Line data Source code
1 : /* Definitions of some C99 math library functions, for those platforms
2 : that don't implement these functions already. */
3 :
4 : #include "Python.h"
5 : #include <float.h>
6 : #include "_math.h"
7 :
8 : /* The following copyright notice applies to the original
9 : implementations of acosh, asinh and atanh. */
10 :
11 : /*
12 : * ====================================================
13 : * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
14 : *
15 : * Developed at SunPro, a Sun Microsystems, Inc. business.
16 : * Permission to use, copy, modify, and distribute this
17 : * software is freely granted, provided that this notice
18 : * is preserved.
19 : * ====================================================
20 : */
21 :
22 : static const double ln2 = 6.93147180559945286227E-01;
23 : static const double two_pow_m28 = 3.7252902984619141E-09; /* 2**-28 */
24 : static const double two_pow_p28 = 268435456.0; /* 2**28 */
25 :
26 : /* acosh(x)
27 : * Method :
28 : * Based on
29 : * acosh(x) = log [ x + sqrt(x*x-1) ]
30 : * we have
31 : * acosh(x) := log(x)+ln2, if x is large; else
32 : * acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
33 : * acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
34 : *
35 : * Special cases:
36 : * acosh(x) is NaN with signal if x<1.
37 : * acosh(NaN) is NaN without signal.
38 : */
39 :
40 : double
41 0 : _Py_acosh(double x)
42 : {
43 0 : if (Py_IS_NAN(x)) {
44 0 : return x+x;
45 : }
46 0 : if (x < 1.) { /* x < 1; return a signaling NaN */
47 0 : errno = EDOM;
48 : #ifdef Py_NAN
49 0 : return Py_NAN;
50 : #else
51 : return (x-x)/(x-x);
52 : #endif
53 : }
54 0 : else if (x >= two_pow_p28) { /* x > 2**28 */
55 0 : if (Py_IS_INFINITY(x)) {
56 0 : return x+x;
57 : }
58 : else {
59 0 : return log(x)+ln2; /* acosh(huge)=log(2x) */
60 : }
61 : }
62 0 : else if (x == 1.) {
63 0 : return 0.0; /* acosh(1) = 0 */
64 : }
65 0 : else if (x > 2.) { /* 2 < x < 2**28 */
66 0 : double t = x*x;
67 0 : return log(2.0*x - 1.0 / (x + sqrt(t - 1.0)));
68 : }
69 : else { /* 1 < x <= 2 */
70 0 : double t = x - 1.0;
71 0 : return m_log1p(t + sqrt(2.0*t + t*t));
72 : }
73 : }
74 :
75 :
76 : /* asinh(x)
77 : * Method :
78 : * Based on
79 : * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
80 : * we have
81 : * asinh(x) := x if 1+x*x=1,
82 : * := sign(x)*(log(x)+ln2)) for large |x|, else
83 : * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
84 : * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
85 : */
86 :
87 : double
88 0 : _Py_asinh(double x)
89 : {
90 : double w;
91 0 : double absx = fabs(x);
92 :
93 0 : if (Py_IS_NAN(x) || Py_IS_INFINITY(x)) {
94 0 : return x+x;
95 : }
96 0 : if (absx < two_pow_m28) { /* |x| < 2**-28 */
97 0 : return x; /* return x inexact except 0 */
98 : }
99 0 : if (absx > two_pow_p28) { /* |x| > 2**28 */
100 0 : w = log(absx)+ln2;
101 : }
102 0 : else if (absx > 2.0) { /* 2 < |x| < 2**28 */
103 0 : w = log(2.0*absx + 1.0 / (sqrt(x*x + 1.0) + absx));
104 : }
105 : else { /* 2**-28 <= |x| < 2= */
106 0 : double t = x*x;
107 0 : w = m_log1p(absx + t / (1.0 + sqrt(1.0 + t)));
108 : }
109 0 : return copysign(w, x);
110 :
111 : }
112 :
113 : /* atanh(x)
114 : * Method :
115 : * 1.Reduced x to positive by atanh(-x) = -atanh(x)
116 : * 2.For x>=0.5
117 : * 1 2x x
118 : * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * -------)
119 : * 2 1 - x 1 - x
120 : *
121 : * For x<0.5
122 : * atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
123 : *
124 : * Special cases:
125 : * atanh(x) is NaN if |x| >= 1 with signal;
126 : * atanh(NaN) is that NaN with no signal;
127 : *
128 : */
129 :
130 : double
131 0 : _Py_atanh(double x)
132 : {
133 : double absx;
134 : double t;
135 :
136 0 : if (Py_IS_NAN(x)) {
137 0 : return x+x;
138 : }
139 0 : absx = fabs(x);
140 0 : if (absx >= 1.) { /* |x| >= 1 */
141 0 : errno = EDOM;
142 : #ifdef Py_NAN
143 0 : return Py_NAN;
144 : #else
145 : return x/0.0;
146 : #endif
147 : }
148 0 : if (absx < two_pow_m28) { /* |x| < 2**-28 */
149 0 : return x;
150 : }
151 0 : if (absx < 0.5) { /* |x| < 0.5 */
152 0 : t = absx+absx;
153 0 : t = 0.5 * m_log1p(t + t*absx / (1.0 - absx));
154 : }
155 : else { /* 0.5 <= |x| <= 1.0 */
156 0 : t = 0.5 * m_log1p((absx + absx) / (1.0 - absx));
157 : }
158 0 : return copysign(t, x);
159 : }
160 :
161 : /* Mathematically, expm1(x) = exp(x) - 1. The expm1 function is designed
162 : to avoid the significant loss of precision that arises from direct
163 : evaluation of the expression exp(x) - 1, for x near 0. */
164 :
165 : double
166 0 : _Py_expm1(double x)
167 : {
168 : /* For abs(x) >= log(2), it's safe to evaluate exp(x) - 1 directly; this
169 : also works fine for infinities and nans.
170 :
171 : For smaller x, we can use a method due to Kahan that achieves close to
172 : full accuracy.
173 : */
174 :
175 0 : if (fabs(x) < 0.7) {
176 : double u;
177 0 : u = exp(x);
178 0 : if (u == 1.0)
179 0 : return x;
180 : else
181 0 : return (u - 1.0) * x / log(u);
182 : }
183 : else
184 0 : return exp(x) - 1.0;
185 : }
186 :
187 : /* log1p(x) = log(1+x). The log1p function is designed to avoid the
188 : significant loss of precision that arises from direct evaluation when x is
189 : small. */
190 :
191 : #ifdef HAVE_LOG1P
192 :
193 : double
194 0 : _Py_log1p(double x)
195 : {
196 : /* Some platforms supply a log1p function but don't respect the sign of
197 : zero: log1p(-0.0) gives 0.0 instead of the correct result of -0.0.
198 :
199 : To save fiddling with configure tests and platform checks, we handle the
200 : special case of zero input directly on all platforms.
201 : */
202 0 : if (x == 0.0) {
203 0 : return x;
204 : }
205 : else {
206 0 : return log1p(x);
207 : }
208 : }
209 :
210 : #else
211 :
212 : double
213 : _Py_log1p(double x)
214 : {
215 : /* For x small, we use the following approach. Let y be the nearest float
216 : to 1+x, then
217 :
218 : 1+x = y * (1 - (y-1-x)/y)
219 :
220 : so log(1+x) = log(y) + log(1-(y-1-x)/y). Since (y-1-x)/y is tiny, the
221 : second term is well approximated by (y-1-x)/y. If abs(x) >=
222 : DBL_EPSILON/2 or the rounding-mode is some form of round-to-nearest
223 : then y-1-x will be exactly representable, and is computed exactly by
224 : (y-1)-x.
225 :
226 : If abs(x) < DBL_EPSILON/2 and the rounding mode is not known to be
227 : round-to-nearest then this method is slightly dangerous: 1+x could be
228 : rounded up to 1+DBL_EPSILON instead of down to 1, and in that case
229 : y-1-x will not be exactly representable any more and the result can be
230 : off by many ulps. But this is easily fixed: for a floating-point
231 : number |x| < DBL_EPSILON/2., the closest floating-point number to
232 : log(1+x) is exactly x.
233 : */
234 :
235 : double y;
236 : if (fabs(x) < DBL_EPSILON/2.) {
237 : return x;
238 : }
239 : else if (-0.5 <= x && x <= 1.) {
240 : /* WARNING: it's possible than an overeager compiler
241 : will incorrectly optimize the following two lines
242 : to the equivalent of "return log(1.+x)". If this
243 : happens, then results from log1p will be inaccurate
244 : for small x. */
245 : y = 1.+x;
246 : return log(y)-((y-1.)-x)/y;
247 : }
248 : else {
249 : /* NaNs and infinities should end up here */
250 : return log(1.+x);
251 : }
252 : }
253 :
254 : #endif /* ifdef HAVE_LOG1P */
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