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