| t2ex/bsd_source/lib/libc/src_bsd/math/e_exp.c | bare source | permlink (0.02 seconds) |
1: /* @(#)e_exp.c 5.1 93/09/24 */ 2: /* 3: * ==================================================== 4: * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5: * 6: * Developed at SunPro, a Sun Microsystems, Inc. business. 7: * Permission to use, copy, modify, and distribute this 8: * software is freely granted, provided that this notice 9: * is preserved. 10: * ==================================================== 11: */ 12: 13: /* LINTLIBRARY */ 14: 15: /* exp(x) 16: * Returns the exponential of x. 17: * 18: * Method 19: * 1. Argument reduction: 20: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 21: * Given x, find r and integer k such that 22: * 23: * x = k*ln2 + r, |r| <= 0.5*ln2. 24: * 25: * Here r will be represented as r = hi-lo for better 26: * accuracy. 27: * 28: * 2. Approximation of exp(r) by a special rational function on 29: * the interval [0,0.34658]: 30: * Write 31: * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 32: * We use a special Remes algorithm on [0,0.34658] to generate 33: * a polynomial of degree 5 to approximate R. The maximum error 34: * of this polynomial approximation is bounded by 2**-59. In 35: * other words, 36: * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 37: * (where z=r*r, and the values of P1 to P5 are listed below) 38: * and 39: * | 5 | -59 40: * | 2.0+P1*z+...+P5*z - R(z) | <= 2 41: * | | 42: * The computation of exp(r) thus becomes 43: * 2*r 44: * exp(r) = 1 + ------- 45: * R - r 46: * r*R1(r) 47: * = 1 + r + ----------- (for better accuracy) 48: * 2 - R1(r) 49: * where 50: * 2 4 10 51: * R1(r) = r - (P1*r + P2*r + ... + P5*r ). 52: * 53: * 3. Scale back to obtain exp(x): 54: * From step 1, we have 55: * exp(x) = 2^k * exp(r) 56: * 57: * Special cases: 58: * exp(INF) is INF, exp(NaN) is NaN; 59: * exp(-INF) is 0, and 60: * for finite argument, only exp(0)=1 is exact. 61: * 62: * Accuracy: 63: * according to an error analysis, the error is always less than 64: * 1 ulp (unit in the last place). 65: * 66: * Misc. info. 67: * For IEEE double 68: * if x > 7.09782712893383973096e+02 then exp(x) overflow 69: * if x < -7.45133219101941108420e+02 then exp(x) underflow 70: * 71: * Constants: 72: * The hexadecimal values are the intended ones for the following 73: * constants. The decimal values may be used, provided that the 74: * compiler will convert from decimal to binary accurately enough 75: * to produce the hexadecimal values shown. 76: */ 77: 78: #include <sys/cdefs.h> 79: #include <float.h> 80: #include <math.h> 81: 82: #include "math_private.h" 83: 84: static const double 85: one = 1.0, 86: halF[2] = {0.5,-0.5,}, 87: huge = 1.0e+300, 88: twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ 89: o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ 90: u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ 91: ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 92: -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ 93: ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 94: -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ 95: invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ 96: P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 97: P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 98: P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 99: P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 100: P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 101: 102: 103: double 104: exp(double x) /* default IEEE double exp */ 105: { 106: double y,hi,lo,c,t; 107: int32_t k,xsb; 108: u_int32_t hx; 109: 110: GET_HIGH_WORD(hx,x); 111: xsb = (hx>>31)&1; /* sign bit of x */ 112: hx &= 0x7fffffff; /* high word of |x| */ 113: 114: /* filter out non-finite argument */ 115: if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 116: if(hx>=0x7ff00000) { 117: u_int32_t lx; 118: GET_LOW_WORD(lx,x); 119: if(((hx&0xfffff)|lx)!=0) 120: return x+x; /* NaN */ 121: else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 122: } 123: if(x > o_threshold) return huge*huge; /* overflow */ 124: if(x < u_threshold) return twom1000*twom1000; /* underflow */ 125: } 126: 127: /* argument reduction */ 128: if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 129: if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 130: hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 131: } else { 132: k = invln2*x+halF[xsb]; 133: t = k; 134: hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 135: lo = t*ln2LO[0]; 136: } 137: x = hi - lo; 138: } 139: else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 140: if(huge+x>one) return one+x;/* trigger inexact */ 141: } 142: else k = 0; 143: 144: /* x is now in primary range */ 145: t = x*x; 146: c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 147: if(k==0) return one-((x*c)/(c-2.0)-x); 148: else y = one-((lo-(x*c)/(2.0-c))-hi); 149: if(k >= -1021) { 150: u_int32_t hy; 151: GET_HIGH_WORD(hy,y); 152: SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */ 153: return y; 154: } else { 155: u_int32_t hy; 156: GET_HIGH_WORD(hy,y); 157: SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */ 158: return y*twom1000; 159: } 160: } 161: 162: #if LDBL_MANT_DIG == 53 163: #ifdef lint 164: /* PROTOLIB1 */ 165: long double expl(long double); 166: #else /* lint */ 167: __weak_alias(expl, exp); 168: #endif /* lint */ 169: #endif /* LDBL_MANT_DIG == 53 */