1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_expm1l.c */ 2 /* 3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 4 * 5 * Permission to use, copy, modify, and distribute this software for any 6 * purpose with or without fee is hereby granted, provided that the above 7 * copyright notice and this permission notice appear in all copies. 8 * 9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 16 */ 17 /* 18 * Exponential function, minus 1 19 * Long double precision 20 * 21 * 22 * SYNOPSIS: 23 * 24 * long double x, y, expm1l(); 25 * 26 * y = expm1l( x ); 27 * 28 * 29 * DESCRIPTION: 30 * 31 * Returns e (2.71828...) raised to the x power, minus 1. 32 * 33 * Range reduction is accomplished by separating the argument 34 * into an integer k and fraction f such that 35 * 36 * x k f 37 * e = 2 e. 38 * 39 * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1 40 * in the basic range [-0.5 ln 2, 0.5 ln 2]. 41 * 42 * 43 * ACCURACY: 44 * 45 * Relative error: 46 * arithmetic domain # trials peak rms 47 * IEEE -45,+maxarg 200,000 1.2e-19 2.5e-20 48 */ 49 50 #include "libm.h" 51 52 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 53 long double expm1l(long double x) 54 { 55 return expm1(x); 56 } 57 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 58 59 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x) 60 -.5 ln 2 < x < .5 ln 2 61 Theoretical peak relative error = 3.4e-22 */ 62 static const long double 63 P0 = -1.586135578666346600772998894928250240826E4L, 64 P1 = 2.642771505685952966904660652518429479531E3L, 65 P2 = -3.423199068835684263987132888286791620673E2L, 66 P3 = 1.800826371455042224581246202420972737840E1L, 67 P4 = -5.238523121205561042771939008061958820811E-1L, 68 Q0 = -9.516813471998079611319047060563358064497E4L, 69 Q1 = 3.964866271411091674556850458227710004570E4L, 70 Q2 = -7.207678383830091850230366618190187434796E3L, 71 Q3 = 7.206038318724600171970199625081491823079E2L, 72 Q4 = -4.002027679107076077238836622982900945173E1L, 73 /* Q5 = 1.000000000000000000000000000000000000000E0 */ 74 /* C1 + C2 = ln 2 */ 75 C1 = 6.93145751953125E-1L, 76 C2 = 1.428606820309417232121458176568075500134E-6L, 77 /* ln 2^-65 */ 78 minarg = -4.5054566736396445112120088E1L, 79 /* ln 2^16384 */ 80 maxarg = 1.1356523406294143949492E4L; 81 82 long double expm1l(long double x) 83 { 84 long double px, qx, xx; 85 int k; 86 87 if (isnan(x)) 88 return x; 89 if (x > maxarg) 90 return x*0x1p16383L; /* overflow, unless x==inf */ 91 if (x == 0.0) 92 return x; 93 if (x < minarg) 94 return -1.0; 95 96 xx = C1 + C2; 97 /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */ 98 px = floorl(0.5 + x / xx); 99 k = px; 100 /* remainder times ln 2 */ 101 x -= px * C1; 102 x -= px * C2; 103 104 /* Approximate exp(remainder ln 2).*/ 105 px = (((( P4 * x + P3) * x + P2) * x + P1) * x + P0) * x; 106 qx = (((( x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0; 107 xx = x * x; 108 qx = x + (0.5 * xx + xx * px / qx); 109 110 /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2). 111 We have qx = exp(remainder ln 2) - 1, so 112 exp(x) - 1 = 2^k (qx + 1) - 1 = 2^k qx + 2^k - 1. */ 113 px = scalbnl(1.0, k); 114 x = px * qx + (px - 1.0); 115 return x; 116 } 117 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 118 // TODO: broken implementation to make things compile 119 long double expm1l(long double x) 120 { 121 return expm1(x); 122 } 123 #endif 124