1*24594Szliu /* 2*24594Szliu * Copyright (c) 1985 Regents of the University of California. 3*24594Szliu * 4*24594Szliu * Use and reproduction of this software are granted in accordance with 5*24594Szliu * the terms and conditions specified in the Berkeley Software License 6*24594Szliu * Agreement (in particular, this entails acknowledgement of the programs' 7*24594Szliu * source, and inclusion of this notice) with the additional understanding 8*24594Szliu * that all recipients should regard themselves as participants in an 9*24594Szliu * ongoing research project and hence should feel obligated to report 10*24594Szliu * their experiences (good or bad) with these elementary function codes, 11*24594Szliu * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12*24594Szliu */ 13*24594Szliu 14*24594Szliu #ifndef lint 15*24594Szliu static char sccsid[] = "@(#)exp__E.c 1.1 (ELEFUNT) 09/06/85"; 16*24594Szliu #endif not lint 17*24594Szliu 18*24594Szliu /* exp__E(x,c) 19*24594Szliu * ASSUMPTION: c << x SO THAT fl(x+c)=x. 20*24594Szliu * (c is the correction term for x) 21*24594Szliu * exp__E RETURNS 22*24594Szliu * 23*24594Szliu * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736 24*24594Szliu * exp__E(x,c) = | 25*24594Szliu * \ 0 , |x| < 1E-19. 26*24594Szliu * 27*24594Szliu * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 28*24594Szliu * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS 29*24594Szliu * CODED IN C BY K.C. NG, 1/31/85; 30*24594Szliu * REVISED BY K.C. NG on 3/16/85, 4/16/85. 31*24594Szliu * 32*24594Szliu * Required system supported function: 33*24594Szliu * copysign(x,y) 34*24594Szliu * 35*24594Szliu * Method: 36*24594Szliu * 1. Rational approximation. Let r=x+c. 37*24594Szliu * Based on 38*24594Szliu * 2 * sinh(r/2) 39*24594Szliu * exp(r) - 1 = ---------------------- , 40*24594Szliu * cosh(r/2) - sinh(r/2) 41*24594Szliu * exp__E(r) is computed using 42*24594Szliu * x*x (x/2)*W - ( Q - ( 2*P + x*P ) ) 43*24594Szliu * --- + (c + x*[---------------------------------- + c ]) 44*24594Szliu * 2 1 - W 45*24594Szliu * where P := p1*x^2 + p2*x^4, 46*24594Szliu * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6) 47*24594Szliu * W := x/2-(Q-x*P), 48*24594Szliu * 49*24594Szliu * (See the listing below for the values of p1,p2,q1,q2,q3. The poly- 50*24594Szliu * nomials P and Q may be regarded as the approximations to sinh 51*24594Szliu * and cosh : 52*24594Szliu * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . ) 53*24594Szliu * 54*24594Szliu * The coefficients were obtained by a special Remez algorithm. 55*24594Szliu * 56*24594Szliu * Approximation error: 57*24594Szliu * 58*24594Szliu * | exp(x) - 1 | 2**(-57), (IEEE double) 59*24594Szliu * | ------------ - (exp__E(x,0)+x)/x | <= 60*24594Szliu * | x | 2**(-69). (VAX D) 61*24594Szliu * 62*24594Szliu * Constants: 63*24594Szliu * The hexadecimal values are the intended ones for the following constants. 64*24594Szliu * The decimal values may be used, provided that the compiler will convert 65*24594Szliu * from decimal to binary accurately enough to produce the hexadecimal values 66*24594Szliu * shown. 67*24594Szliu */ 68*24594Szliu 69*24594Szliu #ifdef VAX /* VAX D format */ 70*24594Szliu /* static double */ 71*24594Szliu /* p1 = 1.5150724356786683059E-2 , Hex 2^ -6 * .F83ABE67E1066A */ 72*24594Szliu /* p2 = 6.3112487873718332688E-5 , Hex 2^-13 * .845B4248CD0173 */ 73*24594Szliu /* q1 = 1.1363478204690669916E-1 , Hex 2^ -3 * .E8B95A44A2EC45 */ 74*24594Szliu /* q2 = 1.2624568129896839182E-3 , Hex 2^ -9 * .A5790572E4F5E7 */ 75*24594Szliu /* q3 = 1.5021856115869022674E-6 ; Hex 2^-19 * .C99EB4604AC395 */ 76*24594Szliu static long p1x[] = { 0x3abe3d78, 0x066a67e1}; 77*24594Szliu static long p2x[] = { 0x5b423984, 0x017348cd}; 78*24594Szliu static long q1x[] = { 0xb95a3ee8, 0xec4544a2}; 79*24594Szliu static long q2x[] = { 0x79053ba5, 0xf5e772e4}; 80*24594Szliu static long q3x[] = { 0x9eb436c9, 0xc395604a}; 81*24594Szliu #define p1 (*(double*)p1x) 82*24594Szliu #define p2 (*(double*)p2x) 83*24594Szliu #define q1 (*(double*)q1x) 84*24594Szliu #define q2 (*(double*)q2x) 85*24594Szliu #define q3 (*(double*)q3x) 86*24594Szliu #else /* IEEE double */ 87*24594Szliu static double 88*24594Szliu p1 = 1.3887401997267371720E-2 , /*Hex 2^ -7 * 1.C70FF8B3CC2CF */ 89*24594Szliu p2 = 3.3044019718331897649E-5 , /*Hex 2^-15 * 1.15317DF4526C4 */ 90*24594Szliu q1 = 1.1110813732786649355E-1 , /*Hex 2^ -4 * 1.C719538248597 */ 91*24594Szliu q2 = 9.9176615021572857300E-4 ; /*Hex 2^-10 * 1.03FC4CB8C98E8 */ 92*24594Szliu #endif 93*24594Szliu 94*24594Szliu double exp__E(x,c) 95*24594Szliu double x,c; 96*24594Szliu { 97*24594Szliu double static zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19; 98*24594Szliu double copysign(),z,p,q,xp,xh,w; 99*24594Szliu if(copysign(x,one)>small) { 100*24594Szliu z = x*x ; 101*24594Szliu p = z*( p1 +z* p2 ); 102*24594Szliu #ifdef VAX 103*24594Szliu q = z*( q1 +z*( q2 +z* q3 )); 104*24594Szliu #else /* IEEE double */ 105*24594Szliu q = z*( q1 +z* q2 ); 106*24594Szliu #endif 107*24594Szliu xp= x*p ; 108*24594Szliu xh= x*half ; 109*24594Szliu w = xh-(q-xp) ; 110*24594Szliu p = p+p; 111*24594Szliu c += x*((xh*w-(q-(p+xp)))/(one-w)+c); 112*24594Szliu return(z*half+c); 113*24594Szliu } 114*24594Szliu /* end of |x| > small */ 115*24594Szliu 116*24594Szliu else { 117*24594Szliu if(x!=zero) one+small; /* raise the inexact flag */ 118*24594Szliu return(copysign(zero,x)); 119*24594Szliu } 120*24594Szliu } 121