'\"
'\" Generated from file 'exact\&.man' by tcllib/doctools with format 'nroff'
'\" Copyright (c) 2015 Kevin B\&. Kenny <kennykb@acm\&.org>
'\" Redistribution permitted under the terms of the Open Publication License <http://www\&.opencontent\&.org/openpub/>
'\"
.TH "math::exact" n 1\&.0\&.2 tcllib "Tcl Math Library"
.\" The -*- nroff -*- definitions below are for supplemental macros used
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.BS
.SH NAME
math::exact \- Exact Real Arithmetic
.SH SYNOPSIS
package require \fBTcl 8\&.6 9\fR
.sp
package require \fBgrammar::aycock 1\&.0\fR
.sp
package require \fBmath::exact 1\&.0\&.2\fR
.sp
\fB::math::exact::exactexpr\fR \fIexpr\fR
.sp
\fInumber\fR \fBref\fR
.sp
\fInumber\fR \fBunref\fR
.sp
\fInumber\fR \fBasPrint\fR \fIprecision\fR
.sp
\fInumber\fR \fBasFloat\fR \fIprecision\fR
.sp
.BE
.SH DESCRIPTION
.PP
The \fBexactexpr\fR command in the \fBmath::exact\fR package
allows for exact computations over the computable real numbers\&.
These are not arbitrary-precision calculations; rather they are
exact, with numbers represented by algorithms that produce successive
approximations\&. At the end of a calculation, the caller can
request a given precision for the end result, and intermediate results are
computed to whatever precision is necessary to satisfy the request\&.
.SH PROCEDURES
The following procedure is the primary entry into the \fBmath::exact\fR
package\&.
.TP
\fB::math::exact::exactexpr\fR \fIexpr\fR
Accepts a mathematical expression in Tcl syntax, and returns an object
that represents the program to calculate successive approximations to
the expression's value\&. The result will be referred to as an
exact real number\&.
.TP
\fInumber\fR \fBref\fR
Increases the reference count of a given exact real number\&.
.TP
\fInumber\fR \fBunref\fR
Decreases the reference count of a given exact real number, and destroys
the number if the reference count is zero\&.
.TP
\fInumber\fR \fBasPrint\fR \fIprecision\fR
Formats the given \fInumber\fR for printing, with the specified \fIprecision\fR\&.
(See below for how \fIprecision\fR is interpreted)\&. Numbers that are known to
be rational are formatted as fractions\&.
.TP
\fInumber\fR \fBasFloat\fR \fIprecision\fR
Formats the given \fInumber\fR for printing, with the specified \fIprecision\fR\&.
(See below for how \fIprecision\fR is interpreted)\&. All numbers are formatted
in floating-point E format\&.
.PP
.SH PARAMETERS
.TP
\fIexpr\fR
Expression to evaluate\&. The syntax for expressions is the same as it is in Tcl,
but the set of operations is smaller\&. See \fBExpressions\fR below
for details\&.
.TP
\fInumber\fR
The object returned by an earlier invocation of \fBmath::exact::exactexpr\fR
.TP
\fIprecision\fR
The requested 'precision' of the result\&. The precision is (approximately)
the absolute value of the binary exponent plus the number of bits of the
binary significand\&. For instance, to return results to IEEE-754 double
precision, 56 bits plus the exponent are required\&. Numbers between 1/2 and 2
will require a precision of 57; numbers between 1/4 and 1/2 or between 2 and 4
will require 58; numbers between 1/8 and 1/4 or between 4 and 8 will require
59; and so on\&.
.PP
.SH EXPRESSIONS
The \fBmath::exact::exactexpr\fR command accepts expressions in a subset
of Tcl's syntax\&. The following components may be used in an expression\&.
.IP \(bu
Decimal integers\&.
.IP \(bu
Variable references with the dollar sign (\fB$\fR)\&.
The value of the variable must be the result of another call to
\fBmath::exact::exactexpr\fR\&. The reference count of the value
will be increased by one for each position at which it appears
in the expression\&.
.IP \(bu
The exponentiation operator (\fB**\fR)\&.
.IP \(bu
Unary plus (\fB+\fR) and minus (\fB-\fR) operators\&.
.IP \(bu
Multiplication (\fB*\fR) and division (\fB/\fR) operators\&.
.IP \(bu
Parentheses used for grouping\&.
.IP \(bu
Functions\&. See \fBFunctions\fR below for the functions that are
available\&.
.PP
.SH FUNCTIONS
The following functions are available for use within exact real expressions\&.
.TP
\fBacos(\fR\fIx\fR\fB)\fR
The inverse cosine of \fIx\fR\&. The result is expressed in radians\&.
The absolute value of \fIx\fR must be less than 1\&.
.TP
\fBacosh(\fR\fIx\fR\fB)\fR
The inverse hyperbolic cosine of \fIx\fR\&.
\fIx\fR must be greater than 1\&.
.TP
\fBasin(\fR\fIx\fR\fB)\fR
The inverse sine of \fIx\fR\&. The result is expressed in radians\&.
The absolute value of \fIx\fR must be less than 1\&.
.TP
\fBasinh(\fR\fIx\fR\fB)\fR
The inverse hyperbolic sine of \fIx\fR\&.
.TP
\fBatan(\fR\fIx\fR\fB)\fR
The inverse tangent of \fIx\fR\&. The result is expressed in radians\&.
.TP
\fBatanh(\fR\fIx\fR\fB)\fR
The inverse hyperbolic tangent of \fIx\fR\&.
The absolute value of \fIx\fR must be less than 1\&.
.TP
\fBcos(\fR\fIx\fR\fB)\fR
The cosine of \fIx\fR\&. \fIx\fR is expressed in radians\&.
.TP
\fBcosh(\fR\fIx\fR\fB)\fR
The hyperbolic cosine of \fIx\fR\&.
.TP
\fBe()\fR
The base of the natural logarithms = \fB2\&.71828\&.\&.\&.\fR
.TP
\fBexp(\fR\fIx\fR\fB)\fR
The exponential function of \fIx\fR\&.
.TP
\fBlog(\fR\fIx\fR\fB)\fR
The natural logarithm of \fIx\fR\&. \fIx\fR must be positive\&.
.TP
\fBpi()\fR
The value of pi = \fB3\&.15159\&.\&.\&.\fR
.TP
\fBsin(\fR\fIx\fR\fB)\fR
The sine of \fIx\fR\&. \fIx\fR is expressed in radians\&.
.TP
\fBsinh(\fR\fIx\fR\fB)\fR
The hyperbolic sine of \fIx\fR\&.
.TP
\fBsqrt(\fR\fIx\fR\fB)\fR
The square root of \fIx\fR\&. \fIx\fR must be positive\&.
.TP
\fBtan(\fR\fIx\fR\fB)\fR
The tangent of \fIx\fR\&. \fIx\fR is expressed in radians\&.
.TP
\fBtanh(\fR\fIx\fR\fB)\fR
The hyperbolic tangent of \fIx\fR\&.
.PP
.SH SUMMARY
The \fBmath::exact::exactexpr\fR command provides a system that
performs exact arithmetic over computable real numbers, representing
the numbers as algorithms for successive approximation\&.
An example, which implements the high-school quadratic formula,
is shown below\&.
.CS


namespace import math::exact::exactexpr
proc exactquad {a b c} {
    set d [[exactexpr {sqrt($b*$b - 4*$a*$c)}] ref]
    set r0 [[exactexpr {(-$b - $d) / (2 * $a)}] ref]
    set r1 [[exactexpr {(-$b + $d) / (2 * $a)}] ref]
    $d unref
    return [list $r0 $r1]
}

set a [[exactexpr 1] ref]
set b [[exactexpr 200] ref]
set c [[exactexpr {(-3/2) * 10**-12}] ref]
lassign [exactquad $a $b $c] r0 r1
$a unref; $b unref; $c unref
puts [list [$r0 asFloat 70] [$r1 asFloat 110]]
$r0 unref; $r1 unref

.CE
The program prints the result:
.CS


-2\&.000000000000000075e2 7\&.499999999999999719e-15

.CE
Note that if IEEE-754 floating point had been used, a catastrophic
roundoff error would yield a smaller root that is a factor of two
too high:
.CS


-200\&.0 1\&.4210854715202004e-14

.CE
The invocations of \fBexactexpr\fR should be fairly self-explanatory\&.
The other commands of note are \fBref\fR and \fBunref\fR\&. It is necessary
for the caller to keep track of references to exact expressions - to call
\fBref\fR every time an exact expression is stored in a variable and
\fBunref\fR every time the variable goes out of scope or is overwritten\&.
The \fBasFloat\fR method emits decimal digits as long as the requested
precision supports them\&. It terminates when the requested precision
yields an uncertainty of more than one unit in the least significant digit\&.
.SH CATEGORY
Mathematics
.SH COPYRIGHT
.nf
Copyright (c) 2015 Kevin B\&. Kenny <kennykb@acm\&.org>
Redistribution permitted under the terms of the Open Publication License <http://www\&.opencontent\&.org/openpub/>

.fi
