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__
111static const double
112#else
113static double
114#endif
115one		= 1.0,
116huge		= 1.0e+300,
117tiny		= 1.0e-300,
118o_threshold	= 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
119ln2_hi		= 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
120ln2_lo		= 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
121invln2		= 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
122	/* scaled coefficients related to expm1 */
123Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
124Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
125Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
126Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
127Q5  =  -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