/* @(#)s_expm1.c 1.5 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.
 * ====================================================
 */
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
   for performance improvement on pipelined processors.
*/

/* expm1(x)
 * Returns exp(x)-1, the exponential of x minus 1.
 *
 * Method
 *       1. Argument reduction:
 *      Given x, find r and integer k such that
 *
 *                               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
 *
 *        Here a correction term c will be computed to compensate
 *      the error in r when rounded to a floating-point number.
 *
 *       2. Approximating expm1(r) by a special rational function on
 *      the interval [0,0.34658]:
 *      Since
 *              r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
 *      we define R1(r*r) by
 *              r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
 *      That is,
 *              R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
 *                       = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
 *                       = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
 *              We use a special Remes algorithm on [0,0.347] to generate
 *      a polynomial of degree 5 in r*r to approximate R1. The
 *      maximum error of this polynomial approximation is bounded
 *      by 2**-61. In other words,
 *              R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
 *      where   Q1      =  -1.6666666666666567384E-2,
 *              Q2      =       3.9682539681370365873E-4,
 *              Q3      =  -9.9206344733435987357E-6,
 *              Q4      =       2.5051361420808517002E-7,
 *              Q5      =  -6.2843505682382617102E-9;
 *      (where z=r*r, and the values of Q1 to Q5 are listed below)
 *      with error bounded by
 *              |                                  5               |     -61
 *              | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
 *              |                                                          |
 *
 *      expm1(r) = exp(r)-1 is then computed by the following
 *      specific way which minimize the accumulation rounding error:
 *                                 2     3
 *                                r     r        [ 3 - (R1 + R1*r/2)  ]
 *                expm1(r) = r + --- + --- * [--------------------]
 *                                2     2        [ 6 - r*(3 - R1*r/2) ]
 *
 *      To compensate the error in the argument reduction, we use
 *              expm1(r+c) = expm1(r) + c + expm1(r)*c
 *                         ~ expm1(r) + c + r*c
 *      Thus c+r*c will be added in as the correction terms for
 *      expm1(r+c). Now rearrange the term to avoid optimization
 *      screw up:
 *                      (          2                                    2 )
 *                      ({      ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
 *       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
 *                      ({      ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
 *                      (                                                 )
 *
 *                 = r - E
 *       3. Scale back to obtain expm1(x):
 *      From step 1, we have
 *         expm1(x) = either 2^k*[expm1(r)+1] - 1
 *                      = or     2^k*[expm1(r) + (1-2^-k)]
 *       4. Implementation notes:
 *      (A). To save one multiplication, we scale the coefficient Qi
 *               to Qi*2^i, and replace z by (x^2)/2.
 *      (B). To achieve maximum accuracy, we compute expm1(x) by
 *        (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
 *        (ii)  if k=0, return r-E
 *        (iii) if k=-1, return 0.5*(r-E)-0.5
 *                (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
 *                         else          return  1.0+2.0*(r-E);
 *        (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
 *        (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
 *        (vii) return 2^k(1-((E+2^-k)-r))
 *
 * Special cases:
 *      expm1(INF) is INF, expm1(NaN) is NaN;
 *      expm1(-INF) is -1, and
 *      for finite argument, only expm1(0)=0 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 expm1(x) overflow
 *
 * 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.
 */

#ifndef __FDLIBM_H__
#include "fdlibm.h"
#endif

#ifndef __have_fpu_expm1

double __ieee754_expm1(double x)
{
	double y, hi, lo, c, t, e, hxs, hfx, r1, h2, h4, R1, R2, R3;
	int32_t k, xsb;
	uint32_t hx;

	static const double hugeval = 1.0e+300;
	static const double o_threshold = 7.09782712893383973096e+02;	/* 0x40862E42, 0xFEFA39EF */
	static const double ln2_hi = 6.93147180369123816490e-01;	/* 0x3fe62e42, 0xfee00000 */
	static const double ln2_lo = 1.90821492927058770002e-10;	/* 0x3dea39ef, 0x35793c76 */
	static const double invln2 = 1.44269504088896338700e+00;	/* 0x3ff71547, 0x652b82fe */
	/* scaled coefficients related to expm1 */
	static const double Q[] = {
		1.0,
#define one Q[0]
		-3.33333333333331316428e-02,		/* BFA11111 111110F4 */
		1.58730158725481460165e-03,			/* 3F5A01A0 19FE5585 */
		-7.93650757867487942473e-05,		/* BF14CE19 9EAADBB7 */
		4.00821782732936239552e-06,			/* 3ED0CFCA 86E65239 */
		-2.01099218183624371326e-07			/* BE8AFDB7 6E09C32D */
	};

	GET_HIGH_WORD(hx, x);
	xsb = hx & IC(0x80000000);				/* sign bit of x */
	if (xsb == 0)
		y = x;
	else
		y = -x;							/* y = |x| */
	hx &= UC(0x7fffffff);					/* high word of |x| */

	/* filter out hugeval and non-finite argument */
	if (hx >= UC(0x4043687A))
	{									/* if |x|>=56*ln2 */
		if (hx >= UC(0x40862E42))
		{								/* if |x|>=709.78... */
			if (hx >= UC(0x7ff00000))
			{
				uint32_t low;

				GET_LOW_WORD(low, x);
				if (((hx & UC(0xfffff)) | low) != 0)
					return x + x;		/* NaN */
				return (xsb == 0) ? x : -one;	/* exp(+-inf)={inf,-1} */
			}
			if (x > o_threshold)
			{
				feraiseexcept(FE_OVERFLOW);
				return HUGE_VAL;		/* overflow */
			}
		}
		if (xsb != 0)
		{								/* x < -56*ln2, return -1.0 with inexact */
			feraiseexcept(FE_INEXACT);	/* raise inexact */
			return -one;				/* return -1 */
		}
	}

	/* argument reduction */
	if (hx > UC(0x3fd62e42))
	{									/* if  |x| > 0.5 ln2 */
		if (hx < UC(0x3FF0A2B2))
		{								/* and |x| < 1.5 ln2 */
			if (xsb == 0)
			{
				hi = x - ln2_hi;
				lo = ln2_lo;
				k = 1;
			} else
			{
				hi = x + ln2_hi;
				lo = -ln2_lo;
				k = -1;
			}
		} else
		{
			k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5);
			t = k;
			hi = x - t * ln2_hi;		/* t*ln2_hi is exact here */
			lo = t * ln2_lo;
		}
		x = hi - lo;
		c = (hi - x) - lo;
	} else if (hx < UC(0x3c900000))
	{									/* when |x|<2**-54, return x */
		t = hugeval + x;				/* return x with inexact flags when x!=0 */
		return x - (t - (hugeval + x));
	} else
	{
		k = 0;
		c = 0;
	}

	/* x is now in primary range */
	hfx = 0.5 * x;
	hxs = x * hfx;
#ifdef DO_NOT_USE_THIS
	r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
#else
	R1 = one + hxs * Q[1];
	h2 = hxs * hxs;
	R2 = Q[2] + hxs * Q[3];
	h4 = h2 * h2;
	R3 = Q[4] + hxs * Q[5];
	r1 = R1 + h2 * R2 + h4 * R3;
#endif
	t = 3.0 - r1 * hfx;
	e = hxs * ((r1 - t) / (6.0 - x * t));
	if (k == 0)
		return x - (x * e - hxs);		/* c is 0 */
	else
	{
		e = (x * (e - c) - c);
		e -= hxs;
		if (k == -1)
			return 0.5 * (x - e) - 0.5;
		if (k == 1)
		{
			if (x < -0.25)
				return -2.0 * (e - (x + 0.5));
			else
				return one + 2.0 * (x - e);
		}
		if (k <= -2 || k > 56)
		{								/* suffice to return exp(x)-1 */
			uint32_t high;

			y = one - (e - x);
			GET_HIGH_WORD(high, y);
			SET_HIGH_WORD(y, high + (k << 20));	/* add k to y's exponent */
			return y - one;
		}
		t = one;
		if (k < 20)
		{
			uint32_t high;

			SET_HIGH_WORD(t, UC(0x3ff00000) - (UC(0x200000) >> k));	/* t=1-2^-k */
			y = t - (e - x);
			GET_HIGH_WORD(high, y);
			SET_HIGH_WORD(y, high + (k << 20));	/* add k to y's exponent */
		} else
		{
			uint32_t high;

			SET_HIGH_WORD(t, ((UC(0x3ff) - k) << 20));	/* 2^-k */
			y = x - (e + t);
			y += one;
			GET_HIGH_WORD(high, y);
			SET_HIGH_WORD(y, high + (k << 20));	/* add k to y's exponent */
		}
	}
	return y;
#undef one
}

#endif

double __expm1(double x)
{
	double z = __ieee754_expm1(x);

	if ((!isfinite(z) || z == -1.0) && isfinite(x) && _LIB_VERSION != _IEEE_)
		return __kernel_standard(x, x, z, signbit(x) ? KMATHERR_EXPM1_UNDERFLOW : KMATHERR_EXPM1_OVERFLOW);

	return z;
}

__typeof(__expm1) expm1 __attribute__((weak, alias("__expm1")));
#ifdef __NO_LONG_DOUBLE_MATH
long double __expm1l(long double x) __attribute__((alias("__expm1")));
__typeof(__expm1l) expm1l __attribute__((weak, alias("__expm1")));
#endif
