1/* @(#)s_expm1.c 5.1 93/09/24 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
13   for performance improvement on pipelined processors.
14*/
15
16#if defined(LIBM_SCCS) && !defined(lint)
17static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $";
18#endif
19
20/* expm1(x)
21 * Returns exp(x)-1, the exponential of x minus 1.
22 *
23 * Method
24 *   1. Argument reduction:
25 *	Given x, find r and integer k such that
26 *
27 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
28 *
29 *      Here a correction term c will be computed to compensate
30 *	the error in r when rounded to a floating-point number.
31 *
32 *   2. Approximating expm1(r) by a special rational function on
33 *	the interval [0,0.34658]:
34 *	Since
35 *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
36 *	we define R1(r*r) by
37 *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
38 *	That is,
39 *	    R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
40 *		     = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
41 *		     = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
42 *      We use a special Reme algorithm on [0,0.347] to generate
43 * 	a polynomial of degree 5 in r*r to approximate R1. The
44 *	maximum error of this polynomial approximation is bounded
45 *	by 2**-61. In other words,
46 *	    R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
47 *	where 	Q1  =  -1.6666666666666567384E-2,
48 * 		Q2  =   3.9682539681370365873E-4,
49 * 		Q3  =  -9.9206344733435987357E-6,
50 * 		Q4  =   2.5051361420808517002E-7,
51 * 		Q5  =  -6.2843505682382617102E-9;
52 *  	(where z=r*r, and the values of Q1 to Q5 are listed below)
53 *	with error bounded by
54 *	    |                  5           |     -61
55 *	    | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
56 *	    |                              |
57 *
58 *	expm1(r) = exp(r)-1 is then computed by the following
59 * 	specific way which minimize the accumulation rounding error:
60 *			       2     3
61 *			      r     r    [ 3 - (R1 + R1*r/2)  ]
62 *	      expm1(r) = r + --- + --- * [--------------------]
63 *		              2     2    [ 6 - r*(3 - R1*r/2) ]
64 *
65 *	To compensate the error in the argument reduction, we use
66 *		expm1(r+c) = expm1(r) + c + expm1(r)*c
67 *			   ~ expm1(r) + c + r*c
68 *	Thus c+r*c will be added in as the correction terms for
69 *	expm1(r+c). Now rearrange the term to avoid optimization
70 * 	screw up:
71 *		        (      2                                    2 )
72 *		        ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
73 *	 expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
74 *	                ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
75 *                      (                                             )
76 *
77 *		   = r - E
78 *   3. Scale back to obtain expm1(x):
79 *	From step 1, we have
80 *	   expm1(x) = either 2^k*[expm1(r)+1] - 1
81 *		    = or     2^k*[expm1(r) + (1-2^-k)]
82 *   4. Implementation notes:
83 *	(A). To save one multiplication, we scale the coefficient Qi
84 *	     to Qi*2^i, and replace z by (x^2)/2.
85 *	(B). To achieve maximum accuracy, we compute expm1(x) by
86 *	  (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
87 *	  (ii)  if k=0, return r-E
88 *	  (iii) if k=-1, return 0.5*(r-E)-0.5
89 *        (iv)	if k=1 if r < -0.25, return 2*((r+0.5)- E)
90 *	       	       else	     return  1.0+2.0*(r-E);
91 *	  (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
92 *	  (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
93 *	  (vii) return 2^k(1-((E+2^-k)-r))
94 *
95 * Special cases:
96 *	expm1(INF) is INF, expm1(NaN) is NaN;
97 *	expm1(-INF) is -1, and
98 *	for finite argument, only expm1(0)=0 is exact.
99 *
100 * Accuracy:
101 *	according to an error analysis, the error is always less than
102 *	1 ulp (unit in the last place).
103 *
104 * Misc. info.
105 *	For IEEE double
106 *	    if x >  7.09782712893383973096e+02 then expm1(x) overflow
107 *
108 * Constants:
109 * The hexadecimal values are the intended ones for the following
110 * constants. The decimal values may be used, provided that the
111 * compiler will convert from decimal to binary accurately enough
112 * to produce the hexadecimal values shown.
113 */
114
115#include "math.h"
116#include "math_private.h"
117#define one Q[0]
118#ifdef __STDC__
119static const double
120#else
121static double
122#endif
123huge		= 1.0e+300,
124tiny		= 1.0e-300,
125o_threshold	= 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
126ln2_hi		= 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
127ln2_lo		= 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
128invln2		= 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
129	/* scaled coefficients related to expm1 */
130Q[]  =  {1.0, -3.33333333333331316428e-02, /* BFA11111 111110F4 */
131   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
132  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
133   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
134  -2.01099218183624371326e-07}; /* BE8AFDB7 6E09C32D */
135
136#ifdef __STDC__
137	double __expm1(double x)
138#else
139	double __expm1(x)
140	double x;
141#endif
142{
143	double y,hi,lo,c,t,e,hxs,hfx,r1,h2,h4,R1,R2,R3;
144	int32_t k,xsb;
145	u_int32_t hx;
146
147	GET_HIGH_WORD(hx,x);
148	xsb = hx&0x80000000;		/* sign bit of x */
149	if(xsb==0) y=x; else y= -x;	/* y = |x| */
150	hx &= 0x7fffffff;		/* high word of |x| */
151
152    /* filter out huge and non-finite argument */
153	if(hx >= 0x4043687A) {			/* if |x|>=56*ln2 */
154	    if(hx >= 0x40862E42) {		/* if |x|>=709.78... */
155                if(hx>=0x7ff00000) {
156		    u_int32_t low;
157		    GET_LOW_WORD(low,x);
158		    if(((hx&0xfffff)|low)!=0)
159		         return x+x; 	 /* NaN */
160		    else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
161	        }
162	        if(x > o_threshold) return huge*huge; /* overflow */
163	    }
164	    if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
165		if(x+tiny<0.0)		/* raise inexact */
166		return tiny-one;	/* return -1 */
167	    }
168	}
169
170    /* argument reduction */
171	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
172	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
173		if(xsb==0)
174		    {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
175		else
176		    {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
177	    } else {
178		k  = invln2*x+((xsb==0)?0.5:-0.5);
179		t  = k;
180		hi = x - t*ln2_hi;	/* t*ln2_hi is exact here */
181		lo = t*ln2_lo;
182	    }
183	    x  = hi - lo;
184	    c  = (hi-x)-lo;
185	}
186	else if(hx < 0x3c900000) {  	/* when |x|<2**-54, return x */
187	    t = huge+x;	/* return x with inexact flags when x!=0 */
188	    return x - (t-(huge+x));
189	}
190	else k = 0;
191
192    /* x is now in primary range */
193	hfx = 0.5*x;
194	hxs = x*hfx;
195#ifdef DO_NOT_USE_THIS
196	r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
197#else
198	R1 = one+hxs*Q[1]; h2 = hxs*hxs;
199	R2 = Q[2]+hxs*Q[3]; h4 = h2*h2;
200	R3 = Q[4]+hxs*Q[5];
201	r1 = R1 + h2*R2 + h4*R3;
202#endif
203	t  = 3.0-r1*hfx;
204	e  = hxs*((r1-t)/(6.0 - x*t));
205	if(k==0) return x - (x*e-hxs);		/* c is 0 */
206	else {
207	    e  = (x*(e-c)-c);
208	    e -= hxs;
209	    if(k== -1) return 0.5*(x-e)-0.5;
210	    if(k==1) {
211	       	if(x < -0.25) return -2.0*(e-(x+0.5));
212	       	else 	      return  one+2.0*(x-e);
213	    }
214	    if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
215	        u_int32_t high;
216	        y = one-(e-x);
217		GET_HIGH_WORD(high,y);
218		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
219	        return y-one;
220	    }
221	    t = one;
222	    if(k<20) {
223	        u_int32_t high;
224	        SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
225	       	y = t-(e-x);
226		GET_HIGH_WORD(high,y);
227		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
228	   } else {
229	        u_int32_t high;
230		SET_HIGH_WORD(t,((0x3ff-k)<<20));	/* 2^-k */
231	       	y = x-(e+t);
232	       	y += one;
233		GET_HIGH_WORD(high,y);
234		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
235	    }
236	}
237	return y;
238}
239weak_alias (__expm1, expm1)
240#ifdef NO_LONG_DOUBLE
241strong_alias (__expm1, __expm1l)
242weak_alias (__expm1, expm1l)
243#endif
244