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 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
23 */
24 /*
25 * Copyright 2005 Sun Microsystems, Inc. All rights reserved.
26 * Use is subject to license terms.
27 */
28
29 #pragma weak __log = log
30
31 /* INDENT OFF */
32 /*
33 * log(x)
34 * Table look-up algorithm with product polynomial approximation.
35 * By K.C. Ng, Oct 23, 2004. Updated Oct 18, 2005.
36 *
37 * (a). For x in [1-0.125, 1+0.1328125], using a special approximation:
38 * Let f = x - 1 and z = f*f.
39 * return f + ((a1*z) *
40 * ((a2 + (a3*f)*(a4+f)) + (f*z)*(a5+f))) *
41 * (((a6 + f*(a7+f)) + (f*z)*(a8+f)) *
42 * ((a9 + (a10*f)*(a11+f)) + (f*z)*(a12+f)))
43 * a1 -6.88821452420390473170286327331268694251775741577e-0002,
44 * a2 1.97493380704769294631262255279580131173133850098e+0000,
45 * a3 2.24963218866067560242072431719861924648284912109e+0000,
46 * a4 -9.02975906958474405783476868236903101205825805664e-0001,
47 * a5 -1.47391630715542865104339398385491222143173217773e+0000,
48 * a6 1.86846544648220058704168877738993614912033081055e+0000,
49 * a7 1.82277370459347465292410106485476717352867126465e+0000,
50 * a8 1.25295479915214102994980294170090928673744201660e+0000,
51 * a9 1.96709676945198275177517643896862864494323730469e+0000,
52 * a10 -4.00127989749189894030934055990655906498432159424e-0001,
53 * a11 3.01675528558798333733648178167641162872314453125e+0000,
54 * a12 -9.52325445049240770778453679668018594384193420410e-0001,
55 *
56 * with remez error |(log(1+f) - P(f))/f| <= 2**-56.81 and
57 *
58 * (b). For 0.09375 <= x < 24
59 * Use an 8-bit table look-up (3-bit for exponent and 5 bit for
60 * significand):
61 * Let ix stands for the high part of x in IEEE double format.
62 * Since 0.09375 <= x < 24, we have
63 * 0x3fb80000 <= ix < 0x40380000.
64 * Let j = (ix - 0x3fb80000) >> 15. Then 0 <= j < 256. Choose
65 * a Y[j] such that HIWORD(Y[j]) ~ 0x3fb8400 + (j<<15) (the middle
66 * number between 0x3fb80000 + (j<<15) and 3fb80000 + ((j+1)<<15)),
67 * and at the same time 1/Y[j] as well as log(Y[j]) are very close
68 * to 53-bits floating point numbers.
69 * A table of Y[j], 1/Y[j], and log(Y[j]) are pre-computed and thus
70 * log(x) = log(Y[j]) + log(1 + (x-Y[j])*(1/Y[j]))
71 * = log(Y[j]) + log(1 + s)
72 * where
73 * s = (x-Y[j])*(1/Y[j])
74 * We compute max (x-Y[j])*(1/Y[j]) for the chosen Y[j] and obtain
75 * |s| < 0.0154. By applying remez algorithm with Product Polynomial
76 * Approximiation, we find the following approximated of log(1+s)
77 * (b1*s)*(b2+s*(b3+s))*((b4+s*b5)+(s*s)*(b6+s))*(b7+s*(b8+s))
78 * with remez error |log(1+s) - P(s)| <= 2**-63.5
79 *
80 * (c). Otherwise, get "n", the exponent of x, and then normalize x to
81 * z in [1,2). Then similar to (b) find a Y[i] that matches z to 5.5
82 * significant bits. Then
83 * log(x) = n*ln2 + log(Y[i]) + log(z/Y[i]).
84 *
85 * Special cases:
86 * log(x) is NaN with signal if x < 0 (including -INF) ;
87 * log(+INF) is +INF; log(0) is -INF with signal;
88 * log(NaN) is that NaN with no signal.
89 *
90 * Maximum error observed: less than 0.90 ulp
91 *
92 * Constants:
93 * The hexadecimal values are the intended ones for the following constants.
94 * The decimal values may be used, provided that the compiler will convert
95 * from decimal to binary accurately enough to produce the hexadecimal values
96 * shown.
97 */
98 /* INDENT ON */
99
100 #include "libm.h"
101
102 extern const double _TBL_log[];
103
104 static const double P[] = {
105 /* ONE */ 1.0,
106 /* TWO52 */ 4503599627370496.0,
107 /* LN2HI */ 6.93147180369123816490e-01, /* 3fe62e42, fee00000 */
108 /* LN2LO */ 1.90821492927058770002e-10, /* 3dea39ef, 35793c76 */
109 /* A1 */ -6.88821452420390473170286327331268694251775741577e-0002,
110 /* A2 */ 1.97493380704769294631262255279580131173133850098e+0000,
111 /* A3 */ 2.24963218866067560242072431719861924648284912109e+0000,
112 /* A4 */ -9.02975906958474405783476868236903101205825805664e-0001,
113 /* A5 */ -1.47391630715542865104339398385491222143173217773e+0000,
114 /* A6 */ 1.86846544648220058704168877738993614912033081055e+0000,
115 /* A7 */ 1.82277370459347465292410106485476717352867126465e+0000,
116 /* A8 */ 1.25295479915214102994980294170090928673744201660e+0000,
117 /* A9 */ 1.96709676945198275177517643896862864494323730469e+0000,
118 /* A10 */ -4.00127989749189894030934055990655906498432159424e-0001,
119 /* A11 */ 3.01675528558798333733648178167641162872314453125e+0000,
120 /* A12 */ -9.52325445049240770778453679668018594384193420410e-0001,
121 /* B1 */ -1.25041641589283658575482149899471551179885864258e-0001,
122 /* B2 */ 1.87161713283355151891381127914642725337613123482e+0000,
123 /* B3 */ -1.89082956295731507978530316904652863740921020508e+0000,
124 /* B4 */ -2.50562891673640253387134180229622870683670043945e+0000,
125 /* B5 */ 1.64822828085258366037635369139024987816810607910e+0000,
126 /* B6 */ -1.24409107065868340669112512841820716857910156250e+0000,
127 /* B7 */ 1.70534231658220414296067701798165217041969299316e+0000,
128 /* B8 */ 1.99196833784655646937267192697618156671524047852e+0000,
129 };
130
131 #define ONE P[0]
132 #define TWO52 P[1]
133 #define LN2HI P[2]
134 #define LN2LO P[3]
135 #define A1 P[4]
136 #define A2 P[5]
137 #define A3 P[6]
138 #define A4 P[7]
139 #define A5 P[8]
140 #define A6 P[9]
141 #define A7 P[10]
142 #define A8 P[11]
143 #define A9 P[12]
144 #define A10 P[13]
145 #define A11 P[14]
146 #define A12 P[15]
147 #define B1 P[16]
148 #define B2 P[17]
149 #define B3 P[18]
150 #define B4 P[19]
151 #define B5 P[20]
152 #define B6 P[21]
153 #define B7 P[22]
154 #define B8 P[23]
155
156 double
157 log(double x) {
158 double *tb, dn, dn1, s, z, r, w;
159 int i, hx, ix, n, lx;
160
161 n = 0;
162 hx = ((int *)&x)[HIWORD];
163 ix = hx & 0x7fffffff;
164 lx = ((int *)&x)[LOWORD];
165
166 /* subnormal,0,negative,inf,nan */
167 if ((hx + 0x100000) < 0x200000) {
168 if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0)) /* nan */
169 return (x * x);
170 if (((hx << 1) | lx) == 0) /* zero */
171 return (_SVID_libm_err(x, x, 16));
172 if (hx < 0) /* negative */
173 return (_SVID_libm_err(x, x, 17));
174 if (((hx - 0x7ff00000) | lx) == 0) /* +inf */
175 return (x);
176
177 /* x must be positive and subnormal */
178 x *= TWO52;
179 n = -52;
180 ix = ((int *)&x)[HIWORD];
181 lx = ((int *)&x)[LOWORD];
182 }
183
184 i = ix >> 19;
185 if (i >= 0x7f7 && i <= 0x806) {
186 /* 0.09375 (0x3fb80000) <= x < 24 (0x40380000) */
187 if (ix >= 0x3fec0000 && ix < 0x3ff22000) {
188 /* 0.875 <= x < 1.125 */
189 s = x - ONE;
190 z = s * s;
191 if (((ix - 0x3ff00000) | lx) == 0) /* x = 1 */
192 return (z);
193 r = (A10 * s) * (A11 + s);
194 w = z * s;
195 return (s + ((A1 * z) *
196 (A2 + ((A3 * s) * (A4 + s) + w * (A5 + s)))) *
197 ((A6 + (s * (A7 + s) + w * (A8 + s))) *
198 (A9 + (r + w * (A12 + s)))));
199 } else {
200 i = (ix - 0x3fb80000) >> 15;
201 tb = (double *)_TBL_log + (i + i + i);
202 s = (x - tb[0]) * tb[1];
203 return (tb[2] + ((B1 * s) * (B2 + s * (B3 + s))) *
204 (((B4 + s * B5) + (s * s) * (B6 + s)) *
205 (B7 + s * (B8 + s))));
206 }
207 } else {
208 dn = (double)(n + ((ix >> 20) - 0x3ff));
209 dn1 = dn * LN2HI;
210 i = (ix & 0x000fffff) | 0x3ff00000; /* scale x to [1,2] */
211 ((int *)&x)[HIWORD] = i;
212 i = (i - 0x3fb80000) >> 15;
213 tb = (double *)_TBL_log + (i + i + i);
214 s = (x - tb[0]) * tb[1];
215 dn = dn * LN2LO + tb[2];
216 return (dn1 + (dn + ((B1 * s) * (B2 + s * (B3 + s))) *
217 (((B4 + s * B5) + (s * s) * (B6 + s)) *
218 (B7 + s * (B8 + s)))));
219 }
220 }