1 /* $NetBSD: n_exp__E.c,v 1.1 1995/10/10 23:36:45 ragge 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. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 */ 34 35 #ifndef lint 36 static char sccsid[] = "@(#)exp__E.c 8.1 (Berkeley) 6/4/93"; 37 #endif /* not lint */ 38 39 /* exp__E(x,c) 40 * ASSUMPTION: c << x SO THAT fl(x+c)=x. 41 * (c is the correction term for x) 42 * exp__E RETURNS 43 * 44 * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736 45 * exp__E(x,c) = | 46 * \ 0 , |x| < 1E-19. 47 * 48 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 49 * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS 50 * CODED IN C BY K.C. NG, 1/31/85; 51 * REVISED BY K.C. NG on 3/16/85, 4/16/85. 52 * 53 * Required system supported function: 54 * copysign(x,y) 55 * 56 * Method: 57 * 1. Rational approximation. Let r=x+c. 58 * Based on 59 * 2 * sinh(r/2) 60 * exp(r) - 1 = ---------------------- , 61 * cosh(r/2) - sinh(r/2) 62 * exp__E(r) is computed using 63 * x*x (x/2)*W - ( Q - ( 2*P + x*P ) ) 64 * --- + (c + x*[---------------------------------- + c ]) 65 * 2 1 - W 66 * where P := p1*x^2 + p2*x^4, 67 * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6) 68 * W := x/2-(Q-x*P), 69 * 70 * (See the listing below for the values of p1,p2,q1,q2,q3. The poly- 71 * nomials P and Q may be regarded as the approximations to sinh 72 * and cosh : 73 * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . ) 74 * 75 * The coefficients were obtained by a special Remez algorithm. 76 * 77 * Approximation error: 78 * 79 * | exp(x) - 1 | 2**(-57), (IEEE double) 80 * | ------------ - (exp__E(x,0)+x)/x | <= 81 * | x | 2**(-69). (VAX D) 82 * 83 * Constants: 84 * The hexadecimal values are the intended ones for the following constants. 85 * The decimal values may be used, provided that the compiler will convert 86 * from decimal to binary accurately enough to produce the hexadecimal values 87 * shown. 88 */ 89 90 #include "mathimpl.h" 91 92 vc(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1, -6, .F83ABE67E1066A) 93 vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173) 94 vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2, -3, .E8B95A44A2EC45) 95 vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4, -9, .A5790572E4F5E7) 96 vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395) 97 98 ic(p1, 1.3887401997267371720E-2, -7, 1.C70FF8B3CC2CF) 99 ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4) 100 ic(q1, 1.1110813732786649355E-1, -4, 1.C719538248597) 101 ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8) 102 103 #ifdef vccast 104 #define p1 vccast(p1) 105 #define p2 vccast(p2) 106 #define q1 vccast(q1) 107 #define q2 vccast(q2) 108 #define q3 vccast(q3) 109 #endif 110 111 double __exp__E(x,c) 112 double x,c; 113 { 114 const static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19; 115 double z,p,q,xp,xh,w; 116 if(copysign(x,one)>small) { 117 z = x*x ; 118 p = z*( p1 +z* p2 ); 119 #if defined(vax)||defined(tahoe) 120 q = z*( q1 +z*( q2 +z* q3 )); 121 #else /* defined(vax)||defined(tahoe) */ 122 q = z*( q1 +z* q2 ); 123 #endif /* defined(vax)||defined(tahoe) */ 124 xp= x*p ; 125 xh= x*half ; 126 w = xh-(q-xp) ; 127 p = p+p; 128 c += x*((xh*w-(q-(p+xp)))/(one-w)+c); 129 return(z*half+c); 130 } 131 /* end of |x| > small */ 132 133 else { 134 if(x!=zero) one+small; /* raise the inexact flag */ 135 return(copysign(zero,x)); 136 } 137 } 138