1math::exact(n) Tcl Math Library math::exact(n)
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6
8 math::exact - Exact Real Arithmetic
9
11 package require Tcl 8.6
12 package require grammar::aycock 1.0
14 package require math::exact 1.0.1
16 ::math::exact::exactexpr expr
18 number ref
20 number unref
22 number asPrint precision
24 number asFloat precision
26______________________________________________________________________________
28
30 The exactexpr command in the math::exact package allows for exact com‐
31 putations over the computable real numbers. These are not arbitrary-
32 precision calculations; rather they are exact, with numbers represented
33 by algorithms that produce successive approximations. At the end of a
34 calculation, the caller can request a given precision for the end
35 result, and intermediate results are computed to whatever precision is
36 necessary to satisfy the request.
37
39 The following procedure is the primary entry into the math::exact pack‐
40 age.
41 ::math::exact::exactexpr expr
43 Accepts a mathematical expression in Tcl syntax, and returns an
44 object that represents the program to calculate successive
45 approximations to the expression's value. The result will be
46 referred to as an exact real number.
47 number ref
49 Increases the reference count of a given exact real number.
50 number unref
52 Decreases the reference count of a given exact real number, and
53 destroys the number if the reference count is zero.
54 number asPrint precision
56 Formats the given number for printing, with the specified preci‐
57 sion. (See below for how precision is interpreted). Numbers
58 that are known to be rational are formatted as fractions.
59 number asFloat precision
61 Formats the given number for printing, with the specified preci‐
62 sion. (See below for how precision is interpreted). All numbers
63 are formatted in floating-point E format.
64
66 expr Expression to evaluate. The syntax for expressions is the same
67 as it is in Tcl, but the set of operations is smaller. See
68 Expressions below for details.
69 number The object returned by an earlier invocation of
71 math::exact::exactexpr
72 precision
74 The requested 'precision' of the result. The precision is
75 (approximately) the absolute value of the binary exponent plus
76 the number of bits of the binary significand. For instance, to
77 return results to IEEE-754 double precision, 56 bits plus the
78 exponent are required. Numbers between 1/2 and 2 will require a
79 precision of 57; numbers between 1/4 and 1/2 or between 2 and 4
80 will require 58; numbers between 1/8 and 1/4 or between 4 and 8
81 will require 59; and so on.
82
84 The math::exact::exactexpr command accepts expressions in a subset of
85 Tcl's syntax. The following components may be used in an expression.
86 · Decimal integers.
88 · Variable references with the dollar sign ($). The value of the
90 variable must be the result of another call to
91 math::exact::exactexpr. The reference count of the value will be
92 increased by one for each position at which it appears in the
93 expression.
94 · The exponentiation operator (**).
96 · Unary plus (+) and minus (-) operators.
98 · Multiplication (*) and division (/) operators.
100 · Parentheses used for grouping.
102 · Functions. See Functions below for the functions that are avail‐
104 able.
105
107 The following functions are available for use within exact real expres‐
108 sions.
109 acos(x)
111 The inverse cosine of x. The result is expressed in radians.
112 The absolute value of x must be less than 1.
113 acosh(x)
115 The inverse hyperbolic cosine of x. x must be greater than 1.
116 asin(x)
118 The inverse sine of x. The result is expressed in radians. The
119 absolute value of x must be less than 1.
120 asinh(x)
122 The inverse hyperbolic sine of x.
123 atan(x)
125 The inverse tangent of x. The result is expressed in radians.
126 atanh(x)
128 The inverse hyperbolic tangent of x. The absolute value of x
129 must be less than 1.
130 cos(x) The cosine of x. x is expressed in radians.
132 cosh(x)
134 The hyperbolic cosine of x.
135 e() The base of the natural logarithms = 2.71828...
137 exp(x) The exponential function of x.
139 log(x) The natural logarithm of x. x must be positive.
141 pi() The value of pi = 3.15159...
143 sin(x) The sine of x. x is expressed in radians.
145 sinh(x)
147 The hyperbolic sine of x.
148 sqrt(x)
150 The square root of x. x must be positive.
151 tan(x) The tangent of x. x is expressed in radians.
153 tanh(x)
155 The hyperbolic tangent of x.
156
158 The math::exact::exactexpr command provides a system that performs
159 exact arithmetic over computable real numbers, representing the numbers
160 as algorithms for successive approximation. An example, which imple‐
161 ments the high-school quadratic formula, is shown below.
162 namespace import math::exact::exactexpr
165 proc exactquad {a b c} {
166 set d [[exactexpr {sqrt($b*$b - 4*$a*$c)}] ref]
167 set r0 [[exactexpr {(-$b - $d) / (2 * $a)}] ref]
168 set r1 [[exactexpr {(-$b + $d) / (2 * $a)}] ref]
169 $d unref
170 return [list $r0 $r1]
171 }
172 set a [[exactexpr 1] ref]
174 set b [[exactexpr 200] ref]
175 set c [[exactexpr {(-3/2) * 10**-12}] ref]
176 lassign [exactquad $a $b $c] r0 r1
177 $a unref; $b unref; $c unref
178 puts [list [$r0 asFloat 70] [$r1 asFloat 110]]
179 $r0 unref; $r1 unref
180 The program prints the result:
182 -2.000000000000000075e2 7.499999999999999719e-15
185 Note that if IEEE-754 floating point had been used, a catastrophic
187 roundoff error would yield a smaller root that is a factor of two too
188 high:
189 -200.0 1.4210854715202004e-14
192 The invocations of exactexpr should be fairly self-explanatory. The
194 other commands of note are ref and unref. It is necessary for the call‐
195 er to keep track of references to exact expressions - to call ref every
196 time an exact expression is stored in a variable and unref every time
197 the variable goes out of scope or is overwritten. The asFloat method
198 emits decimal digits as long as the requested precision supports them.
199 It terminates when the requested precision yields an uncertainty of
200 more than one unit in the least significant digit.
201
203 Mathematics
204
206 Copyright (c) 2015 Kevin B. Kenny <kennykb@acm.org>
207 Redistribution permitted under the terms of the Open Publication License <http://www.opencontent.org/openpub/>
208tcllib 1.0.1 math::exact(n)