projects / unix-history / blame
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|---|---|---|
| 734a37b2 ZAL |
1 | /* |
| 2 | * Copyright (c) 1985 Regents of the University of California. | |
| 3 | * | |
| 4 | * Use and reproduction of this software are granted in accordance with | |
| 5 | * the terms and conditions specified in the Berkeley Software License | |
| 6 | * Agreement (in particular, this entails acknowledgement of the programs' | |
| 7 | * source, and inclusion of this notice) with the additional understanding | |
| 8 | * that all recipients should regard themselves as participants in an | |
| 9 | * ongoing research project and hence should feel obligated to report | |
| 10 | * their experiences (good or bad) with these elementary function codes, | |
| 11 | * using "sendbug 4bsd-bugs@BERKELEY", to the authors. | |
| 12 | */ | |
| 13 | ||
| 14 | #ifndef lint | |
| 15 | static char sccsid[] = "@(#)exp__E.c 1.1 (ELEFUNT) %G%"; | |
| 16 | #endif not lint | |
| 17 | ||
| 18 | /* exp__E(x,c) | |
| 19 | * ASSUMPTION: c << x SO THAT fl(x+c)=x. | |
| 20 | * (c is the correction term for x) | |
| 21 | * exp__E RETURNS | |
| 22 | * | |
| 23 | * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736 | |
| 24 | * exp__E(x,c) = | | |
| 25 | * \ 0 , |x| < 1E-19. | |
| 26 | * | |
| 27 | * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) | |
| 28 | * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS | |
| 29 | * CODED IN C BY K.C. NG, 1/31/85; | |
| 30 | * REVISED BY K.C. NG on 3/16/85, 4/16/85. | |
| 31 | * | |
| 32 | * Required system supported function: | |
| 33 | * copysign(x,y) | |
| 34 | * | |
| 35 | * Method: | |
| 36 | * 1. Rational approximation. Let r=x+c. | |
| 37 | * Based on | |
| 38 | * 2 * sinh(r/2) | |
| 39 | * exp(r) - 1 = ---------------------- , | |
| 40 | * cosh(r/2) - sinh(r/2) | |
| 41 | * exp__E(r) is computed using | |
| 42 | * x*x (x/2)*W - ( Q - ( 2*P + x*P ) ) | |
| 43 | * --- + (c + x*[---------------------------------- + c ]) | |
| 44 | * 2 1 - W | |
| 45 | * where P := p1*x^2 + p2*x^4, | |
| 46 | * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6) | |
| 47 | * W := x/2-(Q-x*P), | |
| 48 | * | |
| 49 | * (See the listing below for the values of p1,p2,q1,q2,q3. The poly- | |
| 50 | * nomials P and Q may be regarded as the approximations to sinh | |
| 51 | * and cosh : | |
| 52 | * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . ) | |
| 53 | * | |
| 54 | * The coefficients were obtained by a special Remez algorithm. | |
| 55 | * | |
| 56 | * Approximation error: | |
| 57 | * | |
| 58 | * | exp(x) - 1 | 2**(-57), (IEEE double) | |
| 59 | * | ------------ - (exp__E(x,0)+x)/x | <= | |
| 60 | * | x | 2**(-69). (VAX D) | |
| 61 | * | |
| 62 | * Constants: | |
| 63 | * The hexadecimal values are the intended ones for the following constants. | |
| 64 | * The decimal values may be used, provided that the compiler will convert | |
| 65 | * from decimal to binary accurately enough to produce the hexadecimal values | |
| 66 | * shown. | |
| 67 | */ | |
| 68 | ||
| 69 | #ifdef VAX /* VAX D format */ | |
| 70 | /* static double */ | |
| 71 | /* p1 = 1.5150724356786683059E-2 , Hex 2^ -6 * .F83ABE67E1066A */ | |
| 72 | /* p2 = 6.3112487873718332688E-5 , Hex 2^-13 * .845B4248CD0173 */ | |
| 73 | /* q1 = 1.1363478204690669916E-1 , Hex 2^ -3 * .E8B95A44A2EC45 */ | |
| 74 | /* q2 = 1.2624568129896839182E-3 , Hex 2^ -9 * .A5790572E4F5E7 */ | |
| 75 | /* q3 = 1.5021856115869022674E-6 ; Hex 2^-19 * .C99EB4604AC395 */ | |
| 76 | static long p1x[] = { 0x3abe3d78, 0x066a67e1}; | |
| 77 | static long p2x[] = { 0x5b423984, 0x017348cd}; | |
| 78 | static long q1x[] = { 0xb95a3ee8, 0xec4544a2}; | |
| 79 | static long q2x[] = { 0x79053ba5, 0xf5e772e4}; | |
| 80 | static long q3x[] = { 0x9eb436c9, 0xc395604a}; | |
| 81 | #define p1 (*(double*)p1x) | |
| 82 | #define p2 (*(double*)p2x) | |
| 83 | #define q1 (*(double*)q1x) | |
| 84 | #define q2 (*(double*)q2x) | |
| 85 | #define q3 (*(double*)q3x) | |
| 86 | #else /* IEEE double */ | |
| 87 | static double | |
| 88 | p1 = 1.3887401997267371720E-2 , /*Hex 2^ -7 * 1.C70FF8B3CC2CF */ | |
| 89 | p2 = 3.3044019718331897649E-5 , /*Hex 2^-15 * 1.15317DF4526C4 */ | |
| 90 | q1 = 1.1110813732786649355E-1 , /*Hex 2^ -4 * 1.C719538248597 */ | |
| 91 | q2 = 9.9176615021572857300E-4 ; /*Hex 2^-10 * 1.03FC4CB8C98E8 */ | |
| 92 | #endif | |
| 93 | ||
| 94 | double exp__E(x,c) | |
| 95 | double x,c; | |
| 96 | { | |
| 97 | double static zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19; | |
| 98 | double copysign(),z,p,q,xp,xh,w; | |
| 99 | if(copysign(x,one)>small) { | |
| 100 | z = x*x ; | |
| 101 | p = z*( p1 +z* p2 ); | |
| 102 | #ifdef VAX | |
| 103 | q = z*( q1 +z*( q2 +z* q3 )); | |
| 104 | #else /* IEEE double */ | |
| 105 | q = z*( q1 +z* q2 ); | |
| 106 | #endif | |
| 107 | xp= x*p ; | |
| 108 | xh= x*half ; | |
| 109 | w = xh-(q-xp) ; | |
| 110 | p = p+p; | |
| 111 | c += x*((xh*w-(q-(p+xp)))/(one-w)+c); | |
| 112 | return(z*half+c); | |
| 113 | } | |
| 114 | /* end of |x| > small */ | |
| 115 | ||
| 116 | else { | |
| 117 | if(x!=zero) one+small; /* raise the inexact flag */ | |
| 118 | return(copysign(zero,x)); | |
| 119 | } | |
| 120 | } |