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+/*
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+ * ====================================================
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+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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+ *
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+ * Developed at SunPro, a Sun Microsystems, Inc. business.
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+ * Permission to use, copy, modify, and distribute this
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+ * software is freely granted, provided that this notice
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+ * is preserved.
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+ * ====================================================
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+ */
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+
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+/* __ieee754_exp(x)
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+ * Returns the exponential of x.
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+ *
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+ * Method
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+ * 1. Argument reduction:
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+ * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
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+ * Given x, find r and integer k such that
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+ *
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+ * x = k*ln2 + r, |r| <= 0.5*ln2.
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+ *
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+ * Here r will be represented as r = hi-lo for better
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+ * accuracy.
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+ *
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+ * 2. Approximation of exp(r) by a special rational function on
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+ * the interval [0,0.34658]:
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+ * Write
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+ * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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+ * We use a special Reme algorithm on [0,0.34658] to generate
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+ * a polynomial of degree 5 to approximate R. The maximum error
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+ * of this polynomial approximation is bounded by 2**-59. In
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+ * other words,
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+ * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
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+ * (where z=r*r, and the values of P1 to P5 are listed below)
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+ * and
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+ * | 5 | -59
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+ * | 2.0+P1*z+...+P5*z - R(z) | <= 2
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+ * | |
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+ * The computation of exp(r) thus becomes
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+ * 2*r
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+ * exp(r) = 1 + -------
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+ * R - r
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+ * r*R1(r)
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+ * = 1 + r + ----------- (for better accuracy)
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+ * 2 - R1(r)
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+ * where
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+ * 2 4 10
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+ * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
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+ *
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+ * 3. Scale back to obtain exp(x):
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+ * From step 1, we have
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+ * exp(x) = 2^k * exp(r)
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+ *
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+ * Special cases:
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+ * exp(INF) is INF, exp(NaN) is NaN;
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+ * exp(-INF) is 0, and
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+ * for finite argument, only exp(0)=1 is exact.
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+ *
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+ * Accuracy:
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+ * according to an error analysis, the error is always less than
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+ * 1 ulp (unit in the last place).
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+ *
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+ * Misc. info.
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+ * For IEEE double
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+ * if x > 7.09782712893383973096e+02 then exp(x) overflow
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+ * if x < -7.45133219101941108420e+02 then exp(x) underflow
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+ *
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+ * Constants:
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+ * The hexadecimal values are the intended ones for the following
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+ * constants. The decimal values may be used, provided that the
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+ * compiler will convert from decimal to binary accurately enough
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+ * to produce the hexadecimal values shown.
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+ */
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+
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+#include "math_libm.h"
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+#include "math_private.h"
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+
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+static const double
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+one = 1.0,
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+halF[2] = {0.5,-0.5,},
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+huge = 1.0e+300,
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+twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
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+o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
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+u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
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+ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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+ -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
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+ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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+ -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
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+invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
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+P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
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+P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
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+P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
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+P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
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+P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
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+
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+double __ieee754_exp(double x) /* default IEEE double exp */
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+{
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+ double y;
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+ double hi = 0.0;
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+ double lo = 0.0;
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+ double c;
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+ double t;
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+ int32_t k=0;
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+ int32_t xsb;
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+ u_int32_t hx;
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+
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+ GET_HIGH_WORD(hx,x);
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+ xsb = (hx>>31)&1; /* sign bit of x */
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+ hx &= 0x7fffffff; /* high word of |x| */
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+
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+ /* filter out non-finite argument */
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+ if(hx >= 0x40862E42) { /* if |x|>=709.78... */
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+ if(hx>=0x7ff00000) {
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+ u_int32_t lx;
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+ GET_LOW_WORD(lx,x);
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+ if(((hx&0xfffff)|lx)!=0)
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+ return x+x; /* NaN */
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+ else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
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+ }
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+ if(x > o_threshold) return huge*huge; /* overflow */
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+ if(x < u_threshold) return twom1000*twom1000; /* underflow */
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+ }
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+
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+ /* argument reduction */
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+ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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+ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
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+ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
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+ } else {
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+ k = invln2*x+halF[xsb];
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+ t = k;
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+ hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
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+ lo = t*ln2LO[0];
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+ }
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+ x = hi - lo;
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+ }
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+ else if(hx < 0x3e300000) { /* when |x|<2**-28 */
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+ if(huge+x>one) return one+x;/* trigger inexact */
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+ }
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+ else k = 0;
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+
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+ /* x is now in primary range */
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+ t = x*x;
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+ c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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+ if(k==0) return one-((x*c)/(c-2.0)-x);
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+ else y = one-((lo-(x*c)/(2.0-c))-hi);
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+ if(k >= -1021) {
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+ u_int32_t hy;
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+ GET_HIGH_WORD(hy,y);
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+ SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
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+ return y;
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+ } else {
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+ u_int32_t hy;
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+ GET_HIGH_WORD(hy,y);
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+ SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
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+ return y*twom1000;
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+ }
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+}
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+
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+/*
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+ * wrapper exp(x)
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+ */
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+#ifndef _IEEE_LIBM
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+double exp(double x)
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+{
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+ static const double o_threshold = 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
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+ static const double u_threshold = -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */
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+
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+ double z = __ieee754_exp(x);
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+ if (_LIB_VERSION == _IEEE_)
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+ return z;
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+ if (isfinite(x)) {
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+ if (x > o_threshold)
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+ return __kernel_standard(x, x, 6); /* exp overflow */
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+ if (x < u_threshold)
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+ return __kernel_standard(x, x, 7); /* exp underflow */
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+ }
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+ return z;
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+}
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+#else
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+strong_alias(__ieee754_exp, exp)
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+#endif
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+libm_hidden_def(exp)
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Ethan Lee