1 /* $OpenBSD: e_expl.c,v 1.3 2013/11/12 20:35:19 martynas Exp $ */ 2 3 /* 4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 5 * 6 * Permission to use, copy, modify, and distribute this software for any 7 * purpose with or without fee is hereby granted, provided that the above 8 * copyright notice and this permission notice appear in all copies. 9 * 10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 17 */ 18 19 /* expl.c 20 * 21 * Exponential function, long double precision 22 * 23 * 24 * 25 * SYNOPSIS: 26 * 27 * long double x, y, expl(); 28 * 29 * y = expl( x ); 30 * 31 * 32 * 33 * DESCRIPTION: 34 * 35 * Returns e (2.71828...) raised to the x power. 36 * 37 * Range reduction is accomplished by separating the argument 38 * into an integer k and fraction f such that 39 * 40 * x k f 41 * e = 2 e. 42 * 43 * A Pade' form of degree 2/3 is used to approximate exp(f) - 1 44 * in the basic range [-0.5 ln 2, 0.5 ln 2]. 45 * 46 * 47 * ACCURACY: 48 * 49 * Relative error: 50 * arithmetic domain # trials peak rms 51 * IEEE +-10000 50000 1.12e-19 2.81e-20 52 * 53 * 54 * Error amplification in the exponential function can be 55 * a serious matter. The error propagation involves 56 * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ), 57 * which shows that a 1 lsb error in representing X produces 58 * a relative error of X times 1 lsb in the function. 59 * While the routine gives an accurate result for arguments 60 * that are exactly represented by a long double precision 61 * computer number, the result contains amplified roundoff 62 * error for large arguments not exactly represented. 63 * 64 * 65 * ERROR MESSAGES: 66 * 67 * message condition value returned 68 * exp underflow x < MINLOG 0.0 69 * exp overflow x > MAXLOG MAXNUM 70 * 71 */ 72 73 /* Exponential function */ 74 75 #include <math.h> 76 77 #include "math_private.h" 78 79 static long double P[3] = { 80 1.2617719307481059087798E-4L, 81 3.0299440770744196129956E-2L, 82 9.9999999999999999991025E-1L, 83 }; 84 static long double Q[4] = { 85 3.0019850513866445504159E-6L, 86 2.5244834034968410419224E-3L, 87 2.2726554820815502876593E-1L, 88 2.0000000000000000000897E0L, 89 }; 90 static const long double C1 = 6.9314575195312500000000E-1L; 91 static const long double C2 = 1.4286068203094172321215E-6L; 92 static const long double MAXLOGL = 1.1356523406294143949492E4L; 93 static const long double MINLOGL = -1.13994985314888605586758E4L; 94 static const long double LOG2EL = 1.4426950408889634073599E0L; 95 96 long double 97 expl(long double x) 98 { 99 long double px, xx; 100 int n; 101 102 if( isnan(x) ) 103 return(x); 104 if( x > MAXLOGL) 105 return( INFINITY ); 106 107 if( x < MINLOGL ) 108 return(0.0L); 109 110 /* Express e**x = e**g 2**n 111 * = e**g e**( n loge(2) ) 112 * = e**( g + n loge(2) ) 113 */ 114 px = floorl( LOG2EL * x + 0.5L ); /* floor() truncates toward -infinity. */ 115 n = px; 116 x -= px * C1; 117 x -= px * C2; 118 119 120 /* rational approximation for exponential 121 * of the fractional part: 122 * e**x = 1 + 2x P(x**2)/( Q(x**2) - P(x**2) ) 123 */ 124 xx = x * x; 125 px = x * __polevll( xx, P, 2 ); 126 x = px/( __polevll( xx, Q, 3 ) - px ); 127 x = 1.0L + ldexpl( x, 1 ); 128 129 x = ldexpl( x, n ); 130 return(x); 131 } 132