'\" '\" Generated from file 'bigfloat\&.man' by tcllib/doctools with format 'nroff' '\" Copyright (c) 2004-2008, by Stephane Arnold '\" .TH "math::bigfloat" 3tcl 2\&.0\&.3 tcllib "Tcl Math Library" .\" The -*- nroff -*- definitions below are for supplemental macros used .\" in Tcl/Tk manual entries. .\" .\" .AP type name in/out ?indent? .\" Start paragraph describing an argument to a library procedure. .\" type is type of argument (int, etc.), in/out is either "in", "out", .\" or "in/out" to describe whether procedure reads or modifies arg, .\" and indent is equivalent to second arg of .IP (shouldn't ever be .\" needed; use .AS below instead) .\" .\" .AS ?type? ?name? .\" Give maximum sizes of arguments for setting tab stops. Type and .\" name are examples of largest possible arguments that will be passed .\" to .AP later. If args are omitted, default tab stops are used. .\" .\" .BS .\" Start box enclosure. From here until next .BE, everything will be .\" enclosed in one large box. .\" .\" .BE .\" End of box enclosure. .\" .\" .CS .\" Begin code excerpt. .\" .\" .CE .\" End code excerpt. .\" .\" .VS ?version? ?br? .\" Begin vertical sidebar, for use in marking newly-changed parts .\" of man pages. The first argument is ignored and used for recording .\" the version when the .VS was added, so that the sidebars can be .\" found and removed when they reach a certain age. If another argument .\" is present, then a line break is forced before starting the sidebar. .\" .\" .VE .\" End of vertical sidebar. .\" .\" .DS .\" Begin an indented unfilled display. .\" .\" .DE .\" End of indented unfilled display. .\" .\" .SO ?manpage? .\" Start of list of standard options for a Tk widget. The manpage .\" argument defines where to look up the standard options; if .\" omitted, defaults to "options". The options follow on successive .\" lines, in three columns separated by tabs. .\" .\" .SE .\" End of list of standard options for a Tk widget. .\" .\" .OP cmdName dbName dbClass .\" Start of description of a specific option. cmdName gives the .\" option's name as specified in the class command, dbName gives .\" the option's name in the option database, and dbClass gives .\" the option's class in the option database. .\" .\" .UL arg1 arg2 .\" Print arg1 underlined, then print arg2 normally. .\" .\" .QW arg1 ?arg2? .\" Print arg1 in quotes, then arg2 normally (for trailing punctuation). .\" .\" .PQ arg1 ?arg2? .\" Print an open parenthesis, arg1 in quotes, then arg2 normally .\" (for trailing punctuation) and then a closing parenthesis. .\" .\" # Set up traps and other miscellaneous stuff for Tcl/Tk man pages. .if t .wh -1.3i ^B .nr ^l \n(.l .ad b .\" # Start an argument description .de AP .ie !"\\$4"" .TP \\$4 .el \{\ . ie !"\\$2"" .TP \\n()Cu . el .TP 15 .\} .ta \\n()Au \\n()Bu .ie !"\\$3"" \{\ \&\\$1 \\fI\\$2\\fP (\\$3) .\".b .\} .el \{\ .br .ie !"\\$2"" \{\ \&\\$1 \\fI\\$2\\fP .\} .el \{\ \&\\fI\\$1\\fP .\} .\} .. .\" # define tabbing values for .AP .de AS .nr )A 10n .if !"\\$1"" .nr )A \\w'\\$1'u+3n .nr )B \\n()Au+15n .\" .if !"\\$2"" .nr )B \\w'\\$2'u+\\n()Au+3n .nr )C \\n()Bu+\\w'(in/out)'u+2n .. .AS Tcl_Interp Tcl_CreateInterp in/out .\" # BS - start boxed text .\" # ^y = starting y location .\" # ^b = 1 .de BS .br .mk ^y .nr ^b 1u .if n .nf .if n .ti 0 .if n \l'\\n(.lu\(ul' .if n .fi .. .\" # BE - end boxed text (draw box now) .de BE .nf .ti 0 .mk ^t .ie n \l'\\n(^lu\(ul' .el \{\ .\" Draw four-sided box normally, but don't draw top of .\" box if the box started on an earlier page. .ie !\\n(^b-1 \{\ \h'-1.5n'\L'|\\n(^yu-1v'\l'\\n(^lu+3n\(ul'\L'\\n(^tu+1v-\\n(^yu'\l'|0u-1.5n\(ul' .\} .el \}\ \h'-1.5n'\L'|\\n(^yu-1v'\h'\\n(^lu+3n'\L'\\n(^tu+1v-\\n(^yu'\l'|0u-1.5n\(ul' .\} .\} .fi .br .nr ^b 0 .. .\" # VS - start vertical sidebar .\" # ^Y = starting y location .\" # ^v = 1 (for troff; for nroff this doesn't matter) .de VS .if !"\\$2"" .br .mk ^Y .ie n 'mc \s12\(br\s0 .el .nr ^v 1u .. .\" # VE - end of vertical sidebar .de VE .ie n 'mc .el \{\ .ev 2 .nf .ti 0 .mk ^t \h'|\\n(^lu+3n'\L'|\\n(^Yu-1v\(bv'\v'\\n(^tu+1v-\\n(^Yu'\h'-|\\n(^lu+3n' .sp -1 .fi .ev .\} .nr ^v 0 .. .\" # Special macro to handle page bottom: finish off current .\" # box/sidebar if in box/sidebar mode, then invoked standard .\" # page bottom macro. .de ^B .ev 2 'ti 0 'nf .mk ^t .if \\n(^b \{\ .\" Draw three-sided box if this is the box's first page, .\" draw two sides but no top otherwise. .ie !\\n(^b-1 \h'-1.5n'\L'|\\n(^yu-1v'\l'\\n(^lu+3n\(ul'\L'\\n(^tu+1v-\\n(^yu'\h'|0u'\c .el \h'-1.5n'\L'|\\n(^yu-1v'\h'\\n(^lu+3n'\L'\\n(^tu+1v-\\n(^yu'\h'|0u'\c .\} .if \\n(^v \{\ .nr ^x \\n(^tu+1v-\\n(^Yu \kx\h'-\\nxu'\h'|\\n(^lu+3n'\ky\L'-\\n(^xu'\v'\\n(^xu'\h'|0u'\c .\} .bp 'fi .ev .if \\n(^b \{\ .mk ^y .nr ^b 2 .\} .if \\n(^v \{\ .mk ^Y .\} .. .\" # DS - begin display .de DS .RS .nf .sp .. .\" # DE - end display .de DE .fi .RE .sp .. .\" # SO - start of list of standard options .de SO 'ie '\\$1'' .ds So \\fBoptions\\fR 'el .ds So \\fB\\$1\\fR .SH "STANDARD OPTIONS" .LP .nf .ta 5.5c 11c .ft B .. .\" # SE - end of list of standard options .de SE .fi .ft R .LP See the \\*(So manual entry for details on the standard options. .. .\" # OP - start of full description for a single option .de OP .LP .nf .ta 4c Command-Line Name: \\fB\\$1\\fR Database Name: \\fB\\$2\\fR Database Class: \\fB\\$3\\fR .fi .IP .. .\" # CS - begin code excerpt .de CS .RS .nf .ta .25i .5i .75i 1i .. .\" # CE - end code excerpt .de CE .fi .RE .. .\" # UL - underline word .de UL \\$1\l'|0\(ul'\\$2 .. .\" # QW - apply quotation marks to word .de QW .ie '\\*(lq'"' ``\\$1''\\$2 .\"" fix emacs highlighting .el \\*(lq\\$1\\*(rq\\$2 .. .\" # PQ - apply parens and quotation marks to word .de PQ .ie '\\*(lq'"' (``\\$1''\\$2)\\$3 .\"" fix emacs highlighting .el (\\*(lq\\$1\\*(rq\\$2)\\$3 .. .\" # QR - quoted range .de QR .ie '\\*(lq'"' ``\\$1''\\-``\\$2''\\$3 .\"" fix emacs highlighting .el \\*(lq\\$1\\*(rq\\-\\*(lq\\$2\\*(rq\\$3 .. .\" # MT - "empty" string .de MT .QW "" .. .BS .SH NAME math::bigfloat \- Arbitrary precision floating-point numbers .SH SYNOPSIS package require \fBTcl 8\&.5\fR .sp package require \fBmath::bigfloat ?2\&.0\&.3?\fR .sp \fBfromstr\fR \fInumber\fR ?\fItrailingZeros\fR? .sp \fBtostr\fR ?\fB-nosci\fR? \fInumber\fR .sp \fBfromdouble\fR \fIdouble\fR ?\fIdecimals\fR? .sp \fBtodouble\fR \fInumber\fR .sp \fBisInt\fR \fInumber\fR .sp \fBisFloat\fR \fInumber\fR .sp \fBint2float\fR \fIinteger\fR ?\fIdecimals\fR? .sp \fBadd\fR \fIx\fR \fIy\fR .sp \fBsub\fR \fIx\fR \fIy\fR .sp \fBmul\fR \fIx\fR \fIy\fR .sp \fBdiv\fR \fIx\fR \fIy\fR .sp \fBmod\fR \fIx\fR \fIy\fR .sp \fBabs\fR \fIx\fR .sp \fBopp\fR \fIx\fR .sp \fBpow\fR \fIx\fR \fIn\fR .sp \fBiszero\fR \fIx\fR .sp \fBequal\fR \fIx\fR \fIy\fR .sp \fBcompare\fR \fIx\fR \fIy\fR .sp \fBsqrt\fR \fIx\fR .sp \fBlog\fR \fIx\fR .sp \fBexp\fR \fIx\fR .sp \fBcos\fR \fIx\fR .sp \fBsin\fR \fIx\fR .sp \fBtan\fR \fIx\fR .sp \fBcotan\fR \fIx\fR .sp \fBacos\fR \fIx\fR .sp \fBasin\fR \fIx\fR .sp \fBatan\fR \fIx\fR .sp \fBcosh\fR \fIx\fR .sp \fBsinh\fR \fIx\fR .sp \fBtanh\fR \fIx\fR .sp \fBpi\fR \fIn\fR .sp \fBrad2deg\fR \fIradians\fR .sp \fBdeg2rad\fR \fIdegrees\fR .sp \fBround\fR \fIx\fR .sp \fBceil\fR \fIx\fR .sp \fBfloor\fR \fIx\fR .sp .BE .SH 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 \fBWHAT ABOUT TCL 8\&.4 ?\fR for more explanations\&. By convention, we will talk about the numbers treated in this library as : .IP \(bu BigFloat for floating-point numbers of arbitrary length\&. .IP \(bu integers for arbitrary length signed integers, just as basic integers since Tcl 8\&.5\&. .PP Each BigFloat is an interval, namely [\fIm-d, m+d\fR], where \fIm\fR is the mantissa and \fId\fR the uncertainty, representing the limitation of that number's precision\&. This is why we call such mathematics \fIinterval computations\fR\&. Just take an example in physics : when you measure a temperature, not all digits you read are \fIsignificant\fR\&. 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 : .IP \(bu perform math operations on BigFloats and (optionnaly) with integers\&. .IP \(bu convert BigFloats from their internal representations to strings, and vice versa\&. .PP .SH INTRODUCTION .TP \fBfromstr\fR \fInumber\fR ?\fItrailingZeros\fR? Converts \fInumber\fR into a BigFloat\&. Its precision is at least the number of digits provided by \fInumber\fR\&. If the \fInumber\fR contains only digits and eventually a minus sign, it is considered as an integer\&. Subsequently, no conversion is done at all\&. .sp \fItrailingZeros\fR - 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\&. .CS # 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 .CE .sp The \fInumber\fR'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 \fBfromstr\fR\&. 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 \fBPRECISION\fR\&. .sp For example : .CS set x [fromstr 1\&.0000000000] # the next line does the same, but smarter set y [fromstr 1\&. 10] .CE .TP \fBtostr\fR ?\fB-nosci\fR? \fInumber\fR Returns a string form of a BigFloat, in which all digits are exacts\&. \fIAll exact digits\fR means a rounding may occur, for example to zero, if the uncertainty interval does not clearly show the true digits\&. \fInumber\fR may be an integer, causing the command to return exactly the input argument\&. With the \fB-nosci\fR option, the number returned is never shown in scientific notation, i\&.e\&. not like '3\&.4523e+5' but like '345230\&.'\&. .CS puts [tostr [fromstr 0\&.99999]] ;# 1\&.0000 puts [tostr [fromstr 1\&.00001]] ;# 1\&.0000 puts [tostr [fromstr 0\&.002]] ;# 0\&.e-2 .CE .IP See \fBPRECISION\fR for that matter\&. See also \fBiszero\fR for how to detect zeros, which is useful when performing a division\&. .TP \fBfromdouble\fR \fIdouble\fR ?\fIdecimals\fR? Converts a double (a simple floating-point value) to a BigFloat, with exactly \fIdecimals\fR digits\&. Without the \fIdecimals\fR argument, it behaves like \fBfromstr\fR\&. Here, the only important feature you might care of is the ability to create BigFloats with a fixed number of \fIdecimals\fR\&. .CS tostr [fromstr 1\&.111 4] # returns : 1\&.111000 (3 zeros) tostr [fromdouble 1\&.111 4] # returns : 1\&.111 .CE .TP \fBtodouble\fR \fInumber\fR Returns a double, that may be used in \fIexpr\fR, from a BigFloat\&. .TP \fBisInt\fR \fInumber\fR Returns 1 if \fInumber\fR is an integer, 0 otherwise\&. .TP \fBisFloat\fR \fInumber\fR Returns 1 if \fInumber\fR is a BigFloat, 0 otherwise\&. .TP \fBint2float\fR \fIinteger\fR ?\fIdecimals\fR? Converts an integer to a BigFloat with \fIdecimals\fR trailing zeros\&. The default, and minimal, number of \fIdecimals\fR is 1\&. When converting back to string, one decimal is lost: .CS 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" .CE .PP .SH ARITHMETICS .TP \fBadd\fR \fIx\fR \fIy\fR .TP \fBsub\fR \fIx\fR \fIy\fR .TP \fBmul\fR \fIx\fR \fIy\fR Return the sum, difference and product of \fIx\fR by \fIy\fR\&. \fIx\fR - may be either a BigFloat or an integer \fIy\fR - may be either a BigFloat or an integer When both are integers, these commands behave like \fBexpr\fR\&. .TP \fBdiv\fR \fIx\fR \fIy\fR .TP \fBmod\fR \fIx\fR \fIy\fR Return the quotient and the rest of \fIx\fR divided by \fIy\fR\&. Each argument (\fIx\fR and \fIy\fR) can be either a BigFloat or an integer, but you cannot divide an integer by a BigFloat Divide by zero throws an error\&. .TP \fBabs\fR \fIx\fR Returns the absolute value of \fIx\fR .TP \fBopp\fR \fIx\fR Returns the opposite of \fIx\fR .TP \fBpow\fR \fIx\fR \fIn\fR Returns \fIx\fR taken to the \fIn\fRth power\&. It only works if \fIn\fR is an integer\&. \fIx\fR might be a BigFloat or an integer\&. .PP .SH COMPARISONS .TP \fBiszero\fR \fIx\fR Returns 1 if \fIx\fR is : .RS .IP \(bu a BigFloat close enough to zero to raise "divide by zero"\&. .IP \(bu the integer 0\&. .RE .IP See here how numbers that are close to zero are converted to strings: .CS 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 .CE .TP \fBequal\fR \fIx\fR \fIy\fR Returns 1 if \fIx\fR and \fIy\fR are equal, 0 elsewhere\&. .TP \fBcompare\fR \fIx\fR \fIy\fR Returns 0 if both BigFloat arguments are equal, 1 if \fIx\fR is greater than \fIy\fR, and -1 if \fIx\fR is lower than \fIy\fR\&. You would not be able to compare an integer to a BigFloat : the operands should be both BigFloats, or both integers\&. .PP .SH ANALYSIS .TP \fBsqrt\fR \fIx\fR .TP \fBlog\fR \fIx\fR .TP \fBexp\fR \fIx\fR .TP \fBcos\fR \fIx\fR .TP \fBsin\fR \fIx\fR .TP \fBtan\fR \fIx\fR .TP \fBcotan\fR \fIx\fR .TP \fBacos\fR \fIx\fR .TP \fBasin\fR \fIx\fR .TP \fBatan\fR \fIx\fR .TP \fBcosh\fR \fIx\fR .TP \fBsinh\fR \fIx\fR .TP \fBtanh\fR \fIx\fR 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 \fIx\fR\&. .TP \fBpi\fR \fIn\fR Returns a BigFloat representing the Pi constant with \fIn\fR digits after the dot\&. \fIn\fR is a positive integer\&. .TP \fBrad2deg\fR \fIradians\fR .TP \fBdeg2rad\fR \fIdegrees\fR \fIradians\fR - angle expressed in radians (BigFloat) .sp \fIdegrees\fR - angle expressed in degrees (BigFloat) .sp Convert an angle from radians to degrees, and \fIvice versa\fR\&. .PP .SH ROUNDING .TP \fBround\fR \fIx\fR .TP \fBceil\fR \fIx\fR .TP \fBfloor\fR \fIx\fR The above functions return the \fIx\fR BigFloat, rounded like with the same mathematical function in \fIexpr\fR, and returns it as an integer\&. .PP .SH PRECISION How do conversions work with precision ? .IP \(bu When a BigFloat is converted from string, the internal representation holds its uncertainty as 1 at the level of the last digit\&. .IP \(bu During computations, the uncertainty of each result is internally computed the closest to the reality, thus saving the memory used\&. .IP \(bu 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\&. .PP Uncertainties are kept in the internal representation of the number ; it is recommended to use \fBtostr\fR only for outputting data (on the screen or in a file), and NEVER call \fBfromstr\fR with the result of \fBtostr\fR\&. 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\&. .PP 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: .CS 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) .CE The uncertainty of the sum, or the difference, of two numbers, is the sum of their respective uncertainties\&. .CS 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 .CE 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: .CS 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]] .CE .PP 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 \fBiszero\fR\&. .CS 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\&.\&.\&. .CE Effects of the precision of a number considered equal to zero to the cos function: .CS 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 .CE BigFloats with different internal representations may be converted to the same string\&. .PP 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, .CS 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\&. .CE .SH "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 \fImath::bigfloat\fR 1\&.2\&. Indeed, an important function introduced in Tcl 8\&.5 is required - the ability to handle bignums, that we can do with \fBexpr\fR\&. Before 8\&.5, this ability was provided by several packages, including the pure-Tcl \fImath::bignum\fR package provided by \fItcllib\fR\&. 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 \fBfromstr\fR and \fBtostr\fR in order to convert them between string and internal representations\&. .CS # # 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]]] .CE .SH "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: .CS namespace import ::math::bigfloat::* .CE If you matter much about avoiding names conflicts, I considere it should be resolved by the following : .CS 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] .CE .SH EXAMPLES Guess what happens when you are doing some astronomy\&. Here is an example : .CS # 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]]" .CE .SH "BUGS, IDEAS, FEEDBACK" This document, and the package it describes, will undoubtedly contain bugs and other problems\&. Please report such in the category \fImath :: bignum :: float\fR of the \fITcllib Trackers\fR [http://core\&.tcl\&.tk/tcllib/reportlist]\&. Please also report any ideas for enhancements you may have for either package and/or documentation\&. .PP When proposing code changes, please provide \fIunified diffs\fR, i\&.e the output of \fBdiff -u\fR\&. .PP Note further that \fIattachments\fR are strongly preferred over inlined patches\&. Attachments can be made by going to the \fBEdit\fR form of the ticket immediately after its creation, and then using the left-most button in the secondary navigation bar\&. .SH KEYWORDS computations, floating-point, interval, math, multiprecision, tcl .SH CATEGORY Mathematics .SH COPYRIGHT .nf Copyright (c) 2004-2008, by Stephane Arnold .fi