sl@0: /* S_EXPM1.C
sl@0:  * 
sl@0:  * Portions Copyright (c) 1993-2005 Nokia Corporation and/or its subsidiary(-ies).
sl@0:  * All rights reserved.
sl@0:  */
sl@0: 
sl@0: 
sl@0: /* @(#)s_expm1.c 5.1 93/09/24 */
sl@0: /*
sl@0:  * ====================================================
sl@0:  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
sl@0:  *
sl@0:  * Developed at SunPro, a Sun Microsystems, Inc. business.
sl@0:  * Permission to use, copy, modify, and distribute this
sl@0:  * software is freely granted, provided that this notice 
sl@0:  * is preserved.
sl@0:  * ====================================================
sl@0:  */
sl@0: 
sl@0: /*
sl@0: FUNCTION
sl@0: 	<<expm1>>, <<expm1f>>---exponential minus 1
sl@0: INDEX
sl@0: 	expm1
sl@0: INDEX
sl@0: 	expm1f
sl@0: 
sl@0: ANSI_SYNOPSIS
sl@0: 	#include <math.h>
sl@0: 	double expm1(double <[x]>);
sl@0: 	float expm1f(float <[x]>);
sl@0: 
sl@0: TRAD_SYNOPSIS
sl@0: 	#include <math.h>
sl@0: 	double expm1(<[x]>);
sl@0: 	double <[x]>;
sl@0: 
sl@0: 	float expm1f(<[x]>);
sl@0: 	float <[x]>;
sl@0: 
sl@0: DESCRIPTION
sl@0: 	<<expm1>> and <<expm1f>> calculate the exponential of <[x]>
sl@0: 	and subtract 1, that is,
sl@0: 	@ifinfo
sl@0: 	e raised to the power <[x]> minus 1 (where e
sl@0: 	@end ifinfo
sl@0: 	@tex
sl@0: 	$e^x - 1$ (where $e$
sl@0: 	@end tex
sl@0: 	is the base of the natural system of logarithms, approximately
sl@0: 	2.71828).  The result is accurate even for small values of
sl@0: 	<[x]>, where using <<exp(<[x]>)-1>> would lose many
sl@0: 	significant digits.
sl@0: 
sl@0: RETURNS
sl@0: 	e raised to the power <[x]>, minus 1.
sl@0: 
sl@0: PORTABILITY
sl@0: 	Neither <<expm1>> nor <<expm1f>> is required by ANSI C or by
sl@0: 	the System V Interface Definition (Issue 2).
sl@0: */
sl@0: 
sl@0: /* expm1(x)
sl@0:  * Returns exp(x)-1, the exponential of x minus 1.
sl@0:  *
sl@0:  * Method
sl@0:  *   1. Argument reduction:
sl@0:  *	Given x, find r and integer k such that
sl@0:  *
sl@0:  *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658  
sl@0:  *
sl@0:  *      Here a correction term c will be computed to compensate 
sl@0:  *	the error in r when rounded to a floating-point number.
sl@0:  *
sl@0:  *   2. Approximating expm1(r) by a special rational function on
sl@0:  *	the interval [0,0.34658]:
sl@0:  *	Since
sl@0:  *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
sl@0:  *	we define R1(r*r) by
sl@0:  *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
sl@0:  *	That is,
sl@0:  *	    R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
sl@0:  *		     = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
sl@0:  *		     = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
sl@0:  *      We use a special Reme algorithm on [0,0.347] to generate 
sl@0:  * 	a polynomial of degree 5 in r*r to approximate R1. The 
sl@0:  *	maximum error of this polynomial approximation is bounded 
sl@0:  *	by 2**-61. In other words,
sl@0:  *	    R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
sl@0:  *	where 	Q1  =  -1.6666666666666567384E-2,
sl@0:  * 		Q2  =   3.9682539681370365873E-4,
sl@0:  * 		Q3  =  -9.9206344733435987357E-6,
sl@0:  * 		Q4  =   2.5051361420808517002E-7,
sl@0:  * 		Q5  =  -6.2843505682382617102E-9;
sl@0:  *  	(where z=r*r, and the values of Q1 to Q5 are listed below)
sl@0:  *	with error bounded by
sl@0:  *	    |                  5           |     -61
sl@0:  *	    | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2 
sl@0:  *	    |                              |
sl@0:  *	
sl@0:  *	expm1(r) = exp(r)-1 is then computed by the following 
sl@0:  * 	specific way which minimize the accumulation rounding error: 
sl@0:  *			       2     3
sl@0:  *			      r     r    [ 3 - (R1 + R1*r/2)  ]
sl@0:  *	      expm1(r) = r + --- + --- * [--------------------]
sl@0:  *		              2     2    [ 6 - r*(3 - R1*r/2) ]
sl@0:  *	
sl@0:  *	To compensate the error in the argument reduction, we use
sl@0:  *		expm1(r+c) = expm1(r) + c + expm1(r)*c 
sl@0:  *			   ~ expm1(r) + c + r*c 
sl@0:  *	Thus c+r*c will be added in as the correction terms for
sl@0:  *	expm1(r+c). Now rearrange the term to avoid optimization 
sl@0:  * 	screw up:
sl@0:  *		        (      2                                    2 )
sl@0:  *		        ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
sl@0:  *	 expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
sl@0:  *	                ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
sl@0:  *                      (                                             )
sl@0:  *    	
sl@0:  *		   = r - E
sl@0:  *   3. Scale back to obtain expm1(x):
sl@0:  *	From step 1, we have
sl@0:  *	   expm1(x) = either 2^k*[expm1(r)+1] - 1
sl@0:  *		    = or     2^k*[expm1(r) + (1-2^-k)]
sl@0:  *   4. Implementation notes:
sl@0:  *	(A). To save one multiplication, we scale the coefficient Qi
sl@0:  *	     to Qi*2^i, and replace z by (x^2)/2.
sl@0:  *	(B). To achieve maximum accuracy, we compute expm1(x) by
sl@0:  *	  (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
sl@0:  *	  (ii)  if k=0, return r-E
sl@0:  *	  (iii) if k=-1, return 0.5*(r-E)-0.5
sl@0:  *        (iv)	if k=1 if r < -0.25, return 2*((r+0.5)- E)
sl@0:  *	       	       else	     return  1.0+2.0*(r-E);
sl@0:  *	  (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
sl@0:  *	  (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
sl@0:  *	  (vii) return 2^k(1-((E+2^-k)-r)) 
sl@0:  *
sl@0:  * Special cases:
sl@0:  *	expm1(INF) is INF, expm1(NaN) is NaN;
sl@0:  *	expm1(-INF) is -1, and
sl@0:  *	for finite argument, only expm1(0)=0 is exact.
sl@0:  *
sl@0:  * Accuracy:
sl@0:  *	according to an error analysis, the error is always less than
sl@0:  *	1 ulp (unit in the last place).
sl@0:  *
sl@0:  * Misc. info.
sl@0:  *	For IEEE double 
sl@0:  *	    if x >  7.09782712893383973096e+02 then expm1(x) overflow
sl@0:  *
sl@0:  * Constants:
sl@0:  * The hexadecimal values are the intended ones for the following 
sl@0:  * constants. The decimal values may be used, provided that the 
sl@0:  * compiler will convert from decimal to binary accurately enough
sl@0:  * to produce the hexadecimal values shown.
sl@0:  */
sl@0: 
sl@0: #include "FDLIBM.H"
sl@0: 
sl@0: static const double
sl@0: one		= 1.0,
sl@0: huge		= 1.0e+300,
sl@0: tiny		= 1.0e-300,
sl@0: o_threshold	= 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
sl@0: ln2_hi		= 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
sl@0: ln2_lo		= 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
sl@0: invln2		= 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
sl@0: 	/* scaled coefficients related to expm1 */
sl@0: Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
sl@0: Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
sl@0: Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
sl@0: Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
sl@0: Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
sl@0: 
sl@0: /**
sl@0: Calculate the exponential of x and subtract 1 
sl@0: that is raised to the power x minus 1
sl@0: @return e raised to the power x, minus 1.
sl@0: @param x e's power.
sl@0: */	
sl@0: EXPORT_C double expm1(double x) __SOFTFP
sl@0: {
sl@0: 	double y,hi,lo,c = 0.0,t,e,hxs,hfx,r1;
sl@0: 	__int32_t k,xsb;
sl@0: 	__uint32_t hx;
sl@0: 
sl@0: 	GET_HIGH_WORD(hx,x);
sl@0: 	xsb = hx&0x80000000;		/* sign bit of x */
sl@0: 	if(xsb==0) y=x; else y= -x;	/* y = |x| */
sl@0: 	hx &= 0x7fffffff;		/* high word of |x| */
sl@0: 
sl@0:     /* filter out huge and non-finite argument */
sl@0: 	if(hx >= 0x4043687A) {			/* if |x|>=56*ln2 */
sl@0: 	    if(hx >= 0x40862E42) {		/* if |x|>=709.78... */
sl@0:                 if(hx>=0x7ff00000) {
sl@0: 		    __uint32_t low;
sl@0: 		    GET_LOW_WORD(low,x);
sl@0: 		    if(((hx&0xfffff)|low)!=0) 
sl@0: 		         return x+x; 	 /* NaN */
sl@0: 		    else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
sl@0: 	        }
sl@0: 	        if(x > o_threshold) return huge*huge; /* overflow */
sl@0: 	    }
sl@0: 	    if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
sl@0: 		if(x+tiny<0.0)		/* raise inexact */
sl@0: 		return tiny-one;	/* return -1 */
sl@0: 	    }
sl@0: 	}
sl@0: 
sl@0:     /* argument reduction */
sl@0: 	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */ 
sl@0: 	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
sl@0: 		if(xsb==0)
sl@0: 		    {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
sl@0: 		else
sl@0: 		    {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
sl@0: 	    } else {
sl@0: 		k  = invln2*x+((xsb==0)?0.5:-0.5);
sl@0: 		t  = k;
sl@0: 		hi = x - t*ln2_hi;	/* t*ln2_hi is exact here */
sl@0: 		lo = t*ln2_lo;
sl@0: 	    }
sl@0: 	    x  = hi - lo;
sl@0: 	    c  = (hi-x)-lo;
sl@0: 	} 
sl@0: 	else if(hx < 0x3c900000) {  	/* when |x|<2**-54, return x */
sl@0: 	    t = huge+x;	/* return x with inexact flags when x!=0 */
sl@0: 	    return x - (t-(huge+x));	
sl@0: 	}
sl@0: 	else k = 0;
sl@0: 
sl@0:     /* x is now in primary range */
sl@0: 	hfx = 0.5*x;
sl@0: 	hxs = x*hfx;
sl@0: 	r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
sl@0: 	t  = 3.0-r1*hfx;
sl@0: 	e  = hxs*((r1-t)/(6.0 - x*t));
sl@0: 	if(k==0) return x - (x*e-hxs);		/* c is 0 */
sl@0: 	else {
sl@0: 	    e  = (x*(e-c)-c);
sl@0: 	    e -= hxs;
sl@0: 	    if(k== -1) return 0.5*(x-e)-0.5;
sl@0: 	    if(k==1) {
sl@0: 	       	if(x < -0.25) return -2.0*(e-(x+0.5));
sl@0: 	       	else 	      return  one+2.0*(x-e);
sl@0: 		}
sl@0: 	    if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
sl@0: 	        __uint32_t high;
sl@0: 	        y = one-(e-x);
sl@0: 		GET_HIGH_WORD(high,y);
sl@0: 		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
sl@0: 	        return y-one;
sl@0: 	    }
sl@0: 	    t = one;
sl@0: 	    if(k<20) {
sl@0: 	        __uint32_t high;
sl@0: 	        SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
sl@0: 	       	y = t-(e-x);
sl@0: 		GET_HIGH_WORD(high,y);
sl@0: 		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
sl@0: 	   } else {
sl@0: 	        __uint32_t high;
sl@0: 		SET_HIGH_WORD(t,((0x3ff-k)<<20));	/* 2^-k */
sl@0: 	       	y = x-(e+t);
sl@0: 	       	y += one;
sl@0: 		GET_HIGH_WORD(high,y);
sl@0: 		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
sl@0: 	    }
sl@0: 	}
sl@0: 	return y;
sl@0: }