1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that the above copyright notice and this paragraph are 7 * duplicated in all such forms and that any documentation, 8 * advertising materials, and other materials related to such 9 * distribution and use acknowledge that the software was developed 10 * by the University of California, Berkeley. The name of the 11 * University may not be used to endorse or promote products derived 12 * from this software without specific prior written permission. 13 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 16 * 17 * All recipients should regard themselves as participants in an ongoing 18 * research project and hence should feel obligated to report their 19 * experiences (good or bad) with these elementary function codes, using 20 * the sendbug(8) program, to the authors. 21 */ 22 23 #ifndef lint 24 static char sccsid[] = "@(#)exp__E.c 5.4 (Berkeley) 09/22/88"; 25 #endif /* not lint */ 26 27 /* exp__E(x,c) 28 * ASSUMPTION: c << x SO THAT fl(x+c)=x. 29 * (c is the correction term for x) 30 * exp__E RETURNS 31 * 32 * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736 33 * exp__E(x,c) = | 34 * \ 0 , |x| < 1E-19. 35 * 36 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 37 * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS 38 * CODED IN C BY K.C. NG, 1/31/85; 39 * REVISED BY K.C. NG on 3/16/85, 4/16/85. 40 * 41 * Required system supported function: 42 * copysign(x,y) 43 * 44 * Method: 45 * 1. Rational approximation. Let r=x+c. 46 * Based on 47 * 2 * sinh(r/2) 48 * exp(r) - 1 = ---------------------- , 49 * cosh(r/2) - sinh(r/2) 50 * exp__E(r) is computed using 51 * x*x (x/2)*W - ( Q - ( 2*P + x*P ) ) 52 * --- + (c + x*[---------------------------------- + c ]) 53 * 2 1 - W 54 * where P := p1*x^2 + p2*x^4, 55 * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6) 56 * W := x/2-(Q-x*P), 57 * 58 * (See the listing below for the values of p1,p2,q1,q2,q3. The poly- 59 * nomials P and Q may be regarded as the approximations to sinh 60 * and cosh : 61 * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . ) 62 * 63 * The coefficients were obtained by a special Remez algorithm. 64 * 65 * Approximation error: 66 * 67 * | exp(x) - 1 | 2**(-57), (IEEE double) 68 * | ------------ - (exp__E(x,0)+x)/x | <= 69 * | x | 2**(-69). (VAX D) 70 * 71 * Constants: 72 * The hexadecimal values are the intended ones for the following constants. 73 * The decimal values may be used, provided that the compiler will convert 74 * from decimal to binary accurately enough to produce the hexadecimal values 75 * shown. 76 */ 77 78 #include "mathimpl.h" 79 80 vc(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1, -6, .F83ABE67E1066A) 81 vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173) 82 vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2, -3, .E8B95A44A2EC45) 83 vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4, -9, .A5790572E4F5E7) 84 vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395) 85 86 ic(p1, 1.3887401997267371720E-2, -7, 1.C70FF8B3CC2CF) 87 ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4) 88 ic(q1, 1.1110813732786649355E-1, -4, 1.C719538248597) 89 ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8) 90 91 #ifdef vccast 92 #define p1 vccast(p1) 93 #define p2 vccast(p2) 94 #define q1 vccast(q1) 95 #define q2 vccast(q2) 96 #define q3 vccast(q3) 97 #endif 98 99 double exp__E(x,c) 100 double x,c; 101 { 102 const static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19; 103 double z,p,q,xp,xh,w; 104 if(copysign(x,one)>small) { 105 z = x*x ; 106 p = z*( p1 +z* p2 ); 107 #if defined(vax)||defined(tahoe) 108 q = z*( q1 +z*( q2 +z* q3 )); 109 #else /* defined(vax)||defined(tahoe) */ 110 q = z*( q1 +z* q2 ); 111 #endif /* defined(vax)||defined(tahoe) */ 112 xp= x*p ; 113 xh= x*half ; 114 w = xh-(q-xp) ; 115 p = p+p; 116 c += x*((xh*w-(q-(p+xp)))/(one-w)+c); 117 return(z*half+c); 118 } 119 /* end of |x| > small */ 120 121 else { 122 if(x!=zero) one+small; /* raise the inexact flag */ 123 return(copysign(zero,x)); 124 } 125 } 126