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path: root/src/math/exp.c
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 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 ``` ``````/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */ /* * ==================================================== * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* exp(x) * Returns the exponential of x. * * Method * 1. Argument reduction: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2. * * Here r will be represented as r = hi-lo for better * accuracy. * * 2. Approximation of exp(r) by a special rational function on * the interval [0,0.34658]: * Write * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... * We use a special Remes algorithm on [0,0.34658] to generate * a polynomial of degree 5 to approximate R. The maximum error * of this polynomial approximation is bounded by 2**-59. In * other words, * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 * (where z=r*r, and the values of P1 to P5 are listed below) * and * | 5 | -59 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 * | | * The computation of exp(r) thus becomes * 2*r * exp(r) = 1 + ------- * R - r * r*R1(r) * = 1 + r + ----------- (for better accuracy) * 2 - R1(r) * where * 2 4 10 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): * From step 1, we have * exp(x) = 2^k * exp(r) * * Special cases: * exp(INF) is INF, exp(NaN) is NaN; * exp(-INF) is 0, and * for finite argument, only exp(0)=1 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then exp(x) overflow * if x < -7.45133219101941108420e+02 then exp(x) underflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include "libm.h" static const double one = 1.0, halF = {0.5,-0.5,}, huge = 1.0e+300, o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ ln2HI = { 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ -6.93147180369123816490e-01},/* 0xbfe62e42, 0xfee00000 */ ln2LO = { 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ -1.90821492927058770002e-10},/* 0xbdea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ static const volatile double twom1000 = 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0 */ double exp(double x) { double y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; uint32_t hx; GET_HIGH_WORD(hx, x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if (hx >= 0x40862E42) { /* if |x| >= 709.78... */ if (hx >= 0x7ff00000) { uint32_t lx; GET_LOW_WORD(lx,x); if (((hx&0xfffff)|lx) != 0) /* NaN */ return x+x; return xsb==0 ? x : 0.0; /* exp(+-inf)={inf,0} */ } if (x > o_threshold) return huge*huge; /* overflow */ if (x < u_threshold) return twom1000*twom1000; /* underflow */ } /* argument reduction */ if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo = ln2LO[xsb]; k = 1 - xsb - xsb; } else { k = (int)(invln2*x+halF[xsb]); t = k; hi = x - t*ln2HI; /* t*ln2HI is exact here */ lo = t*ln2LO; } STRICT_ASSIGN(double, x, hi - lo); } else if(hx < 0x3e300000) { /* |x| < 2**-28 */ /* raise inexact */ if (huge+x > one) return one+x; } else k = 0; /* x is now in primary range */ t = x*x; if (k >= -1021) INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0); else INSERT_WORDS(twopk, 0x3ff00000+((k+1000)<<20), 0); c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); if (k == 0) return one - ((x*c)/(c-2.0) - x); y = one-((lo-(x*c)/(2.0-c))-hi); if (k < -1021) return y*twopk*twom1000; if (k == 1024) return y*2.0*0x1p1023; return y*twopk; } ``````