromstr 0.002]] ;# 0.e-2
See PRECISION for that matter. See also iszero for how to
detect zeros, which is useful when performing a division.
fromdouble double ?decimals?
Converts a double (a simple floating-point value) to a BigFloat,
with exactly decimals digits. Without the decimals argument, it
behaves like fromstr. Here, the only important feature you
might care of is the ability to create BigFloats with a fixed
number of decimals.
tostr [fromstr 1.111 4]
# returns : 1.111000 (3 zeros)
tostr [fromdouble 1.111 4]
# returns : 1.111
todouble number
Returns a double, that may be used in expr, from a BigFloat.
isInt number
Returns 1 if number is an integer, 0 otherwise.
isFloat number
Returns 1 if number is a BigFloat, 0 otherwise.
int2float integer ?decimals?
Converts an integer to a BigFloat with decimals trailing zeros.
The default, and minimal, number of decimals is 1. When con‐
verting back to string, one decimal is lost:
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"
ARITHMETICS
add x y
sub x y
mul x y
Return the sum, difference and product of x by y. x - may be
either a BigFloat or an integer y - may be either a BigFloat or
an integer When both are integers, these commands behave like
expr.
div x y
mod x y
Return the quotient and the rest of x divided by y. Each argu‐
ment (x and y) can be either a BigFloat or an integer, but you
cannot divide an integer by a BigFloat Divide by zero throws an
error.
abs x Returns the absolute value of x
opp x Returns the opposite of x
pow x n
Returns x taken to the nth power. It only works if n is an
integer. x might be a BigFloat or an integer.
COMPARISONS
iszero x
Returns 1 if x is :
· a BigFloat close enough to zero to raise "divide by
zero".
· the integer 0.
See here how numbers that are close to zero are converted to
strings:
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
equal x y
Returns 1 if x and y are equal, 0 elsewhere.
compare x y
Returns 0 if both BigFloat arguments are equal, 1 if x is
greater than y, and -1 if x is lower than y. You would not be
able to compare an integer to a BigFloat : the operands should
be both BigFloats, or both integers.
ANALYSIS
sqrt x
log x
exp x
cos x
sin x
tan x
cotan x
acos x
asin x
atan x
cosh x
sinh x
tanh x The above functions return, respectively, the following : square
root, logarithm, exponential, cosine, sine, tangent, cotangent,
arc cosine, arc sine, arc tangent, hyperbolic cosine, hyperbolic
sine, hyperbolic tangent, of a BigFloat named x.
pi n Returns a BigFloat representing the Pi constant with n digits
after the dot. n is a positive integer.
rad2deg radians
deg2rad degrees
radians - angle expressed in radians (BigFloat)
degrees - angle expressed in degrees (BigFloat)
Convert an angle from radians to degrees, and vice versa.
ROUNDING
round x
ceil x
floor x
The above functions return the x BigFloat, rounded like with the
same mathematical function in expr, and returns it as an inte‐
ger.
PRECISION
How do conversions work with precision ?
· When a BigFloat is converted from string, the internal represen‐
tation holds its uncertainty as 1 at the level of the last
digit.
· During computations, the uncertainty of each result is inter‐
nally computed the closest to the reality, thus saving the mem‐
ory used.
· When converting back to string, the digits that are printed are
not subject to uncertainty. However, some rounding is done, as
not doing so causes severe problems.
Uncertainties are kept in the internal representation of the number ;
it is recommended to use tostr only for outputting data (on the screen
or in a file), and NEVER call fromstr with the result of tostr. It is
better to always keep operands in their internal representation. Due
to the internals of this library, the uncertainty interval may be
slightly wider than expected, but this should not cause false digits.
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)
The uncertainty of the sum, or the difference, of two numbers, is the
sum of their respective uncertainties.
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
But when, for example, we add or substract an integer to a BigFloat,
the relative uncertainty of the result is unchanged. So it is desirable
not to convert integers to BigFloats:
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...
Effects of the precision of a number considered equal to zero to the
cos function:
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
BigFloats with different internal representations may be converted to
the same string.
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.
WHAT ABOUT TCL 8.4 ?
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 pack‐
age 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]]]
NAMESPACES AND OTHER PACKAGES
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::*
If you matter much about avoiding names conflicts, I considere it
should be resolved by the following :
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]
EXAMPLES
Guess what happens when you are doing some astronomy. Here is an exam‐
ple :
# 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]]"
BUGS, IDEAS, FEEDBACK
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 [http://source‐
forge.net/tracker/?group_id=12883]. Please also report any ideas for
enhancements you may have for either package and/or documentation.
KEYWORDS
computations, floating-point, interval, math, multiprecision, tcl
CATEGORY
Mathematics
COPYRIGHT
Copyright (c) 2004-2008, by Stephane Arnold <stephanearnold at yahoo dot fr>
math 2.0.1 math::bigfloat(n)
