1 /* Copyright JS Foundation and other contributors, http://js.foundation
2 *
3 * Licensed under the Apache License, Version 2.0 (the "License");
4 * you may not use this file except in compliance with the License.
5 * You may obtain a copy of the License at
6 *
7 * http://www.apache.org/licenses/LICENSE-2.0
8 *
9 * Unless required by applicable law or agreed to in writing, software
10 * distributed under the License is distributed on an "AS IS" BASIS
11 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12 * See the License for the specific language governing permissions and
13 * limitations under the License.
14 *
15 * This file is based on work under the following copyright and permission
16 * notice:
17 *
18 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
19 *
20 * Developed at SunSoft, a Sun Microsystems, Inc. business.
21 * Permission to use, copy, modify, and distribute this
22 * software is freely granted, provided that this notice
23 * is preserved.
24 *
25 * @(#)e_log.c 1.3 95/01/18
26 */
27
28 #include "jerry-libm-internal.h"
29
30 /* log(x)
31 * Return the logrithm of x
32 *
33 * Method :
34 * 1. Argument Reduction: find k and f such that
35 * x = 2^k * (1+f),
36 * where sqrt(2)/2 < 1+f < sqrt(2) .
37 *
38 * 2. Approximation of log(1+f).
39 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
40 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
41 * = 2s + s*R
42 * We use a special Reme algorithm on [0,0.1716] to generate
43 * a polynomial of degree 14 to approximate R The maximum error
44 * of this polynomial approximation is bounded by 2**-58.45. In
45 * other words,
46 * 2 4 6 8 10 12 14
47 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
48 * (the values of Lg1 to Lg7 are listed in the program)
49 * and
50 * | 2 14 | -58.45
51 * | Lg1*s +...+Lg7*s - R(z) | <= 2
52 * | |
53 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
54 * In order to guarantee error in log below 1ulp, we compute log
55 * by
56 * log(1+f) = f - s*(f - R) (if f is not too large)
57 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
58 *
59 * 3. Finally, log(x) = k*ln2 + log(1+f).
60 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
61 * Here ln2 is split into two floating point number:
62 * ln2_hi + ln2_lo,
63 * where n*ln2_hi is always exact for |n| < 2000.
64 *
65 * Special cases:
66 * log(x) is NaN with signal if x < 0 (including -INF) ;
67 * log(+INF) is +INF; log(0) is -INF with signal;
68 * log(NaN) is that NaN with no signal.
69 *
70 * Accuracy:
71 * according to an error analysis, the error is always less than
72 * 1 ulp (unit in the last place).
73 *
74 * Constants:
75 * The hexadecimal values are the intended ones for the following
76 * constants. The decimal values may be used, provided that the
77 * compiler will convert from decimal to binary accurately enough
78 * to produce the hexadecimal values shown.
79 */
80
81 #define zero 0.0
82 #define ln2_hi 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
83 #define ln2_lo 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
84 #define two54 1.80143985094819840000e+16 /* 43500000 00000000 */
85 #define Lg1 6.666666666666735130e-01 /* 3FE55555 55555593 */
86 #define Lg2 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
87 #define Lg3 2.857142874366239149e-01 /* 3FD24924 94229359 */
88 #define Lg4 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
89 #define Lg5 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
90 #define Lg6 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
91 #define Lg7 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
92
93 double
log(double x)94 log (double x)
95 {
96 double hfsq, f, s, z, R, w, t1, t2, dk;
97 int k, hx, i, j;
98 unsigned lx;
99
100 hx = __HI (x); /* high word of x */
101 lx = __LO (x); /* low word of x */
102
103 k = 0;
104 if (hx < 0x00100000) /* x < 2**-1022 */
105 {
106 if (((hx & 0x7fffffff) | lx) == 0) /* log(+-0) = -inf */
107 {
108 return -two54 / zero;
109 }
110 if (hx < 0) /* log(-#) = NaN */
111 {
112 return (x - x) / zero;
113 }
114 k -= 54;
115 x *= two54; /* subnormal number, scale up x */
116 hx = __HI (x); /* high word of x */
117 }
118 if (hx >= 0x7ff00000)
119 {
120 return x + x;
121 }
122 k += (hx >> 20) - 1023;
123 hx &= 0x000fffff;
124 i = (hx + 0x95f64) & 0x100000;
125
126 double_accessor temp;
127 temp.dbl = x;
128 temp.as_int.hi = hx | (i ^ 0x3ff00000); /* normalize x or x / 2 */
129 k += (i >> 20);
130 f = temp.dbl - 1.0;
131
132 if ((0x000fffff & (2 + hx)) < 3) /* |f| < 2**-20 */
133 {
134 if (f == zero)
135 {
136 if (k == 0)
137 {
138 return zero;
139 }
140 else
141 {
142 dk = (double) k;
143 return dk * ln2_hi + dk * ln2_lo;
144 }
145 }
146 R = f * f * (0.5 - 0.33333333333333333 * f);
147 if (k == 0)
148 {
149 return f - R;
150 }
151 else
152 {
153 dk = (double) k;
154 return dk * ln2_hi - ((R - dk * ln2_lo) - f);
155 }
156 }
157 s = f / (2.0 + f);
158 dk = (double) k;
159 z = s * s;
160 i = hx - 0x6147a;
161 w = z * z;
162 j = 0x6b851 - hx;
163 t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
164 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
165 i |= j;
166 R = t2 + t1;
167 if (i > 0)
168 {
169 hfsq = 0.5 * f * f;
170 if (k == 0)
171 {
172 return f - (hfsq - s * (hfsq + R));
173 }
174 else
175 {
176 return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
177 }
178 }
179 else
180 {
181 if (k == 0)
182 {
183 return f - s * (f - R);
184 }
185 else
186 {
187 return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
188 }
189 }
190 } /* log */
191
192 #undef zero
193 #undef ln2_hi
194 #undef ln2_lo
195 #undef two54
196 #undef Lg1
197 #undef Lg2
198 #undef Lg3
199 #undef Lg4
200 #undef Lg5
201 #undef Lg6
202 #undef Lg7
203