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 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __log2 = log2
32
33
34 /*
35 * log2(x) = log(x)/log2
36 *
37 * Base on Table look-up algorithm with product polynomial
38 * approximation for log(x).
39 *
40 * By K.C. Ng, Nov 29, 2004
41 *
42 * (a). For x in [1-0.125, 1+0.125], from log.c we have
43 * log(x) = f + ((a1*f^2) *
44 * ((a2 + (a3*f)*(a4+f)) + (f^3)*(a5+f))) *
45 * (((a6 + f*(a7+f)) + (f^3)*(a8+f)) *
46 * ((a9 + (a10*f)*(a11+f)) + (f^3)*(a12+f)))
47 * where f = x - 1.
48 * (i) modify a1 <- a1 / log2
49 * (ii) 1/log2 = 1.4426950408889634...
50 * = 1.5 - 0.057304959... (4 bit shift)
51 * Let lv = 1.5 - 1/log2, then
52 * lv = 0.057304959111036592640075318998107956665325,
53 * (iii) f*1.5 is exact because f has 3 trailing zero.
54 * (iv) Thus, log2(x) = f*1.5 - (lv*f - PPoly)
55 *
56 * (b). For 0.09375 <= x < 24
57 * Let j = (ix - 0x3fb80000) >> 15. Look up Y[j], 1/Y[j], and log(Y[j])
58 * from _TBL_log.c. Then
59 * log2(x) = log2(Y[j]) + log2(1 + (x-Y[j])*(1/Y[j]))
60 * = log(Y[j])(1/log2) + log2(1 + s)
61 * where
62 * s = (x-Y[j])*(1/Y[j])
63 * From log.c, we have log(1+s) =
64 * 2 2 2
65 * (b s) (b + b s + s ) [b + b s + s (b + s)] (b + b s + s )
66 * 1 2 3 4 5 6 7 8
67 *
68 * By setting b1 <- b1/log2, we have
69 * log2(x) = 1.5 * T - (lv * T - POLY(s))
70 *
71 * (c). Otherwise, get "n", the exponent of x, and then normalize x to
72 * z in [1,2). Then similar to (b) find a Y[i] that matches z to 5.5
73 * significant bits. Then
74 * log2(x) = n + log2(z).
75 *
76 * Special cases:
77 * log2(x) is NaN with signal if x < 0 (including -INF) ;
78 * log2(+INF) is +INF; log2(0) is -INF with signal;
79 * log2(NaN) is that NaN with no signal.
80 *
81 * Maximum error observed: less than 0.84 ulp
82 *
83 * Constants:
84 * The hexadecimal values are the intended ones for the following constants.
85 * The decimal values may be used, provided that the compiler will convert
86 * from decimal to binary accurately enough to produce the hexadecimal values
87 * shown.
88 */
89
90 #include "libm.h"
91 #include "libm_protos.h"
92
93 extern const double _TBL_log[];
94
95 static const double P[] = {
96 /* ONE */
97 1.0,
98 /* TWO52 */ 4503599627370496.0,
99 /* LN10V */ 1.4426950408889634073599246810018920433347, /* 1/log10 */
100 /* ZERO */ 0.0,
101 /* A1 */ -9.6809362455249638217841932228967194640116e-02,
102 /* A2 */ 1.99628461483039965074226529395673424005508422852e+0000,
103 /* A3 */ 2.26812367662950720159642514772713184356689453125e+0000,
104 /* A4 */ -9.05030639084976384900471657601883634924888610840e-0001,
105 /* A5 */ -1.48275767132434044270894446526654064655303955078e+0000,
106 /* A6 */ 1.88158320939722756293122074566781520843505859375e+0000,
107 /* A7 */ 1.83309386046986411145098827546462416648864746094e+0000,
108 /* A8 */ 1.24847063988317086291601754055591300129890441895e+0000,
109 /* A9 */ 1.98372421445537705508854742220137268304824829102e+0000,
110 /* A10 */ -3.94711735767898475035764249696512706577777862549e-0001,
111 /* A11 */ 3.07890395362954372160402272129431366920471191406e+0000,
112 /* A12 */ -9.60099585275022149311041630426188930869102478027e-0001,
113 /* B1 */ -1.8039695622547469514898963204616532885451e-01,
114 /* B2 */ 1.87161713283355151891381127914642725337613123482e+0000,
115 /* B3 */ -1.89082956295731507978530316904652863740921020508e+0000,
116 /* B4 */ -2.50562891673640253387134180229622870683670043945e+0000,
117 /* B5 */ 1.64822828085258366037635369139024987816810607910e+0000,
118 /* B6 */ -1.24409107065868340669112512841820716857910156250e+0000,
119 /* B7 */ 1.70534231658220414296067701798165217041969299316e+0000,
120 /* B8 */ 1.99196833784655646937267192697618156671524047852e+0000,
121 /* LGH */ 1.5,
122 /* LGL */ 0.057304959111036592640075318998107956665325,
123 };
124
125 #define ONE P[0]
126 #define TWO52 P[1]
127 #define LN10V P[2]
128 #define ZERO P[3]
129 #define A1 P[4]
130 #define A2 P[5]
131 #define A3 P[6]
132 #define A4 P[7]
133 #define A5 P[8]
134 #define A6 P[9]
135 #define A7 P[10]
136 #define A8 P[11]
137 #define A9 P[12]
138 #define A10 P[13]
139 #define A11 P[14]
140 #define A12 P[15]
141 #define B1 P[16]
142 #define B2 P[17]
143 #define B3 P[18]
144 #define B4 P[19]
145 #define B5 P[20]
146 #define B6 P[21]
147 #define B7 P[22]
148 #define B8 P[23]
149 #define LGH P[24]
150 #define LGL P[25]
151
152 double
153 log2(double x)
154 {
155 int i, hx, ix, n, lx;
156
157 n = 0;
158 hx = ((int *)&x)[HIWORD];
159 ix = hx & 0x7fffffff;
160 lx = ((int *)&x)[LOWORD];
161
162 /* subnormal,0,negative,inf,nan */
163 if ((hx + 0x100000) < 0x200000) {
164 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
165 if (ix >= 0x7ff80000) /* assumes sparc-like QNaN */
166 return (x); /* for Cheetah when x is QNaN */
167 #endif
168
169 if (((hx << 1) | lx) == 0) /* log(0.0) = -inf */
170 return (A5 / fabs(x));
171
172 if (hx < 0) { /* x < 0 */
173 if (ix >= 0x7ff00000)
174 return (x - x); /* x is -inf or NaN */
175 else
176 return (ZERO / (x - x));
177 }
178
179 if (((hx - 0x7ff00000) | lx) == 0) /* log(inf) = inf */
180 return (x);
181
182 if (ix >= 0x7ff00000) /* log(NaN) = NaN */
183 return (x - x);
184
185 x *= TWO52;
186 n = -52;
187 hx = ((int *)&x)[HIWORD];
188 ix = hx & 0x7fffffff;
189 lx = ((int *)&x)[LOWORD];
190 }
191
192 /* 0.09375 (0x3fb80000) <= x < 24 (0x40380000) */
193 i = ix >> 19;
194
195 if (i >= 0x7f7 && i <= 0x806) {
196 /* 0.875 <= x < 1.125 */
197 if (ix >= 0x3fec0000 && ix < 0x3ff20000) {
198 double s, z, r, w;
199
200 s = x - ONE;
201 z = s * s;
202 r = (A10 * s) * (A11 + s);
203 w = z * s;
204
205 if (((ix << 12) | lx) == 0) {
206 return (z);
207 } else {
208 return (LGH * s - (LGL * s - ((A1 * z) * ((A2 +
209 (A3 * s) * (A4 + s)) + w * (A5 + s))) *
210 (((A6 + s * (A7 + s)) + w * (A8 + s)) *
211 ((A9 + r) + w * (A12 + s)))));
212 }
213 } else {
214 double *tb, s;
215
216 i = (ix - 0x3fb80000) >> 15;
217 tb = (double *)_TBL_log + (i + i + i);
218
219 if (((ix << 12) | lx) == 0) /* 2's power */
220 return ((double)((ix >> 20) - 0x3ff));
221
222 s = (x - tb[0]) * tb[1];
223 return (LGH * tb[2] - (LGL * tb[2] - ((B1 * s) *
224 (B2 + s * (B3 + s))) * (((B4 + s * B5) + (s * s) *
225 (B6 + s)) * (B7 + s * (B8 + s)))));
226 }
227 } else {
228 double *tb, dn, s;
229
230 dn = (double)(n + ((ix >> 20) - 0x3ff));
231 ix <<= 12;
232
233 if ((ix | lx) == 0)
234 return (dn);
235
236 i = ((unsigned)ix >> 12) | 0x3ff00000; /* scale x to [1,2) */
237 ((int *)&x)[HIWORD] = i;
238 i = (i - 0x3fb80000) >> 15;
239 tb = (double *)_TBL_log + (i + i + i);
240 s = (x - tb[0]) * tb[1];
241 return (dn + (tb[2] * LN10V + ((B1 * s) * (B2 + s * (B3 + s))) *
242 (((B4 + s * B5) + (s * s) * (B6 + s)) *
243 (B7 + s * (B8 + s)))));
244 }
245 }