1 /* $NetBSD: n_exp__E.c,v 1.7 2008/04/29 15:10:02 uwe Exp $ */ 2 /* 3 * Copyright (c) 1985, 1993 4 * The Regents of the University of California. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 3. Neither the name of the University nor the names of its contributors 15 * may be used to endorse or promote products derived from this software 16 * without specific prior written permission. 17 * 18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 28 * SUCH DAMAGE. 29 */ 30 31 #ifndef lint 32 #if 0 33 static char sccsid[] = "@(#)exp__E.c 8.1 (Berkeley) 6/4/93"; 34 #endif 35 #endif /* not lint */ 36 37 /* exp__E(x,c) 38 * ASSUMPTION: c << x SO THAT fl(x+c)=x. 39 * (c is the correction term for x) 40 * exp__E RETURNS 41 * 42 * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736 43 * exp__E(x,c) = | 44 * \ 0 , |x| < 1E-19. 45 * 46 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 47 * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS 48 * CODED IN C BY K.C. NG, 1/31/85; 49 * REVISED BY K.C. NG on 3/16/85, 4/16/85. 50 * 51 * Required system supported function: 52 * copysign(x,y) 53 * 54 * Method: 55 * 1. Rational approximation. Let r=x+c. 56 * Based on 57 * 2 * sinh(r/2) 58 * exp(r) - 1 = ---------------------- , 59 * cosh(r/2) - sinh(r/2) 60 * exp__E(r) is computed using 61 * x*x (x/2)*W - ( Q - ( 2*P + x*P ) ) 62 * --- + (c + x*[---------------------------------- + c ]) 63 * 2 1 - W 64 * where P := p1*x^2 + p2*x^4, 65 * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6) 66 * W := x/2-(Q-x*P), 67 * 68 * (See the listing below for the values of p1,p2,q1,q2,q3. The poly- 69 * nomials P and Q may be regarded as the approximations to sinh 70 * and cosh : 71 * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . ) 72 * 73 * The coefficients were obtained by a special Remez algorithm. 74 * 75 * Approximation error: 76 * 77 * | exp(x) - 1 | 2**(-57), (IEEE double) 78 * | ------------ - (exp__E(x,0)+x)/x | <= 79 * | x | 2**(-69). (VAX D) 80 * 81 * Constants: 82 * The hexadecimal values are the intended ones for the following constants. 83 * The decimal values may be used, provided that the compiler will convert 84 * from decimal to binary accurately enough to produce the hexadecimal values 85 * shown. 86 */ 87 88 #define _LIBM_STATIC 89 #include "mathimpl.h" 90 91 vc(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1, -6, .F83ABE67E1066A) 92 vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173) 93 vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2, -3, .E8B95A44A2EC45) 94 vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4, -9, .A5790572E4F5E7) 95 vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395) 96 97 ic(p1, 1.3887401997267371720E-2, -7, 1.C70FF8B3CC2CF) 98 ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4) 99 ic(q1, 1.1110813732786649355E-1, -4, 1.C719538248597) 100 ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8) 101 102 #ifdef vccast 103 #define p1 vccast(p1) 104 #define p2 vccast(p2) 105 #define q1 vccast(q1) 106 #define q2 vccast(q2) 107 #define q3 vccast(q3) 108 #endif 109 110 double 111 __exp__E(double x, double c) 112 { 113 static const double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19; 114 double z,p,q,xp,xh,w; 115 if(copysign(x,one)>small) { 116 z = x*x ; 117 p = z*( p1 +z* p2 ); 118 #if defined(__vax__)||defined(tahoe) 119 q = z*( q1 +z*( q2 +z* q3 )); 120 #else /* defined(__vax__)||defined(tahoe) */ 121 q = z*( q1 +z* q2 ); 122 #endif /* defined(__vax__)||defined(tahoe) */ 123 xp= x*p ; 124 xh= x*half ; 125 w = xh-(q-xp) ; 126 p = p+p; 127 c += x*((xh*w-(q-(p+xp)))/(one-w)+c); 128 return(z*half+c); 129 } 130 /* end of |x| > small */ 131 132 else { 133 if(x!=zero) w=one+small; /* raise the inexact flag ??? -ragge */ 134 return(copysign(zero,x)); 135 } 136 } 137