1b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 2b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* @(#)e_exp.c 1.6 04/04/22 */ 3b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* 4b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * ==================================================== 5b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 6b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 7b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Permission to use, copy, modify, and distribute this 8b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * software is freely granted, provided that this notice 9b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * is preserved. 10b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * ==================================================== 11b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project */ 12b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 13b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* __ieee754_exp(x) 14b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Returns the exponential of x. 15b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 16b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Method 17b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 1. Argument reduction: 18b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 19b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Given x, find r and integer k such that 20b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 21b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * x = k*ln2 + r, |r| <= 0.5*ln2. 22b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 23b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Here r will be represented as r = hi-lo for better 24b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * accuracy. 25b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 26b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 2. Approximation of ieee_exp(r) by a special rational function on 27b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * the interval [0,0.34658]: 28b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Write 29b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * R(r**2) = r*(ieee_exp(r)+1)/(ieee_exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 30b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * We use a special Remes algorithm on [0,0.34658] to generate 31b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * a polynomial of degree 5 to approximate R. The maximum error 32b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * of this polynomial approximation is bounded by 2**-59. In 33b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * other words, 34b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 35b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * (where z=r*r, and the values of P1 to P5 are listed below) 36b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * and 37b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * | 5 | -59 38b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * | 2.0+P1*z+...+P5*z - R(z) | <= 2 39b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * | | 40b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * The computation of ieee_exp(r) thus becomes 41b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 2*r 42b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * exp(r) = 1 + ------- 43b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * R - r 44b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * r*R1(r) 45b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * = 1 + r + ----------- (for better accuracy) 46b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 2 - R1(r) 47b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * where 48b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 2 4 10 49b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * R1(r) = r - (P1*r + P2*r + ... + P5*r ). 50b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 51b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 3. Scale back to obtain ieee_exp(x): 52b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * From step 1, we have 53b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * ieee_exp(x) = 2^k * ieee_exp(r) 54b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 55b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Special cases: 56b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * exp(INF) is INF, ieee_exp(NaN) is NaN; 57b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * exp(-INF) is 0, and 58b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * for finite argument, only ieee_exp(0)=1 is exact. 59b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 60b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Accuracy: 61b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * according to an error analysis, the error is always less than 62b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 1 ulp (unit in the last place). 63b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 64b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Misc. info. 65b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * For IEEE double 66b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * if x > 7.09782712893383973096e+02 then ieee_exp(x) overflow 67b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * if x < -7.45133219101941108420e+02 then ieee_exp(x) underflow 68b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 69b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Constants: 70b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * The hexadecimal values are the intended ones for the following 71b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * constants. The decimal values may be used, provided that the 72b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * compiler will convert from decimal to binary accurately enough 73b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * to produce the hexadecimal values shown. 74b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project */ 75b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 76b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#include "fdlibm.h" 77b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 78b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#ifdef __STDC__ 79b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projectstatic const double 80b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#else 81b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projectstatic double 82b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#endif 83b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projectone = 1.0, 84b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source ProjecthalF[2] = {0.5,-0.5,}, 85b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projecthuge = 1.0e+300, 86b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projecttwom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ 87b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projecto_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ 88b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projectu_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ 89b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projectln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 90b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ 91b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projectln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 92b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ 93b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projectinvln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ 94b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source ProjectP1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 95b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source ProjectP2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 96b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source ProjectP3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 97b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source ProjectP4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 98b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source ProjectP5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 99b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 100b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 101b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#ifdef __STDC__ 102b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project double __ieee754_exp(double x) /* default IEEE double exp */ 103b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#else 104b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project double __ieee754_exp(x) /* default IEEE double exp */ 105b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project double x; 106b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#endif 107b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project{ 108b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project double y,hi,lo,c,t; 109b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project int k,xsb; 110b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project unsigned hx; 111b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 112b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project hx = __HI(x); /* high word of x */ 113b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project xsb = (hx>>31)&1; /* sign bit of x */ 114b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project hx &= 0x7fffffff; /* high word of |x| */ 115b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 116b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project /* filter out non-finite argument */ 117b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 118b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(hx>=0x7ff00000) { 119b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(((hx&0xfffff)|__LO(x))!=0) 120b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project return x+x; /* NaN */ 121b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project else return (xsb==0)? x:0.0; /* ieee_exp(+-inf)={inf,0} */ 122b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 123b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(x > o_threshold) return huge*huge; /* overflow */ 124b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(x < u_threshold) return twom1000*twom1000; /* underflow */ 125b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 126b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 127b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project /* argument reduction */ 128b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 129b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 130b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 131b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } else { 132b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project k = (int)(invln2*x+halF[xsb]); 133b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project t = k; 134b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 135b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project lo = t*ln2LO[0]; 136b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 137b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project x = hi - lo; 138b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 139b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 140b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(huge+x>one) return one+x;/* trigger inexact */ 141b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 142b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project else k = 0; 143b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 144b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project /* x is now in primary range */ 145b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project t = x*x; 146b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 147b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(k==0) return one-((x*c)/(c-2.0)-x); 148b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project else y = one-((lo-(x*c)/(2.0-c))-hi); 149b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if(k >= -1021) { 150b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project __HI(y) += (k<<20); /* add k to y's exponent */ 151b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project return y; 152b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } else { 153b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project __HI(y) += ((k+1000)<<20);/* add k to y's exponent */ 154b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project return y*twom1000; 155b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 156b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project} 157