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| Rev | Author | Line No. | Line |
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| 2 | pj | 1 | /* @(#)e_exp.c 5.1 93/09/24 */ |
| 2 | /* |
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| 3 | * ==================================================== |
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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| 5 | * |
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| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
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| 7 | * Permission to use, copy, modify, and distribute this |
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| 8 | * software is freely granted, provided that this notice |
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| 9 | * is preserved. |
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| 10 | * ==================================================== |
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| 11 | */ |
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| 12 | |||
| 13 | #ifndef lint |
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| 14 | static char rcsid[] = "$\Id: e_exp.c,v 1.3.2.1 1997/02/23 11:03:02 joerg Exp $"; |
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| 15 | #endif |
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| 16 | |||
| 17 | /* __ieee754_exp(x) |
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| 18 | * Returns the exponential of x. |
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| 19 | * |
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| 20 | * Method |
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| 21 | * 1. Argument reduction: |
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| 22 | * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. |
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| 23 | * Given x, find r and integer k such that |
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| 24 | * |
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| 25 | * x = k*ln2 + r, |r| <= 0.5*ln2. |
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| 26 | * |
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| 27 | * Here r will be represented as r = hi-lo for better |
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| 28 | * accuracy. |
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| 29 | * |
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| 30 | * 2. Approximation of exp(r) by a special rational function on |
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| 31 | * the interval [0,0.34658]: |
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| 32 | * Write |
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| 33 | * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... |
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| 34 | * We use a special Reme algorithm on [0,0.34658] to generate |
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| 35 | * a polynomial of degree 5 to approximate R. The maximum error |
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| 36 | * of this polynomial approximation is bounded by 2**-59. In |
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| 37 | * other words, |
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| 38 | * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 |
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| 39 | * (where z=r*r, and the values of P1 to P5 are listed below) |
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| 40 | * and |
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| 41 | * | 5 | -59 |
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| 42 | * | 2.0+P1*z+...+P5*z - R(z) | <= 2 |
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| 43 | * | | |
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| 44 | * The computation of exp(r) thus becomes |
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| 45 | * 2*r |
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| 46 | * exp(r) = 1 + ------- |
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| 47 | * R - r |
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| 48 | * r*R1(r) |
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| 49 | * = 1 + r + ----------- (for better accuracy) |
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| 50 | * 2 - R1(r) |
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| 51 | * where |
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| 52 | * 2 4 10 |
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| 53 | * R1(r) = r - (P1*r + P2*r + ... + P5*r ). |
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| 54 | * |
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| 55 | * 3. Scale back to obtain exp(x): |
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| 56 | * From step 1, we have |
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| 57 | * exp(x) = 2^k * exp(r) |
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| 58 | * |
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| 59 | * Special cases: |
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| 60 | * exp(INF) is INF, exp(NaN) is NaN; |
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| 61 | * exp(-INF) is 0, and |
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| 62 | * for finite argument, only exp(0)=1 is exact. |
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| 63 | * |
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| 64 | * Accuracy: |
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| 65 | * according to an error analysis, the error is always less than |
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| 66 | * 1 ulp (unit in the last place). |
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| 67 | * |
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| 68 | * Misc. info. |
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| 69 | * For IEEE double |
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| 70 | * if x > 7.09782712893383973096e+02 then exp(x) overflow |
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| 71 | * if x < -7.45133219101941108420e+02 then exp(x) underflow |
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| 72 | * |
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| 73 | * Constants: |
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| 74 | * The hexadecimal values are the intended ones for the following |
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| 75 | * constants. The decimal values may be used, provided that the |
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| 76 | * compiler will convert from decimal to binary accurately enough |
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| 77 | * to produce the hexadecimal values shown. |
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| 78 | */ |
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| 79 | |||
| 80 | #include "math.h" |
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| 81 | #include "math_private.h" |
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| 82 | |||
| 83 | #ifdef __STDC__ |
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| 84 | static const double |
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| 85 | #else |
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| 86 | static double |
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| 87 | #endif |
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| 88 | one = 1.0, |
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| 89 | halF[2] = {0.5,-0.5,}, |
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| 90 | huge = 1.0e+300, |
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| 91 | twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ |
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| 92 | o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ |
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| 93 | u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ |
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| 94 | ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ |
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| 95 | -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ |
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| 96 | ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ |
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| 97 | -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ |
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| 98 | invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ |
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| 99 | P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ |
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| 100 | P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ |
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| 101 | P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ |
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| 102 | P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ |
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| 103 | P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ |
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| 104 | |||
| 105 | |||
| 106 | #ifdef __STDC__ |
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| 107 | double __generic___ieee754_exp(double x) /* default IEEE double exp */ |
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| 108 | #else |
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| 109 | double __generic___ieee754_exp(x) /* default IEEE double exp */ |
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| 110 | double x; |
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| 111 | #endif |
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| 112 | { |
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| 113 | double y,hi=0.0,lo=0.0,c,t; |
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| 114 | int32_t k=0,xsb; |
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| 115 | u_int32_t hx; |
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| 116 | |||
| 117 | GET_HIGH_WORD(hx,x); |
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| 118 | xsb = (hx>>31)&1; /* sign bit of x */ |
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| 119 | hx &= 0x7fffffff; /* high word of |x| */ |
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| 120 | |||
| 121 | /* filter out non-finite argument */ |
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| 122 | if(hx >= 0x40862E42) { /* if |x|>=709.78... */ |
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| 123 | if(hx>=0x7ff00000) { |
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| 124 | u_int32_t lx; |
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| 125 | GET_LOW_WORD(lx,x); |
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| 126 | if(((hx&0xfffff)|lx)!=0) |
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| 127 | return x+x; /* NaN */ |
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| 128 | else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ |
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| 129 | } |
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| 130 | if(x > o_threshold) return huge*huge; /* overflow */ |
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| 131 | if(x < u_threshold) return twom1000*twom1000; /* underflow */ |
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| 132 | } |
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| 133 | |||
| 134 | /* argument reduction */ |
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| 135 | if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ |
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| 136 | if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ |
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| 137 | hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; |
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| 138 | } else { |
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| 139 | k = invln2*x+halF[xsb]; |
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| 140 | t = k; |
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| 141 | hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ |
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| 142 | lo = t*ln2LO[0]; |
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| 143 | } |
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| 144 | x = hi - lo; |
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| 145 | } |
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| 146 | else if(hx < 0x3e300000) { /* when |x|<2**-28 */ |
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| 147 | if(huge+x>one) return one+x;/* trigger inexact */ |
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| 148 | } |
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| 149 | else k = 0; |
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| 150 | |||
| 151 | /* x is now in primary range */ |
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| 152 | t = x*x; |
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| 153 | c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); |
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| 154 | if(k==0) return one-((x*c)/(c-2.0)-x); |
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| 155 | else y = one-((lo-(x*c)/(2.0-c))-hi); |
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| 156 | if(k >= -1021) { |
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| 157 | u_int32_t hy; |
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| 158 | GET_HIGH_WORD(hy,y); |
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| 159 | SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */ |
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| 160 | return y; |
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| 161 | } else { |
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| 162 | u_int32_t hy; |
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| 163 | GET_HIGH_WORD(hy,y); |
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| 164 | SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */ |
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| 165 | return y*twom1000; |
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| 166 | } |
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| 167 | } |