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 * Copyright 2006 Sun Microsystems, Inc. All rights reserved. 27 * Use is subject to license terms. 28 */ 29 30 #ifdef __LITTLE_ENDIAN 31 #define H0(x) *(3 + (int *) &x) 32 #define H1(x) *(2 + (int *) &x) 33 #define H2(x) *(1 + (int *) &x) 34 #define H3(x) *(int *) &x 35 #else 36 #define H0(x) *(int *) &x 37 #define H1(x) *(1 + (int *) &x) 38 #define H2(x) *(2 + (int *) &x) 39 #define H3(x) *(3 + (int *) &x) 40 #endif 41 42 /* 43 * log1pl(x) 44 * Table look-up algorithm by modifying logl.c 45 * By K.C. Ng, July 6, 1995 46 * 47 * (a). For 1+x in [31/33,33/31], using a special approximation: 48 * s = x/(2.0+x); ... here |s| <= 0.03125 49 * z = s*s; 50 * return x-s*(x-z*(B1+z*(B2+z*(B3+z*(B4+...+z*B9)...)))); 51 * (i.e., x is in [-2/33,2/31]) 52 * 53 * (b). Otherwise, normalize 1+x = 2^n * 1.f. 54 * Here we may need a correction term for 1+x rounded. 55 * Use a 6-bit table look-up: find a 6 bit g that match f to 6.5 bits, 56 * then 57 * log(1+x) = n*ln2 + log(1.g) + log(1.f/1.g). 58 * Here the leading and trailing values of log(1.g) are obtained from 59 * a size-64 table. 60 * For log(1.f/1.g), let s = (1.f-1.g)/(1.f+1.g). Note that 61 * 1.f = 2^-n(1+x) 62 * 63 * then 64 * log(1.f/1.g) = log((1+s)/(1-s)) = 2s + 2/3 s^3 + 2/5 s^5 +... 65 * Note that |s|<2**-8=0.00390625. We use an odd s-polynomial 66 * approximation to compute log(1.f/1.g): 67 * s*(A1+s^2*(A2+s^2*(A3+s^2*(A4+s^2*(A5+s^2*(A6+s^2*A7)))))) 68 * (Precision is 2**-136.91 bits, absolute error) 69 * 70 * CAUTION: 71 * For x>=1, compute 1+x will lost one bit (OK). 72 * For x in [-0.5,-1), 1+x is exact. 73 * For x in (-0.5,-2/33]U[2/31,1), up to 4 last bits of x will be lost 74 * in 1+x. Therefore, to recover the lost bits, one need to compute 75 * 1.f-1.g accurately. 76 * 77 * Let hx = HI(x), m = (hx>>16)-0x3fff (=ilogbl(x)), note that 78 * -2/33 = -0.0606...= 2^-5 * 1.939..., 79 * 2/31 = 0.09375 = 2^-4 * 1.500..., 80 * so for x in (-0.5,-2/33], -5<=m<=-2, n= -1, 1+f=2*(1+x) 81 * for x in [2/33,1), -4<=m<=-1, n= 0, f=x 82 * 83 * In short: 84 * if x>0, let g: hg= ((hx + (0x200<<(-m)))>>(10-m))<<(10-m) 85 * then 1.f-1.g = x-g 86 * if x<0, let g': hg' =((ix-(0x200)<<(-m-1))>>(9-m))<<(9-m) 87 * (ix=hx&0x7fffffff) 88 * then 1.f-1.g = 2*(g'+x), 89 * 90 * (c). The final result is computed by 91 * (n*ln2_hi+_TBL_logl_hi[j]) + 92 * ( (n*ln2_lo+_TBL_logl_lo[j]) + s*(A1+...) ) 93 * 94 * Note. 95 * For ln2_hi and _TBL_logl_hi[j], we force their last 32 bit to be zero 96 * so that n*ln2_hi + _TBL_logl_hi[j] is exact. Here 97 * _TBL_logl_hi[j] + _TBL_logl_lo[j] match log(1+j*2**-6) to 194 bits 98 * 99 * 100 * Special cases: 101 * log(x) is NaN with signal if x < 0 (including -INF) ; 102 * log(+INF) is +INF; log(0) is -INF with signal; 103 * log(NaN) is that NaN with no signal. 104 * 105 * Constants: 106 * The hexadecimal values are the intended ones for the following constants. 107 * The decimal values may be used, provided that the compiler will convert 108 * from decimal to binary accurately enough to produce the hexadecimal values 109 * shown. 110 */ 111 112 #pragma weak __log1pl = log1pl 113 114 #include "libm.h" 115 116 extern const long double _TBL_logl_hi[], _TBL_logl_lo[]; 117 118 static const long double 119 zero = 0.0L, 120 one = 1.0L, 121 two = 2.0L, 122 ln2hi = 6.931471805599453094172319547495844850203e-0001L, 123 ln2lo = 1.667085920830552208890449330400379754169e-0025L, 124 A1 = 2.000000000000000000000000000000000000024e+0000L, 125 A2 = 6.666666666666666666666666666666091393804e-0001L, 126 A3 = 4.000000000000000000000000407167070220671e-0001L, 127 A4 = 2.857142857142857142730077490612903681164e-0001L, 128 A5 = 2.222222222222242577702836920812882605099e-0001L, 129 A6 = 1.818181816435493395985912667105885828356e-0001L, 130 A7 = 1.538537835211839751112067512805496931725e-0001L, 131 B1 = 6.666666666666666666666666666666961498329e-0001L, 132 B2 = 3.999999999999999999999999990037655042358e-0001L, 133 B3 = 2.857142857142857142857273426428347457918e-0001L, 134 B4 = 2.222222222222222221353229049747910109566e-0001L, 135 B5 = 1.818181818181821503532559306309070138046e-0001L, 136 B6 = 1.538461538453809210486356084587356788556e-0001L, 137 B7 = 1.333333344463358756121456892645178795480e-0001L, 138 B8 = 1.176460904783899064854645174603360383792e-0001L, 139 B9 = 1.057293869956598995326368602518056990746e-0001L; 140 141 long double 142 log1pl(long double x) { 143 long double f, s, z, qn, h, t, y, g; 144 int i, j, ix, iy, n, hx, m; 145 146 hx = H0(x); 147 ix = hx & 0x7fffffff; 148 if (ix < 0x3ffaf07c) { /* |x|<2/33 */ 149 if (ix <= 0x3f8d0000) { /* x <= 2**-114, return x */ 150 if ((int) x == 0) 151 return (x); 152 } 153 s = x / (two + x); /* |s|<2**-8 */ 154 z = s * s; 155 return (x - s * (x - z * (B1 + z * (B2 + z * (B3 + z * (B4 + 156 z * (B5 + z * (B6 + z * (B7 + z * (B8 + z * B9)))))))))); 157 } 158 if (ix >= 0x7fff0000) { /* x is +inf or NaN */ 159 return (x + fabsl(x)); 160 } 161 if (hx < 0 && ix >= 0x3fff0000) { 162 if (ix > 0x3fff0000 || (H1(x) | H2(x) | H3(x)) != 0) 163 x = zero; 164 return (x / zero); /* log1p(x) is NaN if x<-1 */ 165 /* log1p(-1) is -inf */ 166 } 167 if (ix >= 0x7ffeffff) 168 y = x; /* avoid spurious overflow */ 169 else 170 y = one + x; 171 iy = H0(y); 172 n = ((iy + 0x200) >> 16) - 0x3fff; 173 iy = (iy & 0x0000ffff) | 0x3fff0000; /* scale 1+x to [1,2] */ 174 H0(y) = iy; 175 z = zero; 176 m = (ix >> 16) - 0x3fff; 177 /* HI(1+x) = (((hx&0xffff)|0x10000)>>(-m))|0x3fff0000 */ 178 if (n == 0) { /* x in [2/33,1) */ 179 g = zero; 180 H0(g) = ((hx + (0x200 << (-m))) >> (10 - m)) << (10 - m); 181 t = x - g; 182 i = (((((hx & 0xffff) | 0x10000) >> (-m)) | 0x3fff0000) + 183 0x200) >> 10; 184 H0(z) = i << 10; 185 186 } else if ((1 + n) == 0 && (ix < 0x3ffe0000)) { /* x in (-0.5,-2/33] */ 187 g = zero; 188 H0(g) = ((ix + (0x200 << (-m - 1))) >> (9 - m)) << (9 - m); 189 t = g + x; 190 t = t + t; 191 /* 192 * HI(2*(1+x)) = 193 * ((0x10000-(((hx&0xffff)|0x10000)>>(-m)))<<1)|0x3fff0000 194 */ 195 /* 196 * i = 197 * ((((0x10000-(((hx&0xffff)|0x10000)>>(-m)))<<1)|0x3fff0000)+ 198 * 0x200)>>10; H0(z)=i<<10; 199 */ 200 z = two * (one - g); 201 i = H0(z) >> 10; 202 } else { 203 i = (iy + 0x200) >> 10; 204 H0(z) = i << 10; 205 t = y - z; 206 } 207 208 s = t / (y + z); 209 j = i & 0x3f; 210 z = s * s; 211 qn = (long double) n; 212 t = qn * ln2lo + _TBL_logl_lo[j]; 213 h = qn * ln2hi + _TBL_logl_hi[j]; 214 f = t + s * (A1 + z * (A2 + z * (A3 + z * (A4 + z * (A5 + z * (A6 + 215 z * A7)))))); 216 return (h + f); 217 } 218