1math::bigfloat(n)              Tcl Math Library              math::bigfloat(n)
2
3
4
5______________________________________________________________________________
6

NAME

8       math::bigfloat - Arbitrary precision floating-point numbers
9

SYNOPSIS

11       package require Tcl  8.5
12
13       package require math::bigfloat  ?2.0.1?
14
15       fromstr number ?trailingZeros?
16
17       tostr ?-nosci? number
18
19       fromdouble double ?decimals?
20
21       todouble number
22
23       isInt number
24
25       isFloat number
26
27       int2float integer ?decimals?
28
29       add x y
30
31       sub x y
32
33       mul x y
34
35       div x y
36
37       mod x y
38
39       abs x
40
41       opp x
42
43       pow x n
44
45       iszero x
46
47       equal x y
48
49       compare x y
50
51       sqrt x
52
53       log x
54
55       exp x
56
57       cos x
58
59       sin x
60
61       tan x
62
63       cotan x
64
65       acos x
66
67       asin x
68
69       atan x
70
71       cosh x
72
73       sinh x
74
75       tanh x
76
77       pi n
78
79       rad2deg radians
80
81       deg2rad degrees
82
83       round x
84
85       ceil x
86
87       floor x
88
89______________________________________________________________________________
90

DESCRIPTION

92       The  bigfloat  package provides arbitrary precision floating-point math
93       capabilities to the Tcl language. It is designed to work with Tcl  8.5,
94       but  for  Tcl  8.4 is provided an earlier version of this package.  See
95       WHAT ABOUT TCL 8.4 ? for more explanations.   By  convention,  we  will
96       talk about the numbers treated in this library as :
97
98       ·      BigFloat for floating-point numbers of arbitrary length.
99
100       ·      integers  for  arbitrary  length  signed integers, just as basic
101              integers since Tcl 8.5.
102
103       Each BigFloat is an interval, namely [m-d, m+d], where m  is  the  man‐
104       tissa  and  d the uncertainty, representing the limitation of that num‐
105       ber's precision.  This is why we call such mathematics interval  compu‐
106       tations.  Just take an example in physics : when you measure a tempera‐
107       ture, not all digits you read are significant. Sometimes you just  can‐
108       not  trust  all  digits  - not to mention if doubles (f.p. numbers) can
109       handle all these digits.  BigFloat can handle this problem  -  trusting
110       the  digits  you  get - plus the ability to store numbers with an arbi‐
111       trary precision.  BigFloats are internally represented  at  Tcl  lists:
112       this  package provides a set of procedures operating against the inter‐
113       nal representation in order to :
114
115       ·      perform math operations on BigFloats and (optionnaly) with inte‐
116              gers.
117
118       ·      convert   BigFloats   from  their  internal  representations  to
119              strings, and vice versa.
120

INTRODUCTION

122       fromstr number ?trailingZeros?
123              Converts number into a BigFloat. Its precision is at  least  the
124              number  of  digits  provided  by number.  If the number contains
125              only digits and eventually a minus sign, it is considered as  an
126              integer. Subsequently, no conversion is done at all.
127
128              trailingZeros  - the number of zeros to append at the end of the
129              floating-point number  to  get  more  precision.  It  cannot  be
130              applied to an integer.
131
132
133              # x and y are BigFloats : the first string contained a dot, and the second an e sign
134              set x [fromstr -1.000000]
135              set y [fromstr 2000e30]
136              # let's see how we get integers
137              set t 20000000000000
138              # the old way (package 1.2) is still supported for backwards compatibility :
139              set m [fromstr 10000000000]
140              # but we do not need fromstr for integers anymore
141              set n -39
142              # t, m and n are integers
143
144
145       The  number's  last  digit is considered by the procedure to be true at
146       +/-1, For example, 1.00 is the interval  [0.99,  1.01],  and  0.43  the
147       interval [0.42, 0.44].  The Pi constant may be approximated by the num‐
148       ber "3.1415".  This string could be considered as the interval  [3.1414
149       ,  3.1416] by fromstr.  So, when you mean 1.0 as a double, you may have
150       to write 1.000000 to get enough precision.  To learn  more  about  this
151       subject, see PRECISION.
152
153       For example :
154
155
156              set x [fromstr 1.0000000000]
157              # the next line does the same, but smarter
158              set y [fromstr 1. 10]
159
160
161       tostr ?-nosci? number
162              Returns  a  string  form  of a BigFloat, in which all digits are
163              exacts.  All exact digits means a rounding may occur, for  exam‐
164              ple  to  zero, if the uncertainty interval does not clearly show
165              the true digits.  number may be an integer, causing the  command
166              to  return  exactly the input argument.  With the -nosci option,
167              the number returned is never shown in scientific notation,  i.e.
168              not like '3.4523e+5' but like '345230.'.
169
170
171              puts [tostr [fromstr 0.99999]] ;# 1.0000
172              puts [tostr [fromstr 1.00001]] ;# 1.0000
173              puts [tostr [fromstr 0.002]] ;# 0.e-2
174
175
176              See  PRECISION  for  that  matter.   See  also iszero for how to
177              detect zeros, which is useful when performing a division.
178
179       fromdouble double ?decimals?
180              Converts a double (a simple floating-point value) to a BigFloat,
181              with exactly decimals digits.  Without the decimals argument, it
182              behaves like fromstr.  Here,  the  only  important  feature  you
183              might  care  of  is the ability to create BigFloats with a fixed
184              number of decimals.
185
186
187              tostr [fromstr 1.111 4]
188              # returns : 1.111000 (3 zeros)
189              tostr [fromdouble 1.111 4]
190              # returns : 1.111
191
192
193       todouble number
194              Returns a double, that may be used in expr, from a BigFloat.
195
196       isInt number
197              Returns 1 if number is an integer, 0 otherwise.
198
199       isFloat number
200              Returns 1 if number is a BigFloat, 0 otherwise.
201
202       int2float integer ?decimals?
203              Converts an integer to a BigFloat with decimals trailing  zeros.
204              The  default,  and  minimal, number of decimals is 1.  When con‐
205              verting back to string, one decimal is lost:
206
207
208              set n 10
209              set x [int2float $n]; # like fromstr 10.0
210              puts [tostr $x]; # prints "10."
211              set x [int2float $n 3]; # like fromstr 10.000
212              puts [tostr $x]; # prints "10.00"
213
214

ARITHMETICS

216       add x y
217
218       sub x y
219
220       mul x y
221              Return the sum, difference and product of x by y.  x  -  may  be
222              either  a BigFloat or an integer y - may be either a BigFloat or
223              an integer When both are integers, these  commands  behave  like
224              expr.
225
226       div x y
227
228       mod x y
229              Return  the quotient and the rest of x divided by y.  Each argu‐
230              ment (x and y) can be either a BigFloat or an integer,  but  you
231              cannot  divide an integer by a BigFloat Divide by zero throws an
232              error.
233
234       abs x  Returns the absolute value of x
235
236       opp x  Returns the opposite of x
237
238       pow x n
239              Returns x taken to the nth power.  It only  works  if  n  is  an
240              integer.  x might be a BigFloat or an integer.
241

COMPARISONS

243       iszero x
244              Returns 1 if x is :
245
246              ·      a  BigFloat  close  enough  to  zero  to raise "divide by
247                     zero".
248
249              ·      the integer 0.
250
251              See here how numbers that are close to  zero  are  converted  to
252              strings:
253
254
255              tostr [fromstr 0.001] ; # -> 0.e-2
256              tostr [fromstr 0.000000] ; # -> 0.e-5
257              tostr [fromstr -0.000001] ; # -> 0.e-5
258              tostr [fromstr 0.0] ; # -> 0.
259              tostr [fromstr 0.002] ; # -> 0.e-2
260
261              set a [fromstr 0.002] ; # uncertainty interval : 0.001, 0.003
262              tostr  $a ; # 0.e-2
263              iszero $a ; # false
264
265              set a [fromstr 0.001] ; # uncertainty interval : 0.000, 0.002
266              tostr  $a ; # 0.e-2
267              iszero $a ; # true
268
269
270       equal x y
271              Returns 1 if x and y are equal, 0 elsewhere.
272
273       compare x y
274              Returns  0  if  both  BigFloat  arguments  are  equal, 1 if x is
275              greater than y, and -1 if x is lower than y.  You would  not  be
276              able  to  compare an integer to a BigFloat : the operands should
277              be both BigFloats, or both integers.
278

ANALYSIS

280       sqrt x
281
282       log x
283
284       exp x
285
286       cos x
287
288       sin x
289
290       tan x
291
292       cotan x
293
294       acos x
295
296       asin x
297
298       atan x
299
300       cosh x
301
302       sinh x
303
304       tanh x The above functions return, respectively, the following : square
305              root,  logarithm, exponential, cosine, sine, tangent, cotangent,
306              arc cosine, arc sine, arc tangent, hyperbolic cosine, hyperbolic
307              sine, hyperbolic tangent, of a BigFloat named x.
308
309       pi n   Returns  a  BigFloat  representing the Pi constant with n digits
310              after the dot.  n is a positive integer.
311
312       rad2deg radians
313
314       deg2rad degrees
315              radians - angle expressed in radians (BigFloat)
316
317              degrees - angle expressed in degrees (BigFloat)
318
319              Convert an angle from radians to degrees, and vice versa.
320

ROUNDING

322       round x
323
324       ceil x
325
326       floor x
327              The above functions return the x BigFloat, rounded like with the
328              same  mathematical  function in expr, and returns it as an inte‐
329              ger.
330

PRECISION

332       How do conversions work with precision ?
333
334       ·      When a BigFloat is converted from string, the internal represen‐
335              tation  holds  its  uncertainty  as  1  at the level of the last
336              digit.
337
338       ·      During computations, the uncertainty of each  result  is  inter‐
339              nally  computed the closest to the reality, thus saving the mem‐
340              ory used.
341
342       ·      When converting back to string, the digits that are printed  are
343              not  subject  to uncertainty. However, some rounding is done, as
344              not doing so causes severe problems.
345
346       Uncertainties are kept in the internal representation of the  number  ;
347       it  is recommended to use tostr only for outputting data (on the screen
348       or in a file), and NEVER call fromstr with the result of tostr.  It  is
349       better  to  always keep operands in their internal representation.  Due
350       to the internals of this  library,  the  uncertainty  interval  may  be
351       slightly wider than expected, but this should not cause false digits.
352
353       Now  you may ask this question : What precision am I going to get after
354       calling add, sub, mul or div?  First you set a number from  the  string
355       representation and, by the way, its uncertainty is set:
356
357
358              set a [fromstr 1.230]
359              # $a belongs to [1.229, 1.231]
360              set a [fromstr 1.000]
361              # $a belongs to [0.999, 1.001]
362              # $a has a relative uncertainty of 0.1% : 0.001(the uncertainty)/1.000(the medium value)
363
364       The  uncertainty  of the sum, or the difference, of two numbers, is the
365       sum of their respective uncertainties.
366
367
368              set a [fromstr 1.230]
369              set b [fromstr 2.340]
370              set sum [add $a $b]]
371              # the result is : [3.568, 3.572] (the last digit is known with an uncertainty of 2)
372              tostr $sum ; # 3.57
373
374       But when, for example, we add or substract an integer  to  a  BigFloat,
375       the relative uncertainty of the result is unchanged. So it is desirable
376       not to convert integers to BigFloats:
377
378
379              set a [fromstr 0.999999999]
380              # now something dangerous
381              set b [fromstr 2.000]
382              # the result has only 3 digits
383              tostr [add $a $b]
384
385              # how to keep precision at its maximum
386              puts [tostr [add $a 2]]
387
388
389       For multiplication and division,  the  relative  uncertainties  of  the
390       product  or  the  quotient, is the sum of the relative uncertainties of
391       the operands.  Take care of division by zero : check each divider  with
392       iszero.
393
394
395              set num [fromstr 4.00]
396              set denom [fromstr 0.01]
397
398              puts [iszero $denom];# true
399              set quotient [div $num $denom];# error : divide by zero
400
401              # opposites of our operands
402              puts [compare $num [opp $num]]; # 1
403              puts [compare $denom [opp $denom]]; # 0 !!!
404              # No suprise ! 0 and its opposite are the same...
405
406       Effects  of  the  precision of a number considered equal to zero to the
407       cos function:
408
409
410              puts [tostr [cos [fromstr 0. 10]]]; # -> 1.000000000
411              puts [tostr [cos [fromstr 0. 5]]]; # -> 1.0000
412              puts [tostr [cos [fromstr 0e-10]]]; # -> 1.000000000
413              puts [tostr [cos [fromstr 1e-10]]]; # -> 1.000000000
414
415       BigFloats with different internal representations may be  converted  to
416       the same string.
417
418       For  most  analysis  functions  (cosine, square root, logarithm, etc.),
419       determining the precision of the result is difficult.  It seems however
420       that  in  many  cases, the loss of precision in the result is of one or
421       two digits.  There are some exceptions : for example,
422
423
424              tostr [exp [fromstr 100.0 10]]
425              # returns : 2.688117142e+43 which has only 10 digits of precision, although the entry
426              # has 14 digits of precision.
427
428

WHAT ABOUT TCL 8.4 ?

430       If your setup do not provide Tcl 8.5 but supports 8.4, the package  can
431       still  be  loaded,  switching  back  to  math::bigfloat 1.2. Indeed, an
432       important function introduced in Tcl 8.5 is required - the  ability  to
433       handle bignums, that we can do with expr.  Before 8.5, this ability was
434       provided by several packages, including the pure-Tcl math::bignum pack‐
435       age  provided  by  tcllib.  In this case, all you need to know, is that
436       arguments to the commands explained here, are expected to be  in  their
437       internal  representation.  So even with integers, you will need to call
438       fromstr and tostr in order to convert them between string and  internal
439       representations.
440
441
442              #
443              # with Tcl 8.5
444              # ============
445              set a [pi 20]
446              # round returns an integer and 'everything is a string' applies to integers
447              # whatever big they are
448              puts [round [mul $a 10000000000]]
449              #
450              # the same with Tcl 8.4
451              # =====================
452              set a [pi 20]
453              # bignums (arbitrary length integers) need a conversion hook
454              set b [fromstr 10000000000]
455              # round returns a bignum:
456              # before printing it, we need to convert it with 'tostr'
457              puts [tostr [round [mul $a $b]]]
458
459

NAMESPACES AND OTHER PACKAGES

461       We  have not yet discussed about namespaces because we assumed that you
462       had imported public commands into the global namespace, like this:
463
464
465              namespace import ::math::bigfloat::*
466
467       If you matter much about  avoiding  names  conflicts,  I  considere  it
468       should be resolved by the following :
469
470
471              package require math::bigfloat
472              # beware: namespace ensembles are not available in Tcl 8.4
473              namespace eval ::math::bigfloat {namespace ensemble create -command ::bigfloat}
474              # from now, the bigfloat command takes as subcommands all original math::bigfloat::* commands
475              set a [bigfloat sub [bigfloat fromstr 2.000] [bigfloat fromstr 0.530]]
476              puts [bigfloat tostr $a]
477
478

EXAMPLES

480       Guess  what happens when you are doing some astronomy. Here is an exam‐
481       ple :
482
483
484              # convert acurrate angles with a millisecond-rated accuracy
485              proc degree-angle {degrees minutes seconds milliseconds} {
486                  set result 0
487                  set div 1
488                  foreach factor {1 1000 60 60} var [list $milliseconds $seconds $minutes $degrees] {
489                      # we convert each entry var into milliseconds
490                      set div [expr {$div*$factor}]
491                      incr result [expr {$var*$div}]
492                  }
493                  return [div [int2float $result] $div]
494              }
495              # load the package
496              package require math::bigfloat
497              namespace import ::math::bigfloat::*
498              # work with angles : a standard formula for navigation (taking bearings)
499              set angle1 [deg2rad [degree-angle 20 30 40   0]]
500              set angle2 [deg2rad [degree-angle 21  0 50 500]]
501              set opposite3 [deg2rad [degree-angle 51  0 50 500]]
502              set sinProduct [mul [sin $angle1] [sin $angle2]]
503              set cosProduct [mul [cos $angle1] [cos $angle2]]
504              set angle3 [asin [add [mul $sinProduct [cos $opposite3]] $cosProduct]]
505              puts "angle3 : [tostr [rad2deg $angle3]]"
506
507

BUGS, IDEAS, FEEDBACK

509       This document, and the package it describes, will  undoubtedly  contain
510       bugs  and  other  problems.  Please report such in the category math ::
511       bignum      ::      float      of       the       Tcllib       Trackers
512       [http://core.tcl.tk/tcllib/reportlist].   Please  also report any ideas
513       for enhancements you may have for either package and/or documentation.
514
515       When proposing code changes, please provide unified diffs, i.e the out‐
516       put of diff -u.
517
518       Note  further  that  attachments  are  strongly  preferred over inlined
519       patches. Attachments can be made by going  to  the  Edit  form  of  the
520       ticket  immediately  after  its  creation, and then using the left-most
521       button in the secondary navigation bar.
522

KEYWORDS

524       computations, floating-point, interval, math, multiprecision, tcl
525

CATEGORY

527       Mathematics
528
530       Copyright (c) 2004-2008, by Stephane Arnold <stephanearnold at yahoo dot fr>
531
532
533
534
535tcllib                               2.0.1                   math::bigfloat(n)
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