/* 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);
}