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 diff --git a/src/math/e_exp.c b/src/math/e_exp.cnew file mode 100644index 00000000..66107b95--- /dev/null+++ b/src/math/e_exp.c@@ -0,0 +1,155 @@++/* @(#)e_exp.c 1.6 04/04/22 */+/*+ * ====================================================+ * 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 +#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+exp(double x) /* default IEEE double exp */+{+ double y,hi=0.0,lo=0.0,c,t;+ 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)+ 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 = (int)(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) {+ uint32_t hy;+ GET_HIGH_WORD(hy,y);+ SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */+ return y;+ } else {+ uint32_t hy;+ GET_HIGH_WORD(hy,y);+ SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */+ return y*twom1000;+ }+}