/* @(#)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.

 * ====================================================

 */

/* 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.

 */

#include "fdlibm.h"

#ifdef __STDC__

static const double

#else

static double

#endif

one		= 1.0,

huge		= 1.0e+300,

tiny		= 1.0e-300,

o_threshold	= 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */

ln2_hi		= 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */

ln2_lo		= 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */

invln2		= 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */

	/* scaled coefficients related to expm1 */

Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */

Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */

Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */

Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */

Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */

#ifdef __STDC__

	double expm1(double x)

#else

	double expm1(x)

	double x;

#endif

{

	double y,hi,lo,c,t,e,hxs,hfx,r1;

	int k,xsb;

	unsigned hx;

	hx  = __HI(x);	/* high word of x */

	xsb = hx&0x80000000;		/* sign bit of x */

	if(xsb==0) y=x; else y= -x;	/* y = |x| */

	hx &= 0x7fffffff;		/* high word of |x| */

    /* filter out huge and non-finite argument */

	if(hx >= 0x4043687A) {			/* if |x|>=56*ln2 */

	    if(hx >= 0x40862E42) {		/* if |x|>=709.78... */

                if(hx>=0x7ff00000) {

		    if(((hx&0xfffff)|__LO(x))!=0) 

		         return x+x; 	 /* NaN */

		    else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */

	        }

	        if(x > o_threshold) return huge*huge; /* overflow */

	    }

	    if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */

		if(x+tiny<0.0)		/* raise inexact */

		return tiny-one;	/* return -1 */

	    }

	}

    /* argument reduction */

	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */ 

	    if(hx < 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 < 0x3c900000) {  	/* when |x|<2**-54, return x */

	    t = huge+x;	/* return x with inexact flags when x!=0 */

	    return x - (t-(huge+x));	

	}

	else k = 0;

    /* x is now in primary range */

	hfx = 0.5*x;

	hxs = x*hfx;

	r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));

	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 */

	        y = one-(e-x);

	        __HI(y) += (k<<20);	/* add k to y's exponent */

	        return y-one;

	    }

	    t = one;

	    if(k<20) {

	       	__HI(t) = 0x3ff00000 - (0x200000>>k);  /* t=1-2^-k */

	       	y = t-(e-x);

	       	__HI(y) += (k<<20);	/* add k to y's exponent */

	   } else {

	       	__HI(t)  = ((0x3ff-k)<<20);	/* 2^-k */

	       	y = x-(e+t);

	       	y += one;

	       	__HI(y) += (k<<20);	/* add k to y's exponent */

	    }

	}

	return y;

}