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[] = 16 "@(#)exp__E.c 1.2 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) 09/11/85"; 17 #endif not lint 18 19 /* exp__E(x,c) 20 * ASSUMPTION: c << x SO THAT fl(x+c)=x. 21 * (c is the correction term for x) 22 * exp__E RETURNS 23 * 24 * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736 25 * exp__E(x,c) = | 26 * \ 0 , |x| < 1E-19. 27 * 28 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 29 * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS 30 * CODED IN C BY K.C. NG, 1/31/85; 31 * REVISED BY K.C. NG on 3/16/85, 4/16/85. 32 * 33 * Required system supported function: 34 * copysign(x,y) 35 * 36 * Method: 37 * 1. Rational approximation. Let r=x+c. 38 * Based on 39 * 2 * sinh(r/2) 40 * exp(r) - 1 = ---------------------- , 41 * cosh(r/2) - sinh(r/2) 42 * exp__E(r) is computed using 43 * x*x (x/2)*W - ( Q - ( 2*P + x*P ) ) 44 * --- + (c + x*[---------------------------------- + c ]) 45 * 2 1 - W 46 * where P := p1*x^2 + p2*x^4, 47 * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6) 48 * W := x/2-(Q-x*P), 49 * 50 * (See the listing below for the values of p1,p2,q1,q2,q3. The poly- 51 * nomials P and Q may be regarded as the approximations to sinh 52 * and cosh : 53 * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . ) 54 * 55 * The coefficients were obtained by a special Remez algorithm. 56 * 57 * Approximation error: 58 * 59 * | exp(x) - 1 | 2**(-57), (IEEE double) 60 * | ------------ - (exp__E(x,0)+x)/x | <= 61 * | x | 2**(-69). (VAX D) 62 * 63 * Constants: 64 * The hexadecimal values are the intended ones for the following constants. 65 * The decimal values may be used, provided that the compiler will convert 66 * from decimal to binary accurately enough to produce the hexadecimal values 67 * shown. 68 */ 69 70 #ifdef VAX /* VAX D format */ 71 /* static double */ 72 /* p1 = 1.5150724356786683059E-2 , Hex 2^ -6 * .F83ABE67E1066A */ 73 /* p2 = 6.3112487873718332688E-5 , Hex 2^-13 * .845B4248CD0173 */ 74 /* q1 = 1.1363478204690669916E-1 , Hex 2^ -3 * .E8B95A44A2EC45 */ 75 /* q2 = 1.2624568129896839182E-3 , Hex 2^ -9 * .A5790572E4F5E7 */ 76 /* q3 = 1.5021856115869022674E-6 ; Hex 2^-19 * .C99EB4604AC395 */ 77 static long p1x[] = { 0x3abe3d78, 0x066a67e1}; 78 static long p2x[] = { 0x5b423984, 0x017348cd}; 79 static long q1x[] = { 0xb95a3ee8, 0xec4544a2}; 80 static long q2x[] = { 0x79053ba5, 0xf5e772e4}; 81 static long q3x[] = { 0x9eb436c9, 0xc395604a}; 82 #define p1 (*(double*)p1x) 83 #define p2 (*(double*)p2x) 84 #define q1 (*(double*)q1x) 85 #define q2 (*(double*)q2x) 86 #define q3 (*(double*)q3x) 87 #else /* IEEE double */ 88 static double 89 p1 = 1.3887401997267371720E-2 , /*Hex 2^ -7 * 1.C70FF8B3CC2CF */ 90 p2 = 3.3044019718331897649E-5 , /*Hex 2^-15 * 1.15317DF4526C4 */ 91 q1 = 1.1110813732786649355E-1 , /*Hex 2^ -4 * 1.C719538248597 */ 92 q2 = 9.9176615021572857300E-4 ; /*Hex 2^-10 * 1.03FC4CB8C98E8 */ 93 #endif 94 95 double exp__E(x,c) 96 double x,c; 97 { 98 double static zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19; 99 double copysign(),z,p,q,xp,xh,w; 100 if(copysign(x,one)>small) { 101 z = x*x ; 102 p = z*( p1 +z* p2 ); 103 #ifdef VAX 104 q = z*( q1 +z*( q2 +z* q3 )); 105 #else /* IEEE double */ 106 q = z*( q1 +z* q2 ); 107 #endif 108 xp= x*p ; 109 xh= x*half ; 110 w = xh-(q-xp) ; 111 p = p+p; 112 c += x*((xh*w-(q-(p+xp)))/(one-w)+c); 113 return(z*half+c); 114 } 115 /* end of |x| > small */ 116 117 else { 118 if(x!=zero) one+small; /* raise the inexact flag */ 119 return(copysign(zero,x)); 120 } 121 } 122