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 * Permission to use, copy, modify, and distribute this
21 * software is freely granted, provided that this notice
22 * is preserved.
23 *
24 * @(#)s_log1p.c 5.1 93/09/24
25 */
26
27 #include "jerry-libm-internal.h"
28
29 /* log1p(x)
30 * Method :
31 * 1. Argument Reduction: find k and f such that
32 * 1+x = 2^k * (1+f),
33 * where sqrt(2)/2 < 1+f < sqrt(2) .
34 *
35 * Note. If k=0, then f=x is exact. However, if k!=0, then f
36 * may not be representable exactly. In that case, a correction
37 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
38 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
39 * and add back the correction term c/u.
40 * (Note: when x > 2**53, one can simply return log(x))
41 *
42 * 2. Approximation of log1p(f).
43 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
44 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
45 * = 2s + s*R
46 * We use a special Reme algorithm on [0,0.1716] to generate
47 * a polynomial of degree 14 to approximate R The maximum error
48 * of this polynomial approximation is bounded by 2**-58.45. In
49 * other words,
50 * 2 4 6 8 10 12 14
51 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
52 * (the values of Lp1 to Lp7 are listed in the program)
53 * and
54 * | 2 14 | -58.45
55 * | Lp1*s +...+Lp7*s - R(z) | <= 2
56 * | |
57 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
58 * In order to guarantee error in log below 1ulp, we compute log
59 * by
60 * log1p(f) = f - (hfsq - s*(hfsq+R)).
61 *
62 * 3. Finally, log1p(x) = k*ln2 + log1p(f).
63 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
64 * Here ln2 is split into two floating point number:
65 * ln2_hi + ln2_lo,
66 * where n*ln2_hi is always exact for |n| < 2000.
67 *
68 * Special cases:
69 * log1p(x) is NaN with signal if x < -1 (including -INF) ;
70 * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
71 * log1p(NaN) is that NaN with no signal.
72 *
73 * Accuracy:
74 * according to an error analysis, the error is always less than
75 * 1 ulp (unit in the last place).
76 *
77 * Constants:
78 * The hexadecimal values are the intended ones for the following
79 * constants. The decimal values may be used, provided that the
80 * compiler will convert from decimal to binary accurately enough
81 * to produce the hexadecimal values shown.
82 *
83 * Note: Assuming log() return accurate answer, the following
84 * algorithm can be used to compute log1p(x) to within a few ULP:
85 *
86 * u = 1+x;
87 * if(u==1.0) return x ; else
88 * return log(u)*(x/(u-1.0));
89 *
90 * See HP-15C Advanced Functions Handbook, p.193.
91 */
92
93 #define zero 0.0
94 #define ln2_hi 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
95 #define ln2_lo 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
96 #define two54 1.80143985094819840000e+16 /* 43500000 00000000 */
97 #define Lp1 6.666666666666735130e-01 /* 3FE55555 55555593 */
98 #define Lp2 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
99 #define Lp3 2.857142874366239149e-01 /* 3FD24924 94229359 */
100 #define Lp4 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
101 #define Lp5 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
102 #define Lp6 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
103 #define Lp7 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
104
105 double
log1p(double x)106 log1p (double x)
107 {
108 double hfsq, f, c, s, z, R;
109 double_accessor u;
110 int k, hx, hu, ax;
111
112 hx = __HI (x);
113 ax = hx & 0x7fffffff;
114 c = 0;
115 k = 1;
116 if (hx < 0x3FDA827A)
117 {
118 /* 1+x < sqrt(2)+ */
119 if (ax >= 0x3ff00000)
120 {
121 /* x <= -1.0 */
122 if (x == -1.0)
123 {
124 /* log1p(-1) = +inf */
125 return -two54 / zero;
126 }
127 else
128 {
129 /* log1p(x<-1) = NaN */
130 return NAN;
131 }
132 }
133 if (ax < 0x3e200000)
134 { /* |x| < 2**-29 */
135 if ((two54 + x > zero) /* raise inexact */
136 && (ax < 0x3c900000)) /* |x| < 2**-54 */
137 {
138 return x;
139 }
140 else
141 {
142 return x - x * x * 0.5;
143 }
144 }
145 if ((hx > 0) || hx <= ((int) 0xbfd2bec4))
146 {
147 /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
148 k = 0;
149 f = x;
150 hu = 1;
151 }
152 }
153 if (hx >= 0x7ff00000)
154 {
155 return x + x;
156 }
157 if (k != 0)
158 {
159 if (hx < 0x43400000)
160 {
161 u.dbl = 1.0 + x;
162 hu = u.as_int.hi;
163 k = (hu >> 20) - 1023;
164 c = (k > 0) ? 1.0 - (u.dbl - x) : x - (u.dbl - 1.0); /* correction term */
165 c /= u.dbl;
166 }
167 else
168 {
169 u.dbl = x;
170 hu = u.as_int.hi;
171 k = (hu >> 20) - 1023;
172 c = 0;
173 }
174 hu &= 0x000fffff;
175 /*
176 * The approximation to sqrt(2) used in thresholds is not
177 * critical. However, the ones used above must give less
178 * strict bounds than the one here so that the k==0 case is
179 * never reached from here, since here we have committed to
180 * using the correction term but don't use it if k==0.
181 */
182 if (hu < 0x6a09e)
183 {
184 /* u ~< sqrt(2) */
185 u.as_int.hi = hu | 0x3ff00000; /* normalize u */
186 }
187 else
188 {
189 k += 1;
190 u.as_int.hi = hu | 0x3fe00000; /* normalize u/2 */
191 hu = (0x00100000 - hu) >> 2;
192 }
193 f = u.dbl - 1.0;
194 }
195 hfsq = 0.5 * f * f;
196 if (hu == 0)
197 {
198 /* |f| < 2**-20 */
199 if (f == zero)
200 {
201 if (k == 0)
202 {
203 return zero;
204 }
205 else
206 {
207 c += k * ln2_lo;
208 return k * ln2_hi + c;
209 }
210 }
211 R = hfsq * (1.0 - 0.66666666666666666 * f);
212 if (k == 0)
213 {
214 return f - R;
215 }
216 else
217 {
218 return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
219 }
220 }
221 s = f / (2.0 + f);
222 z = s * s;
223 R = z * (Lp1 +
224 z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
225 if (k == 0)
226 {
227 return f - (hfsq - s * (hfsq + R));
228 }
229 else
230 {
231 return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
232 }
233 } /* log1p */
234
235 #undef zero
236 #undef ln2_hi
237 #undef ln2_lo
238 #undef two54
239 #undef Lp1
240 #undef Lp2
241 #undef Lp3
242 #undef Lp4
243 #undef Lp5
244 #undef Lp6
245 #undef Lp7
246