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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 ``` ``````/* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* __ieee754_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 Reme 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 "math.h" #include "math_private.h" static const double one = 1.0, halF = {0.5,-0.5,}, huge = 1.0e+300, twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ 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 */ double attribute_hidden __ieee754_exp(double x) /* default IEEE double exp */ { double y; double hi = 0.0; double lo = 0.0; double c; double t; int32_t k=0; int32_t xsb; u_int32_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) { u_int32_t lx; GET_LOW_WORD(lx,x); if(((hx&0xfffff)|lx)!=0) return x+x; /* NaN */ else 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 = invln2*x+halF[xsb]; t = k; hi = x - t*ln2HI; /* t*ln2HI is exact here */ lo = t*ln2LO; } x = hi - lo; } else if(hx < 0x3e300000) { /* when |x|<2**-28 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); if(k==0) return one-((x*c)/(c-2.0)-x); else y = one-((lo-(x*c)/(2.0-c))-hi); if(k >= -1021) { u_int32_t hy; GET_HIGH_WORD(hy,y); SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */ return y; } else { u_int32_t hy; GET_HIGH_WORD(hy,y); SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */ return y*twom1000; } } /* * wrapper exp(x) */ #ifndef _IEEE_LIBM double exp(double x) { static const double o_threshold = 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */ static const double u_threshold = -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */ double z = __ieee754_exp(x); if (_LIB_VERSION == _IEEE_) return z; if (isfinite(x)) { if (x > o_threshold) return __kernel_standard(x, x, 6); /* exp overflow */ if (x < u_threshold) return __kernel_standard(x, x, 7); /* exp underflow */ } return z; } #else strong_alias(__ieee754_exp, exp) #endif libm_hidden_def(exp) ``````