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Diffstat (limited to 'lib/mlibc/options/ansi/musl-generic-math/exp.c')
| -rw-r--r-- | lib/mlibc/options/ansi/musl-generic-math/exp.c | 134 |
1 files changed, 134 insertions, 0 deletions
diff --git a/lib/mlibc/options/ansi/musl-generic-math/exp.c b/lib/mlibc/options/ansi/musl-generic-math/exp.c new file mode 100644 index 0000000..9ea672f --- /dev/null +++ b/lib/mlibc/options/ansi/musl-generic-math/exp.c @@ -0,0 +1,134 @@ +/* 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 Remez 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 + * r*c(r) + * = 1 + r + ----------- (for better accuracy) + * 2 - c(r) + * where + * 2 4 10 + * c(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 > 709.782712893383973096 then exp(x) overflows + * if x < -745.133219101941108420 then exp(x) underflows + */ + +#include "libm.h" + +static const double +half[2] = {0.5,-0.5}, +ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ +ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 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) +{ + double_t hi, lo, c, xx, y; + int k, sign; + uint32_t hx; + + GET_HIGH_WORD(hx, x); + sign = hx>>31; + hx &= 0x7fffffff; /* high word of |x| */ + + /* special cases */ + if (hx >= 0x4086232b) { /* if |x| >= 708.39... */ + if (isnan(x)) + return x; + if (x > 709.782712893383973096) { + /* overflow if x!=inf */ + x *= 0x1p1023; + return x; + } + if (x < -708.39641853226410622) { + /* underflow if x!=-inf */ + FORCE_EVAL((float)(-0x1p-149/x)); + if (x < -745.13321910194110842) + return 0; + } + } + + /* argument reduction */ + if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ + if (hx >= 0x3ff0a2b2) /* if |x| >= 1.5 ln2 */ + k = (int)(invln2*x + half[sign]); + else + k = 1 - sign - sign; + hi = x - k*ln2hi; /* k*ln2hi is exact here */ + lo = k*ln2lo; + x = hi - lo; + } else if (hx > 0x3e300000) { /* if |x| > 2**-28 */ + k = 0; + hi = x; + lo = 0; + } else { + /* inexact if x!=0 */ + FORCE_EVAL(0x1p1023 + x); + return 1 + x; + } + + /* x is now in primary range */ + xx = x*x; + c = x - xx*(P1+xx*(P2+xx*(P3+xx*(P4+xx*P5)))); + y = 1 + (x*c/(2-c) - lo + hi); + if (k == 0) + return y; + return scalbn(y, k); +} |