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
13#include <sys/cdefs.h>
14__FBSDID("$FreeBSD$");
15
16/* expm1(x)
17 * Returns exp(x)-1, the exponential of x minus 1.
18 *
19 * Method
20 *   1. Argument reduction:
21 *	Given x, find r and integer k such that
22 *
23 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
24 *
25 *      Here a correction term c will be computed to compensate
26 *	the error in r when rounded to a floating-point number.
27 *
28 *   2. Approximating expm1(r) by a special rational function on
29 *	the interval [0,0.34658]:
30 *	Since
31 *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
32 *	we define R1(r*r) by
33 *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
34 *	That is,
35 *	    R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
36 *		     = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
37 *		     = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
38 *      We use a special Reme algorithm on [0,0.347] to generate
39 * 	a polynomial of degree 5 in r*r to approximate R1. The
40 *	maximum error of this polynomial approximation is bounded
41 *	by 2**-61. In other words,
42 *	    R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
43 *	where 	Q1  =  -1.6666666666666567384E-2,
44 * 		Q2  =   3.9682539681370365873E-4,
45 * 		Q3  =  -9.9206344733435987357E-6,
46 * 		Q4  =   2.5051361420808517002E-7,
47 * 		Q5  =  -6.2843505682382617102E-9;
48 *		z   =  r*r,
49 *	with error bounded by
50 *	    |                  5           |     -61
51 *	    | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
52 *	    |                              |
53 *
54 *	expm1(r) = exp(r)-1 is then computed by the following
55 * 	specific way which minimize the accumulation rounding error:
56 *			       2     3
57 *			      r     r    [ 3 - (R1 + R1*r/2)  ]
58 *	      expm1(r) = r + --- + --- * [--------------------]
59 *		              2     2    [ 6 - r*(3 - R1*r/2) ]
60 *
61 *	To compensate the error in the argument reduction, we use
62 *		expm1(r+c) = expm1(r) + c + expm1(r)*c
63 *			   ~ expm1(r) + c + r*c
64 *	Thus c+r*c will be added in as the correction terms for
65 *	expm1(r+c). Now rearrange the term to avoid optimization
66 * 	screw up:
67 *		        (      2                                    2 )
68 *		        ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
69 *	 expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
70 *	                ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
71 *                      (                                             )
72 *
73 *		   = r - E
74 *   3. Scale back to obtain expm1(x):
75 *	From step 1, we have
76 *	   expm1(x) = either 2^k*[expm1(r)+1] - 1
77 *		    = or     2^k*[expm1(r) + (1-2^-k)]
78 *   4. Implementation notes:
79 *	(A). To save one multiplication, we scale the coefficient Qi
80 *	     to Qi*2^i, and replace z by (x^2)/2.
81 *	(B). To achieve maximum accuracy, we compute expm1(x) by
82 *	  (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
83 *	  (ii)  if k=0, return r-E
84 *	  (iii) if k=-1, return 0.5*(r-E)-0.5
85 *        (iv)	if k=1 if r < -0.25, return 2*((r+0.5)- E)
86 *	       	       else	     return  1.0+2.0*(r-E);
87 *	  (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
88 *	  (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
89 *	  (vii) return 2^k(1-((E+2^-k)-r))
90 *
91 * Special cases:
92 *	expm1(INF) is INF, expm1(NaN) is NaN;
93 *	expm1(-INF) is -1, and
94 *	for finite argument, only expm1(0)=0 is exact.
95 *
96 * Accuracy:
97 *	according to an error analysis, the error is always less than
98 *	1 ulp (unit in the last place).
99 *
100 * Misc. info.
101 *	For IEEE double
102 *	    if x >  7.09782712893383973096e+02 then expm1(x) overflow
103 *
104 * Constants:
105 * The hexadecimal values are the intended ones for the following
106 * constants. The decimal values may be used, provided that the
107 * compiler will convert from decimal to binary accurately enough
108 * to produce the hexadecimal values shown.
109 */
110
111#include "math.h"
112#include "math_private.h"
113
114static const double
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 Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
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
129double
130expm1(double x)
131{
132	double y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
133	int32_t k,xsb;
134	u_int32_t hx;
135
136	GET_HIGH_WORD(hx,x);
137	xsb = hx&0x80000000;		/* sign bit of x */
138	if(xsb==0) y=x; else y= -x;	/* y = |x| */
139	hx &= 0x7fffffff;		/* high word of |x| */
140
141    /* filter out huge and non-finite argument */
142	if(hx >= 0x4043687A) {			/* if |x|>=56*ln2 */
143	    if(hx >= 0x40862E42) {		/* if |x|>=709.78... */
144                if(hx>=0x7ff00000) {
145		    u_int32_t low;
146		    GET_LOW_WORD(low,x);
147		    if(((hx&0xfffff)|low)!=0)
148		         return x+x; 	 /* NaN */
149		    else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
150	        }
151	        if(x > o_threshold) return huge*huge; /* overflow */
152	    }
153	    if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
154		if(x+tiny<0.0)		/* raise inexact */
155		return tiny-one;	/* return -1 */
156	    }
157	}
158
159    /* argument reduction */
160	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
161	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
162		if(xsb==0)
163		    {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
164		else
165		    {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
166	    } else {
167		k  = invln2*x+((xsb==0)?0.5:-0.5);
168		t  = k;
169		hi = x - t*ln2_hi;	/* t*ln2_hi is exact here */
170		lo = t*ln2_lo;
171	    }
172	    x  = hi - lo;
173	    c  = (hi-x)-lo;
174	}
175	else if(hx < 0x3c900000) {  	/* when |x|<2**-54, return x */
176	    t = huge+x;	/* return x with inexact flags when x!=0 */
177	    return x - (t-(huge+x));
178	}
179	else k = 0;
180
181    /* x is now in primary range */
182	hfx = 0.5*x;
183	hxs = x*hfx;
184	r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
185	t  = 3.0-r1*hfx;
186	e  = hxs*((r1-t)/(6.0 - x*t));
187	if(k==0) return x - (x*e-hxs);		/* c is 0 */
188	else {
189	    INSERT_WORDS(twopk,0x3ff00000+(k<<20),0);	/* 2^k */
190	    e  = (x*(e-c)-c);
191	    e -= hxs;
192	    if(k== -1) return 0.5*(x-e)-0.5;
193	    if(k==1) {
194	       	if(x < -0.25) return -2.0*(e-(x+0.5));
195	       	else 	      return  one+2.0*(x-e);
196	    }
197	    if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
198	        y = one-(e-x);
199		if (k == 1024) y = y*2.0*0x1p1023;
200		else y = y*twopk;
201	        return y-one;
202	    }
203	    t = one;
204	    if(k<20) {
205	        SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
206	       	y = t-(e-x);
207		y = y*twopk;
208	   } else {
209		SET_HIGH_WORD(t,((0x3ff-k)<<20));	/* 2^-k */
210	       	y = x-(e+t);
211	       	y += one;
212		y = y*twopk;
213	    }
214	}
215	return y;
216}
217