projects / unix-history / blame
| Commit | Line | Data |
|---|---|---|
| 9b525f39 | 1 | /* |
| 041b27b4 KB |
2 | * Copyright (c) 1985, 1993 |
| 3 | * The Regents of the University of California. All rights reserved. | |
| 9b525f39 | 4 | * |
| 7e3eac84 | 5 | * %sccs.include.redist.c% |
| 734a37b2 ZAL |
6 | */ |
| 7 | ||
| 8 | #ifndef lint | |
| 041b27b4 | 9 | static char sccsid[] = "@(#)exp__E.c 8.1 (Berkeley) %G%"; |
| 9b525f39 | 10 | #endif /* not lint */ |
| 734a37b2 ZAL |
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 | ||
| 9eda3584 KB |
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 | |
| 734a37b2 | 83 | |
| 5acba3ee | 84 | double __exp__E(x,c) |
| 734a37b2 ZAL |
85 | double x,c; |
| 86 | { | |
| 9eda3584 KB |
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; | |
| 734a37b2 ZAL |
89 | if(copysign(x,one)>small) { |
| 90 | z = x*x ; | |
| 91 | p = z*( p1 +z* p2 ); | |
| 859dc438 | 92 | #if defined(vax)||defined(tahoe) |
| 734a37b2 | 93 | q = z*( q1 +z*( q2 +z* q3 )); |
| 859dc438 | 94 | #else /* defined(vax)||defined(tahoe) */ |
| 734a37b2 | 95 | q = z*( q1 +z* q2 ); |
| 859dc438 | 96 | #endif /* defined(vax)||defined(tahoe) */ |
| 734a37b2 ZAL |
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 | } |