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 #pragma weak __expm1l = expm1l 31 32 #if !defined(__sparc) 33 #error Unsupported architecture 34 #endif 35 36 /* 37 * expm1l(x) 38 * 39 * Table driven method 40 * Written by K.C. Ng, June 1995. 41 * Algorithm : 42 * 1. expm1(x) = x if x<2**-114 43 * 2. if |x| <= 0.0625 = 1/16, use approximation 44 * expm1(x) = x + x*P/(2-P) 45 * where 46 * P = x - z*(P1+z*(P2+z*(P3+z*(P4+z*(P5+z*P6+z*P7))))), z = x*x; 47 * (this formula is derived from 48 * 2-P+x = R = x*(exp(x)+1)/(exp(x)-1) ~ 2 + x*x/6 - x^4/360 + ...) 49 * 50 * P1 = 1.66666666666666666666666666666638500528074603030e-0001 51 * P2 = -2.77777777777777777777777759668391122822266551158e-0003 52 * P3 = 6.61375661375661375657437408890138814721051293054e-0005 53 * P4 = -1.65343915343915303310185228411892601606669528828e-0006 54 * P5 = 4.17535139755122945763580609663414647067443411178e-0008 55 * P6 = -1.05683795988668526689182102605260986731620026832e-0009 56 * P7 = 2.67544168821852702827123344217198187229611470514e-0011 57 * 58 * Accuracy: |R-x*(exp(x)+1)/(exp(x)-1)|<=2**-119.13 59 * 60 * 3. For 1/16 < |x| < 1.125, choose x(+-i) ~ +-(i+4.5)/64, i=0,..,67 61 * since 62 * exp(x) = exp(xi+(x-xi))= exp(xi)*exp((x-xi)) 63 * we have 64 * expm1(x) = expm1(xi)+(exp(xi))*(expm1(x-xi)) 65 * where 66 * |s=x-xi| <= 1/128 67 * and 68 * expm1(s)=2s/(2-R), R= s-s^2*(T1+s^2*(T2+s^2*(T3+s^2*(T4+s^2*T5)))) 69 * 70 * T1 = 1.666666666666666666666666666660876387437e-1L, 71 * T2 = -2.777777777777777777777707812093173478756e-3L, 72 * T3 = 6.613756613756613482074280932874221202424e-5L, 73 * T4 = -1.653439153392139954169609822742235851120e-6L, 74 * T5 = 4.175314851769539751387852116610973796053e-8L; 75 * 76 * 4. For |x| >= 1.125, return exp(x)-1. 77 * (see algorithm for exp) 78 * 79 * Special cases: 80 * expm1l(INF) is INF, expm1l(NaN) is NaN; 81 * expm1l(-INF)= -1; 82 * for finite argument, only expm1l(0)=0 is exact. 83 * 84 * Accuracy: 85 * according to an error analysis, the error is always less than 86 * 2 ulp (unit in the last place). 87 * 88 * Misc. info. 89 * For 113 bit long double 90 * if x > 1.135652340629414394949193107797076342845e+4 91 * then expm1l(x) overflow; 92 * 93 * Constants: 94 * Only decimal values are given. We assume that the compiler will convert 95 * from decimal to binary accurately enough to produce the correct 96 * hexadecimal values. 97 */ 98 99 #include "libm.h" 100 101 extern const long double _TBL_expl_hi[], _TBL_expl_lo[]; 102 extern const long double _TBL_expm1lx[], _TBL_expm1l[]; 103 104 static const long double 105 zero = +0.0L, 106 one = +1.0L, 107 two = +2.0L, 108 ln2_64 = +1.083042469624914545964425189778400898568e-2L, 109 ovflthreshold = +1.135652340629414394949193107797076342845e+4L, 110 invln2_32 = +4.616624130844682903551758979206054839765e+1L, 111 ln2_32hi = +2.166084939249829091928849858592451515688e-2L, 112 ln2_32lo = +5.209643502595475652782654157501186731779e-27L, 113 huge = +1.0e4000L, 114 tiny = +1.0e-4000L, 115 P1 = +1.66666666666666666666666666666638500528074603030e-0001L, 116 P2 = -2.77777777777777777777777759668391122822266551158e-0003L, 117 P3 = +6.61375661375661375657437408890138814721051293054e-0005L, 118 P4 = -1.65343915343915303310185228411892601606669528828e-0006L, 119 P5 = +4.17535139755122945763580609663414647067443411178e-0008L, 120 P6 = -1.05683795988668526689182102605260986731620026832e-0009L, 121 P7 = +2.67544168821852702827123344217198187229611470514e-0011L, 122 /* rational approximation coeffs for [-(ln2)/64,(ln2)/64] */ 123 T1 = +1.666666666666666666666666666660876387437e-1L, 124 T2 = -2.777777777777777777777707812093173478756e-3L, 125 T3 = +6.613756613756613482074280932874221202424e-5L, 126 T4 = -1.653439153392139954169609822742235851120e-6L, 127 T5 = +4.175314851769539751387852116610973796053e-8L; 128 129 long double 130 expm1l(long double x) { 131 int hx, ix, j, k, m; 132 long double t, r, s, w; 133 134 hx = ((int *) &x)[HIXWORD]; 135 ix = hx & ~0x80000000; 136 if (ix >= 0x7fff0000) { 137 if (x != x) 138 return (x + x); /* NaN */ 139 if (x < zero) 140 return (-one); /* -inf */ 141 return (x); /* +inf */ 142 } 143 if (ix < 0x3fff4000) { /* |x| < 1.25 */ 144 if (ix < 0x3ffb0000) { /* |x| < 0.0625 */ 145 if (ix < 0x3f8d0000) { 146 if ((int) x == 0) 147 return (x); /* |x|<2^-114 */ 148 } 149 t = x * x; 150 r = (x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * 151 (P5 + t * (P6 + t * P7))))))); 152 return (x + (x * r) / (two - r)); 153 } 154 /* compute i = [64*x] */ 155 m = 0x4009 - (ix >> 16); 156 j = ((ix & 0x0000ffff) | 0x10000) >> m; /* j=4,...,67 */ 157 if (hx < 0) 158 j += 82; /* negative */ 159 s = x - _TBL_expm1lx[j]; 160 t = s * s; 161 r = s - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5)))); 162 r = (s + s) / (two - r); 163 w = _TBL_expm1l[j]; 164 return (w + (w + one) * r); 165 } 166 if (hx > 0) { 167 if (x > ovflthreshold) 168 return (huge * huge); 169 k = (int) (invln2_32 * (x + ln2_64)); 170 } else { 171 if (x < -80.0) 172 return (tiny - x / x); 173 k = (int) (invln2_32 * (x - ln2_64)); 174 } 175 j = k & 0x1f; 176 m = k >> 5; 177 t = (long double) k; 178 x = (x - t * ln2_32hi) - t * ln2_32lo; 179 t = x * x; 180 r = (x - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5))))) - two; 181 x = _TBL_expl_hi[j] - ((_TBL_expl_hi[j] * (x + x)) / r - 182 _TBL_expl_lo[j]); 183 return (scalbnl(x, m) - one); 184 } 185