/* S_EXPM1.C

  * 

  * Portions Copyright (c) 1993-2005 Nokia Corporation and/or its subsidiary(-ies).

  * All rights reserved.

  */

 /* @(#)s_expm1.c 5.1 93/09/24 */

 /*

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

  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.

  *

  * Developed at SunPro, a Sun Microsystems, Inc. business.

  * Permission to use, copy, modify, and distribute this

  * software is freely granted, provided that this notice 

  * is preserved.

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

  */

 /*

 FUNCTION

 	<<expm1>>, <<expm1f>>---exponential minus 1

 INDEX

 	expm1

 INDEX

 	expm1f

 ANSI_SYNOPSIS

 	#include <math.h>

 	double expm1(double <[x]>);

 	float expm1f(float <[x]>);

 TRAD_SYNOPSIS

 	#include <math.h>

 	double expm1(<[x]>);

 	double <[x]>;

 	float expm1f(<[x]>);

 	float <[x]>;

 DESCRIPTION

 	<<expm1>> and <<expm1f>> calculate the exponential of <[x]>

 	and subtract 1, that is,

 	@ifinfo

 	e raised to the power <[x]> minus 1 (where e

 	@end ifinfo

 	@tex

 	$e^x - 1$ (where $e$

 	@end tex

 	is the base of the natural system of logarithms, approximately

 	2.71828).  The result is accurate even for small values of

 	<[x]>, where using <<exp(<[x]>)-1>> would lose many

 	significant digits.

 RETURNS

 	e raised to the power <[x]>, minus 1.

 PORTABILITY

 	Neither <<expm1>> nor <<expm1f>> is required by ANSI C or by

 	the System V Interface Definition (Issue 2).

 */

 /* 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 Reme 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"

 static const double

 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 */

 /**

 Calculate the exponential of x and subtract 1 

 that is raised to the power x minus 1

 @return e raised to the power x, minus 1.

 @param x e's power.

 */	

 EXPORT_C double expm1(double x) __SOFTFP

 {

 	double y,hi,lo,c = 0.0,t,e,hxs,hfx,r1;

 	__int32_t k,xsb;

 	__uint32_t hx;

 	GET_HIGH_WORD(hx,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) {

 		    __uint32_t low;

 		    GET_LOW_WORD(low,x);

 		    if(((hx&0xfffff)|low)!=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 */

 	        __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,0x3ff00000 - (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,((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;

 }