1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2005 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __log1p = log1p
32
33
34 /*
35 * Method :
36 * 1. Argument Reduction: find k and f such that
37 * 1+x = 2^k * (1+f),
38 * where sqrt(2)/2 < 1+f < sqrt(2) .
39 *
40 * Note. If k=0, then f=x is exact. However, if k != 0, then f
41 * may not be representable exactly. In that case, a correction
42 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
43 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
44 * and add back the correction term c/u.
45 * (Note: when x > 2**53, one can simply return log(x))
46 *
47 * 2. Approximation of log1p(f).
48 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
49 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
50 * = 2s + s*R
51 * We use a special Reme algorithm on [0,0.1716] to generate
52 * a polynomial of degree 14 to approximate R The maximum error
53 * of this polynomial approximation is bounded by 2**-58.45. In
54 * other words,
55 * 2 4 6 8 10 12 14
56 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
57 * (the values of Lp1 to Lp7 are listed in the program)
58 * and
59 * | 2 14 | -58.45
60 * | Lp1*s +...+Lp7*s - R(z) | <= 2
61 * | |
62 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
63 * In order to guarantee error in log below 1ulp, we compute log
64 * by
65 * log1p(f) = f - (hfsq - s*(hfsq+R)).
66 *
67 * 3. Finally, log1p(x) = k*ln2 + log1p(f).
68 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
69 * Here ln2 is splitted into two floating point number:
70 * ln2_hi + ln2_lo,
71 * where n*ln2_hi is always exact for |n| < 2000.
72 *
73 * Special cases:
74 * log1p(x) is NaN with signal if x < -1 (including -INF) ;
75 * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
76 * log1p(NaN) is that NaN with no signal.
77 *
78 * Accuracy:
79 * according to an error analysis, the error is always less than
80 * 1 ulp (unit in the last place).
81 *
82 * Constants:
83 * The hexadecimal values are the intended ones for the following
84 * constants. The decimal values may be used, provided that the
85 * compiler will convert from decimal to binary accurately enough
86 * to produce the hexadecimal values shown.
87 *
88 * Note: Assuming log() return accurate answer, the following
89 * algorithm can be used to compute log1p(x) to within a few ULP:
90 *
91 * u = 1+x;
92 * if (u == 1.0) return x ; else
93 * return log(u)*(x/(u-1.0));
94 *
95 * See HP-15C Advanced Functions Handbook, p.193.
96 */
97
98 #include "libm.h"
99
100 static const double xxx[] = {
101 /* ln2_hi */
102 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
103 /* ln2_lo */ 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
104 /* two54 */ 1.80143985094819840000e+16, /* 43500000 00000000 */
105 /* Lp1 */ 6.666666666666735130e-01, /* 3FE55555 55555593 */
106 /* Lp2 */ 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
107 /* Lp3 */ 2.857142874366239149e-01, /* 3FD24924 94229359 */
108 /* Lp4 */ 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
109 /* Lp5 */ 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
110 /* Lp6 */ 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
111 /* Lp7 */ 1.479819860511658591e-01, /* 3FC2F112 DF3E5244 */
112 /* zero */ 0.0
113 };
114
115 #define ln2_hi xxx[0]
116 #define ln2_lo xxx[1]
117 #define two54 xxx[2]
118 #define Lp1 xxx[3]
119 #define Lp2 xxx[4]
120 #define Lp3 xxx[5]
121 #define Lp4 xxx[6]
122 #define Lp5 xxx[7]
123 #define Lp6 xxx[8]
124 #define Lp7 xxx[9]
125 #define zero xxx[10]
126
127 double
128 log1p(double x)
129 {
130 double hfsq, f, c = 0.0, s, z, R, u;
131 int k, hx, hu, ax;
132
133 hx = ((int *)&x)[HIWORD]; /* high word of x */
134 ax = hx & 0x7fffffff;
135
136 if (ax >= 0x7ff00000) { /* x is inf or nan */
137 if (((hx - 0xfff00000) | ((int *)&x)[LOWORD]) == 0) /* -inf */
138 return (_SVID_libm_err(x, x, 44));
139
140 return (x * x);
141 }
142
143 k = 1;
144
145 if (hx < 0x3FDA827A) { /* x < 0.41422 */
146 if (ax >= 0x3ff00000) /* x <= -1.0 */
147 return (_SVID_libm_err(x, x, x == -1.0 ? 43 : 44));
148
149 if (ax < 0x3e200000) { /* |x| < 2**-29 */
150 if (two54 + x > zero && /* raise inexact */
151 ax < 0x3c900000) /* |x| < 2**-54 */
152 return (x);
153 else
154 return (x - x * x * 0.5);
155 }
156
157 if (hx > 0 || hx <= (int)0xbfd2bec3) { /* -0.2929<x<0.41422 */
158 k = 0;
159 f = x;
160 hu = 1;
161 }
162 }
163
164 /* We will initialize 'c' here. */
165 if (k != 0) {
166 if (hx < 0x43400000) {
167 u = 1.0 + x;
168 hu = ((int *)&u)[HIWORD]; /* high word of u */
169 k = (hu >> 20) - 1023;
170
171 /*
172 * correction term
173 */
174 c = k > 0 ? 1.0 - (u - x) : x - (u - 1.0);
175 c /= u;
176 } else {
177 u = x;
178 hu = ((int *)&u)[HIWORD]; /* high word of u */
179 k = (hu >> 20) - 1023;
180 c = 0;
181 }
182
183 hu &= 0x000fffff;
184
185 if (hu < 0x6a09e) { /* normalize u */
186 ((int *)&u)[HIWORD] = hu | 0x3ff00000;
187 } else { /* normalize u/2 */
188 k += 1;
189 ((int *)&u)[HIWORD] = hu | 0x3fe00000;
190 hu = (0x00100000 - hu) >> 2;
191 }
192
193 f = u - 1.0;
194 }
195
196 hfsq = 0.5 * f * f;
197
198 if (hu == 0) { /* |f| < 2**-20 */
199 if (f == zero) {
200 if (k == 0)
201 return (zero);
202
203 /* We already initialized 'c' before, when (k != 0) */
204 c += k * ln2_lo;
205 return (k * ln2_hi + c);
206 }
207
208 R = hfsq * (1.0 - 0.66666666666666666 * f);
209
210 if (k == 0)
211 return (f - R);
212
213 return (k * ln2_hi - ((R - (k * ln2_lo + c)) - f));
214 }
215
216 s = f / (2.0 + f);
217 z = s * s;
218 R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 +
219 z * Lp7))))));
220
221 if (k == 0)
222 return (f - (hfsq - s * (hfsq + R)));
223
224 return (k * ln2_hi -
225 ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f));
226 }