summaryrefslogtreecommitdiffstats
path: root/lib/msun/src/s_expm1.c
blob: adc879b0ecbbf6049643d23d77fdedbfb076ba49 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
/* @(#)s_expm1.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#ifndef lint
static char rcsid[] = "$FreeBSD$";
#endif

/* expm1(x)
 * Returns exp(x)-1, the exponential of x minus 1.
 *
 * Method
 *   1. Argument reduction:
 *	Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
 *
 *      Here a correction term c will be computed to compensate
 *	the error in r when rounded to a floating-point number.
 *
 *   2. Approximating expm1(r) by a special rational function on
 *	the interval [0,0.34658]:
 *	Since
 *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
 *	we define R1(r*r) by
 *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
 *	That is,
 *	    R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
 *		     = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
 *		     = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
 *      We use a special Reme algorithm on [0,0.347] to generate
 * 	a polynomial of degree 5 in r*r to approximate R1. The
 *	maximum error of this polynomial approximation is bounded
 *	by 2**-61. In other words,
 *	    R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
 *	where 	Q1  =  -1.6666666666666567384E-2,
 * 		Q2  =   3.9682539681370365873E-4,
 * 		Q3  =  -9.9206344733435987357E-6,
 * 		Q4  =   2.5051361420808517002E-7,
 * 		Q5  =  -6.2843505682382617102E-9;
 *  	(where z=r*r, and the values of Q1 to Q5 are listed below)
 *	with error bounded by
 *	    |                  5           |     -61
 *	    | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
 *	    |                              |
 *
 *	expm1(r) = exp(r)-1 is then computed by the following
 * 	specific way which minimize the accumulation rounding error:
 *			       2     3
 *			      r     r    [ 3 - (R1 + R1*r/2)  ]
 *	      expm1(r) = r + --- + --- * [--------------------]
 *		              2     2    [ 6 - r*(3 - R1*r/2) ]
 *
 *	To compensate the error in the argument reduction, we use
 *		expm1(r+c) = expm1(r) + c + expm1(r)*c
 *			   ~ expm1(r) + c + r*c
 *	Thus c+r*c will be added in as the correction terms for
 *	expm1(r+c). Now rearrange the term to avoid optimization
 * 	screw up:
 *		        (      2                                    2 )
 *		        ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
 *	 expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
 *	                ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
 *                      (                                             )
 *
 *		   = r - E
 *   3. Scale back to obtain expm1(x):
 *	From step 1, we have
 *	   expm1(x) = either 2^k*[expm1(r)+1] - 1
 *		    = or     2^k*[expm1(r) + (1-2^-k)]
 *   4. Implementation notes:
 *	(A). To save one multiplication, we scale the coefficient Qi
 *	     to Qi*2^i, and replace z by (x^2)/2.
 *	(B). To achieve maximum accuracy, we compute expm1(x) by
 *	  (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
 *	  (ii)  if k=0, return r-E
 *	  (iii) if k=-1, return 0.5*(r-E)-0.5
 *        (iv)	if k=1 if r < -0.25, return 2*((r+0.5)- E)
 *	       	       else	     return  1.0+2.0*(r-E);
 *	  (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
 *	  (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
 *	  (vii) return 2^k(1-((E+2^-k)-r))
 *
 * Special cases:
 *	expm1(INF) is INF, expm1(NaN) is NaN;
 *	expm1(-INF) is -1, and
 *	for finite argument, only expm1(0)=0 is exact.
 *
 * Accuracy:
 *	according to an error analysis, the error is always less than
 *	1 ulp (unit in the last place).
 *
 * Misc. info.
 *	For IEEE double
 *	    if x >  7.09782712893383973096e+02 then expm1(x) overflow
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

#include "math.h"
#include "math_private.h"

#ifdef __STDC__
static const double
#else
static double
#endif
one		= 1.0,
huge		= 1.0e+300,
tiny		= 1.0e-300,
o_threshold	= 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
ln2_hi		= 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
ln2_lo		= 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
invln2		= 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
	/* scaled coefficients related to expm1 */
Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */

#ifdef __STDC__
	double expm1(double x)
#else
	double expm1(x)
	double x;
#endif
{
	double y,hi,lo,c,t,e,hxs,hfx,r1;
	int32_t k,xsb;
	u_int32_t hx;

	GET_HIGH_WORD(hx,x);
	xsb = hx&0x80000000;		/* sign bit of x */
	if(xsb==0) y=x; else y= -x;	/* y = |x| */
	hx &= 0x7fffffff;		/* high word of |x| */

    /* filter out huge and non-finite argument */
	if(hx >= 0x4043687A) {			/* if |x|>=56*ln2 */
	    if(hx >= 0x40862E42) {		/* if |x|>=709.78... */
                if(hx>=0x7ff00000) {
		    u_int32_t low;
		    GET_LOW_WORD(low,x);
		    if(((hx&0xfffff)|low)!=0)
		         return x+x; 	 /* NaN */
		    else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
	        }
	        if(x > o_threshold) return huge*huge; /* overflow */
	    }
	    if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
		if(x+tiny<0.0)		/* raise inexact */
		return tiny-one;	/* return -1 */
	    }
	}

    /* argument reduction */
	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
		if(xsb==0)
		    {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
		else
		    {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
	    } else {
		k  = invln2*x+((xsb==0)?0.5:-0.5);
		t  = k;
		hi = x - t*ln2_hi;	/* t*ln2_hi is exact here */
		lo = t*ln2_lo;
	    }
	    x  = hi - lo;
	    c  = (hi-x)-lo;
	}
	else if(hx < 0x3c900000) {  	/* when |x|<2**-54, return x */
	    t = huge+x;	/* return x with inexact flags when x!=0 */
	    return x - (t-(huge+x));
	}
	else k = 0;

    /* x is now in primary range */
	hfx = 0.5*x;
	hxs = x*hfx;
	r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
	t  = 3.0-r1*hfx;
	e  = hxs*((r1-t)/(6.0 - x*t));
	if(k==0) return x - (x*e-hxs);		/* c is 0 */
	else {
	    e  = (x*(e-c)-c);
	    e -= hxs;
	    if(k== -1) return 0.5*(x-e)-0.5;
	    if(k==1)
	       	if(x < -0.25) return -2.0*(e-(x+0.5));
	       	else 	      return  one+2.0*(x-e);
	    if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
	        u_int32_t high;
	        y = one-(e-x);
		GET_HIGH_WORD(high,y);
		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
	        return y-one;
	    }
	    t = one;
	    if(k<20) {
	        u_int32_t high;
	        SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
	       	y = t-(e-x);
		GET_HIGH_WORD(high,y);
		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
	   } else {
	        u_int32_t high;
		SET_HIGH_WORD(t,((0x3ff-k)<<20));	/* 2^-k */
	       	y = x-(e+t);
	       	y += one;
		GET_HIGH_WORD(high,y);
		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
	    }
	}
	return y;
}
OpenPOWER on IntegriCloud