1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)exp__E.c 5.7 (Berkeley) 01/10/93"; 10 #endif /* not lint */ 11 12 /* exp__E(x,c) 13 * ASSUMPTION: c << x SO THAT fl(x+c)=x. 14 * (c is the correction term for x) 15 * exp__E RETURNS 16 * 17 * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736 18 * exp__E(x,c) = | 19 * \ 0 , |x| < 1E-19. 20 * 21 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 22 * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS 23 * CODED IN C BY K.C. NG, 1/31/85; 24 * REVISED BY K.C. NG on 3/16/85, 4/16/85. 25 * 26 * Required system supported function: 27 * copysign(x,y) 28 * 29 * Method: 30 * 1. Rational approximation. Let r=x+c. 31 * Based on 32 * 2 * sinh(r/2) 33 * exp(r) - 1 = ---------------------- , 34 * cosh(r/2) - sinh(r/2) 35 * exp__E(r) is computed using 36 * x*x (x/2)*W - ( Q - ( 2*P + x*P ) ) 37 * --- + (c + x*[---------------------------------- + c ]) 38 * 2 1 - W 39 * where P := p1*x^2 + p2*x^4, 40 * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6) 41 * W := x/2-(Q-x*P), 42 * 43 * (See the listing below for the values of p1,p2,q1,q2,q3. The poly- 44 * nomials P and Q may be regarded as the approximations to sinh 45 * and cosh : 46 * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . ) 47 * 48 * The coefficients were obtained by a special Remez algorithm. 49 * 50 * Approximation error: 51 * 52 * | exp(x) - 1 | 2**(-57), (IEEE double) 53 * | ------------ - (exp__E(x,0)+x)/x | <= 54 * | x | 2**(-69). (VAX D) 55 * 56 * Constants: 57 * The hexadecimal values are the intended ones for the following constants. 58 * The decimal values may be used, provided that the compiler will convert 59 * from decimal to binary accurately enough to produce the hexadecimal values 60 * shown. 61 */ 62 63 #include "mathimpl.h" 64 65 vc(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1, -6, .F83ABE67E1066A) 66 vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173) 67 vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2, -3, .E8B95A44A2EC45) 68 vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4, -9, .A5790572E4F5E7) 69 vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395) 70 71 ic(p1, 1.3887401997267371720E-2, -7, 1.C70FF8B3CC2CF) 72 ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4) 73 ic(q1, 1.1110813732786649355E-1, -4, 1.C719538248597) 74 ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8) 75 76 #ifdef vccast 77 #define p1 vccast(p1) 78 #define p2 vccast(p2) 79 #define q1 vccast(q1) 80 #define q2 vccast(q2) 81 #define q3 vccast(q3) 82 #endif 83 84 double __exp__E(x,c) 85 double x,c; 86 { 87 const static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19; 88 double z,p,q,xp,xh,w; 89 if(copysign(x,one)>small) { 90 z = x*x ; 91 p = z*( p1 +z* p2 ); 92 #if defined(vax)||defined(tahoe) 93 q = z*( q1 +z*( q2 +z* q3 )); 94 #else /* defined(vax)||defined(tahoe) */ 95 q = z*( q1 +z* q2 ); 96 #endif /* defined(vax)||defined(tahoe) */ 97 xp= x*p ; 98 xh= x*half ; 99 w = xh-(q-xp) ; 100 p = p+p; 101 c += x*((xh*w-(q-(p+xp)))/(one-w)+c); 102 return(z*half+c); 103 } 104 /* end of |x| > small */ 105 106 else { 107 if(x!=zero) one+small; /* raise the inexact flag */ 108 return(copysign(zero,x)); 109 } 110 } 111