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      1 
      2 /* @(#)s_expm1.c 1.5 04/04/22 */
      3 /*
      4  * ====================================================
      5  * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
      6  *
      7  * Permission to use, copy, modify, and distribute this
      8  * software is freely granted, provided that this notice
      9  * is preserved.
     10  * ====================================================
     11  */
     12 
     13 /* ieee_expm1(x)
     14  * Returns ieee_exp(x)-1, the exponential of x minus 1.
     15  *
     16  * Method
     17  *   1. Argument reduction:
     18  *	Given x, find r and integer k such that
     19  *
     20  *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
     21  *
     22  *      Here a correction term c will be computed to compensate
     23  *	the error in r when rounded to a floating-point number.
     24  *
     25  *   2. Approximating ieee_expm1(r) by a special rational function on
     26  *	the interval [0,0.34658]:
     27  *	Since
     28  *	    r*(ieee_exp(r)+1)/(ieee_exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
     29  *	we define R1(r*r) by
     30  *	    r*(ieee_exp(r)+1)/(ieee_exp(r)-1) = 2+ r^2/6 * R1(r*r)
     31  *	That is,
     32  *	    R1(r**2) = 6/r *((ieee_exp(r)+1)/(ieee_exp(r)-1) - 2/r)
     33  *		     = 6/r * ( 1 + 2.0*(1/(ieee_exp(r)-1) - 1/r))
     34  *		     = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
     35  *      We use a special Remes algorithm on [0,0.347] to generate
     36  * 	a polynomial of degree 5 in r*r to approximate R1. The
     37  *	maximum error of this polynomial approximation is bounded
     38  *	by 2**-61. In other words,
     39  *	    R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
     40  *	where 	Q1  =  -1.6666666666666567384E-2,
     41  * 		Q2  =   3.9682539681370365873E-4,
     42  * 		Q3  =  -9.9206344733435987357E-6,
     43  * 		Q4  =   2.5051361420808517002E-7,
     44  * 		Q5  =  -6.2843505682382617102E-9;
     45  *  	(where z=r*r, and the values of Q1 to Q5 are listed below)
     46  *	with error bounded by
     47  *	    |                  5           |     -61
     48  *	    | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
     49  *	    |                              |
     50  *
     51  *	expm1(r) = ieee_exp(r)-1 is then computed by the following
     52  * 	specific way which minimize the accumulation rounding error:
     53  *			       2     3
     54  *			      r     r    [ 3 - (R1 + R1*r/2)  ]
     55  *	      ieee_expm1(r) = r + --- + --- * [--------------------]
     56  *		              2     2    [ 6 - r*(3 - R1*r/2) ]
     57  *
     58  *	To compensate the error in the argument reduction, we use
     59  *		expm1(r+c) = ieee_expm1(r) + c + ieee_expm1(r)*c
     60  *			   ~ ieee_expm1(r) + c + r*c
     61  *	Thus c+r*c will be added in as the correction terms for
     62  *	expm1(r+c). Now rearrange the term to avoid optimization
     63  * 	screw up:
     64  *		        (      2                                    2 )
     65  *		        ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
     66  *	 ieee_expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
     67  *	                ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
     68  *                      (                                             )
     69  *
     70  *		   = r - E
     71  *   3. Scale back to obtain ieee_expm1(x):
     72  *	From step 1, we have
     73  *	   ieee_expm1(x) = either 2^k*[expm1(r)+1] - 1
     74  *		    = or     2^k*[expm1(r) + (1-2^-k)]
     75  *   4. Implementation notes:
     76  *	(A). To save one multiplication, we scale the coefficient Qi
     77  *	     to Qi*2^i, and replace z by (x^2)/2.
     78  *	(B). To achieve maximum accuracy, we compute ieee_expm1(x) by
     79  *	  (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
     80  *	  (ii)  if k=0, return r-E
     81  *	  (iii) if k=-1, return 0.5*(r-E)-0.5
     82  *        (iv)	if k=1 if r < -0.25, return 2*((r+0.5)- E)
     83  *	       	       else	     return  1.0+2.0*(r-E);
     84  *	  (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or ieee_exp(x)-1)
     85  *	  (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
     86  *	  (vii) return 2^k(1-((E+2^-k)-r))
     87  *
     88  * Special cases:
     89  *	expm1(INF) is INF, ieee_expm1(NaN) is NaN;
     90  *	expm1(-INF) is -1, and
     91  *	for finite argument, only ieee_expm1(0)=0 is exact.
     92  *
     93  * Accuracy:
     94  *	according to an error analysis, the error is always less than
     95  *	1 ulp (unit in the last place).
     96  *
     97  * Misc. info.
     98  *	For IEEE double
     99  *	    if x >  7.09782712893383973096e+02 then ieee_expm1(x) overflow
    100  *
    101  * Constants:
    102  * The hexadecimal values are the intended ones for the following
    103  * constants. The decimal values may be used, provided that the
    104  * compiler will convert from decimal to binary accurately enough
    105  * to produce the hexadecimal values shown.
    106  */
    107 
    108 #include "fdlibm.h"
    109 
    110 #ifdef __STDC__
    111 static const double
    112 #else
    113 static double
    114 #endif
    115 one		= 1.0,
    116 huge		= 1.0e+300,
    117 tiny		= 1.0e-300,
    118 o_threshold	= 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
    119 ln2_hi		= 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
    120 ln2_lo		= 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
    121 invln2		= 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
    122 	/* scaled coefficients related to expm1 */
    123 Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
    124 Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
    125 Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
    126 Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
    127 Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
    128 
    129 #ifdef __STDC__
    130 	double ieee_expm1(double x)
    131 #else
    132 	double ieee_expm1(x)
    133 	double x;
    134 #endif
    135 {
    136 	double y,hi,lo,c,t,e,hxs,hfx,r1;
    137 	int k,xsb;
    138 	unsigned hx;
    139 
    140 	hx  = __HI(x);	/* high word of x */
    141 	xsb = hx&0x80000000;		/* sign bit of x */
    142 	if(xsb==0) y=x; else y= -x;	/* y = |x| */
    143 	hx &= 0x7fffffff;		/* high word of |x| */
    144 
    145     /* filter out huge and non-finite argument */
    146 	if(hx >= 0x4043687A) {			/* if |x|>=56*ln2 */
    147 	    if(hx >= 0x40862E42) {		/* if |x|>=709.78... */
    148                 if(hx>=0x7ff00000) {
    149 		    if(((hx&0xfffff)|__LO(x))!=0)
    150 		         return x+x; 	 /* NaN */
    151 		    else return (xsb==0)? x:-1.0;/* ieee_exp(+-inf)={inf,-1} */
    152 	        }
    153 	        if(x > o_threshold) return huge*huge; /* overflow */
    154 	    }
    155 	    if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
    156 		if(x+tiny<0.0)		/* raise inexact */
    157 		return tiny-one;	/* return -1 */
    158 	    }
    159 	}
    160 
    161     /* argument reduction */
    162 	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
    163 	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
    164 		if(xsb==0)
    165 		    {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
    166 		else
    167 		    {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
    168 	    } else {
    169 		k  = invln2*x+((xsb==0)?0.5:-0.5);
    170 		t  = k;
    171 		hi = x - t*ln2_hi;	/* t*ln2_hi is exact here */
    172 		lo = t*ln2_lo;
    173 	    }
    174 	    x  = hi - lo;
    175 	    c  = (hi-x)-lo;
    176 	}
    177 	else if(hx < 0x3c900000) {  	/* when |x|<2**-54, return x */
    178 	    // t = huge+x;	/* return x with inexact flags when x!=0 */
    179 	    // return x - (t-(huge+x));
    180 	    return x;	// inexact flag is not set, but Java ignors this flag anyway
    181 	}
    182 	else k = 0;
    183 
    184     /* x is now in primary range */
    185 	hfx = 0.5*x;
    186 	hxs = x*hfx;
    187 	r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
    188 	t  = 3.0-r1*hfx;
    189 	e  = hxs*((r1-t)/(6.0 - x*t));
    190 	if(k==0) return x - (x*e-hxs);		/* c is 0 */
    191 	else {
    192 	    e  = (x*(e-c)-c);
    193 	    e -= hxs;
    194 	    if(k== -1) return 0.5*(x-e)-0.5;
    195 	    if(k==1)
    196 	       	if(x < -0.25) return -2.0*(e-(x+0.5));
    197 	       	else 	      return  one+2.0*(x-e);
    198 	    if (k <= -2 || k>56) {   /* suffice to return ieee_exp(x)-1 */
    199 	        y = one-(e-x);
    200 	        __HI(y) += (k<<20);	/* add k to y's exponent */
    201 	        return y-one;
    202 	    }
    203 	    t = one;
    204 	    if(k<20) {
    205 	       	__HI(t) = 0x3ff00000 - (0x200000>>k);  /* t=1-2^-k */
    206 	       	y = t-(e-x);
    207 	       	__HI(y) += (k<<20);	/* add k to y's exponent */
    208 	   } else {
    209 	       	__HI(t)  = ((0x3ff-k)<<20);	/* 2^-k */
    210 	       	y = x-(e+t);
    211 	       	y += one;
    212 	       	__HI(y) += (k<<20);	/* add k to y's exponent */
    213 	    }
    214 	}
    215 	return y;
    216 }
    217