math::bigfloat - Arbitrary precision floating-point numbers
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 mantissa and d the uncertainty, representing the limitation of that number's precision. This is why we call such mathematics interval computations. Just take an example in physics : when you measure a temperature, not all digits you read are significant. Sometimes you just cannot 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 arbitrary precision. BigFloats are internally represented at Tcl lists: this package provides a set of procedures operating against the internal representation in order to :
perform math operations on BigFloats and (optionnaly) with integers.
convert BigFloats from their internal representations to strings, and vice versa.
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 number "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]
Returns a string form of a BigFloat, in which all digits are exacts. All exact digits means a rounding may occur, for example 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.
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
Returns a double, that may be used in expr, from a BigFloat.
Returns 1 if number is an integer, 0 otherwise.
Returns 1 if number is a BigFloat, 0 otherwise.
Converts an integer to a BigFloat with decimals trailing zeros. The default, and minimal, number of decimals is 1. When converting 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"
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.
Return the quotient and the rest of x divided by y. Each argument (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.
Returns the absolute value of x
Returns the opposite of x
Returns x taken to the nth power. It only works if n is an integer. x might be a BigFloat or an integer.
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
Returns 1 if x and y are equal, 0 elsewhere.
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.
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.
Returns a BigFloat representing the Pi constant with n digits after the dot. n is a positive integer.
radians - angle expressed in radians (BigFloat)
degrees - angle expressed in degrees (BigFloat)
Convert an angle from radians to degrees, and vice versa.
How do conversions work with precision ?
When a BigFloat is converted from string, the internal representation holds its uncertainty as 1 at the level of the last digit.
During computations, the uncertainty of each result is internally computed the closest to the reality, thus saving the memory 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.
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::*
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]
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 Trackers. Please also report any ideas for enhancements you may have for either package and/or documentation.
When proposing code changes, please provide unified diffs, i.e the output of diff -u.
Note further that attachments are strongly preferred over inlined patches. Attachments can be made by going to the Edit form of the ticket immediately after its creation, and then using the left-most button in the secondary navigation bar.
Mathematics
Copyright © 2004-2008, by Stephane Arnold <stephanearnold at yahoo dot fr>