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 this notice is preserved and that due credit is given 7 * to the University of California at Berkeley. The name of the University 8 * may not be used to endorse or promote products derived from this 9 * software without specific prior written permission. This software 10 * is provided ``as is'' without express or implied warranty. 11 * 12 * All recipients should regard themselves as participants in an ongoing 13 * research project and hence should feel obligated to report their 14 * experiences (good or bad) with these elementary function codes, using 15 * the sendbug(8) program, to the authors. 16 */ 17 18 #ifndef lint 19 static char sccsid[] = "@(#)exp__E.c 5.2 (Berkeley) 04/29/88"; 20 #endif /* not lint */ 21 22 /* exp__E(x,c) 23 * ASSUMPTION: c << x SO THAT fl(x+c)=x. 24 * (c is the correction term for x) 25 * exp__E RETURNS 26 * 27 * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736 28 * exp__E(x,c) = | 29 * \ 0 , |x| < 1E-19. 30 * 31 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 32 * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS 33 * CODED IN C BY K.C. NG, 1/31/85; 34 * REVISED BY K.C. NG on 3/16/85, 4/16/85. 35 * 36 * Required system supported function: 37 * copysign(x,y) 38 * 39 * Method: 40 * 1. Rational approximation. Let r=x+c. 41 * Based on 42 * 2 * sinh(r/2) 43 * exp(r) - 1 = ---------------------- , 44 * cosh(r/2) - sinh(r/2) 45 * exp__E(r) is computed using 46 * x*x (x/2)*W - ( Q - ( 2*P + x*P ) ) 47 * --- + (c + x*[---------------------------------- + c ]) 48 * 2 1 - W 49 * where P := p1*x^2 + p2*x^4, 50 * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6) 51 * W := x/2-(Q-x*P), 52 * 53 * (See the listing below for the values of p1,p2,q1,q2,q3. The poly- 54 * nomials P and Q may be regarded as the approximations to sinh 55 * and cosh : 56 * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . ) 57 * 58 * The coefficients were obtained by a special Remez algorithm. 59 * 60 * Approximation error: 61 * 62 * | exp(x) - 1 | 2**(-57), (IEEE double) 63 * | ------------ - (exp__E(x,0)+x)/x | <= 64 * | x | 2**(-69). (VAX D) 65 * 66 * Constants: 67 * The hexadecimal values are the intended ones for the following constants. 68 * The decimal values may be used, provided that the compiler will convert 69 * from decimal to binary accurately enough to produce the hexadecimal values 70 * shown. 71 */ 72 73 #if defined(vax)||defined(tahoe) /* VAX D format */ 74 #ifdef vax 75 #define _0x(A,B) 0x/**/A/**/B 76 #else /* vax */ 77 #define _0x(A,B) 0x/**/B/**/A 78 #endif /* vax */ 79 /* static double */ 80 /* p1 = 1.5150724356786683059E-2 , Hex 2^ -6 * .F83ABE67E1066A */ 81 /* p2 = 6.3112487873718332688E-5 , Hex 2^-13 * .845B4248CD0173 */ 82 /* q1 = 1.1363478204690669916E-1 , Hex 2^ -3 * .E8B95A44A2EC45 */ 83 /* q2 = 1.2624568129896839182E-3 , Hex 2^ -9 * .A5790572E4F5E7 */ 84 /* q3 = 1.5021856115869022674E-6 ; Hex 2^-19 * .C99EB4604AC395 */ 85 static long p1x[] = { _0x(3abe,3d78), _0x(066a,67e1)}; 86 static long p2x[] = { _0x(5b42,3984), _0x(0173,48cd)}; 87 static long q1x[] = { _0x(b95a,3ee8), _0x(ec45,44a2)}; 88 static long q2x[] = { _0x(7905,3ba5), _0x(f5e7,72e4)}; 89 static long q3x[] = { _0x(9eb4,36c9), _0x(c395,604a)}; 90 #define p1 (*(double*)p1x) 91 #define p2 (*(double*)p2x) 92 #define q1 (*(double*)q1x) 93 #define q2 (*(double*)q2x) 94 #define q3 (*(double*)q3x) 95 #else /* defined(vax)||defined(tahoe) */ 96 static double 97 p1 = 1.3887401997267371720E-2 , /*Hex 2^ -7 * 1.C70FF8B3CC2CF */ 98 p2 = 3.3044019718331897649E-5 , /*Hex 2^-15 * 1.15317DF4526C4 */ 99 q1 = 1.1110813732786649355E-1 , /*Hex 2^ -4 * 1.C719538248597 */ 100 q2 = 9.9176615021572857300E-4 ; /*Hex 2^-10 * 1.03FC4CB8C98E8 */ 101 #endif /* defined(vax)||defined(tahoe) */ 102 103 double exp__E(x,c) 104 double x,c; 105 { 106 static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19; 107 double copysign(),z,p,q,xp,xh,w; 108 if(copysign(x,one)>small) { 109 z = x*x ; 110 p = z*( p1 +z* p2 ); 111 #if defined(vax)||defined(tahoe) 112 q = z*( q1 +z*( q2 +z* q3 )); 113 #else /* defined(vax)||defined(tahoe) */ 114 q = z*( q1 +z* q2 ); 115 #endif /* defined(vax)||defined(tahoe) */ 116 xp= x*p ; 117 xh= x*half ; 118 w = xh-(q-xp) ; 119 p = p+p; 120 c += x*((xh*w-(q-(p+xp)))/(one-w)+c); 121 return(z*half+c); 122 } 123 /* end of |x| > small */ 124 125 else { 126 if(x!=zero) one+small; /* raise the inexact flag */ 127 return(copysign(zero,x)); 128 } 129 } 130