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 re‐
35 sult, 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 ap‐
45 proximations to the expression's value. The result will be re‐
46 ferred 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 Ex‐
68 pressions below for details.
69 number The object returned by an earlier invocation of math::exact::ex‐
71 actexpr
72 precision
74 The requested 'precision' of the result. The precision is (ap‐
75 proximately) the absolute value of the binary exponent plus the
76 number of bits of the binary significand. For instance, to re‐
77 turn results to IEEE-754 double precision, 56 bits plus the ex‐
78 ponent 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 math::exact::ex‐
91 actexpr. The reference count of the value will be increased by
92 one for each position at which it appears in the expression.
93 • The exponentiation operator (**).
95 • Unary plus (+) and minus (-) operators.
97 • Multiplication (*) and division (/) operators.
99 • Parentheses used for grouping.
101 • Functions. See Functions below for the functions that are avail‐
103 able.
104
106 The following functions are available for use within exact real expres‐
107 sions.
108 acos(x)
110 The inverse cosine of x. The result is expressed in radians.
111 The absolute value of x must be less than 1.
112 acosh(x)
114 The inverse hyperbolic cosine of x. x must be greater than 1.
115 asin(x)
117 The inverse sine of x. The result is expressed in radians. The
118 absolute value of x must be less than 1.
119 asinh(x)
121 The inverse hyperbolic sine of x.
122 atan(x)
124 The inverse tangent of x. The result is expressed in radians.
125 atanh(x)
127 The inverse hyperbolic tangent of x. The absolute value of x
128 must be less than 1.
129 cos(x) The cosine of x. x is expressed in radians.
131 cosh(x)
133 The hyperbolic cosine of x.
134 e() The base of the natural logarithms = 2.71828...
136 exp(x) The exponential function of x.
138 log(x) The natural logarithm of x. x must be positive.
140 pi() The value of pi = 3.15159...
142 sin(x) The sine of x. x is expressed in radians.
144 sinh(x)
146 The hyperbolic sine of x.
147 sqrt(x)
149 The square root of x. x must be positive.
150 tan(x) The tangent of x. x is expressed in radians.
152 tanh(x)
154 The hyperbolic tangent of x.
155
157 The math::exact::exactexpr command provides a system that performs ex‐
158 act arithmetic over computable real numbers, representing the numbers
159 as algorithms for successive approximation. An example, which imple‐
160 ments the high-school quadratic formula, is shown below.
161 namespace import math::exact::exactexpr
164 proc exactquad {a b c} {
165 set d [[exactexpr {sqrt($b*$b - 4*$a*$c)}] ref]
166 set r0 [[exactexpr {(-$b - $d) / (2 * $a)}] ref]
167 set r1 [[exactexpr {(-$b + $d) / (2 * $a)}] ref]
168 $d unref
169 return [list $r0 $r1]
170 }
171 set a [[exactexpr 1] ref]
173 set b [[exactexpr 200] ref]
174 set c [[exactexpr {(-3/2) * 10**-12}] ref]
175 lassign [exactquad $a $b $c] r0 r1
176 $a unref; $b unref; $c unref
177 puts [list [$r0 asFloat 70] [$r1 asFloat 110]]
178 $r0 unref; $r1 unref
179 The program prints the result:
181 -2.000000000000000075e2 7.499999999999999719e-15
184 Note that if IEEE-754 floating point had been used, a catastrophic
186 roundoff error would yield a smaller root that is a factor of two too
187 high:
188 -200.0 1.4210854715202004e-14
191 The invocations of exactexpr should be fairly self-explanatory. The
193 other commands of note are ref and unref. It is necessary for the
194 caller to keep track of references to exact expressions - to call ref
195 every time an exact expression is stored in a variable and unref every
196 time the variable goes out of scope or is overwritten. The asFloat
197 method emits decimal digits as long as the requested precision supports
198 them. It terminates when the requested precision yields an uncertainty
199 of more than one unit in the least significant digit.
200
202 Mathematics
203
205 Copyright (c) 2015 Kevin B. Kenny <kennykb@acm.org>
206 Redistribution permitted under the terms of the Open Publication License <http://www.opencontent.org/openpub/>
207tcllib 1.0.1 math::exact(n)