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---
s_expm1.c (7516B)
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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 #ifndef lint
14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_expm1.c,v 1.7 2002/05/28 18:15:04 alfred Exp $";
15 #endif
16
17 /* expm1(x)
18 * Returns exp(x)-1, the exponential of x minus 1.
19 *
20 * Method
21 * 1. Argument reduction:
22 * Given x, find r and integer k such that
23 *
24 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
25 *
26 * Here a correction term c will be computed to compensate
27 * the error in r when rounded to a floating-point number.
28 *
29 * 2. Approximating expm1(r) by a special rational function on
30 * the interval [0,0.34658]:
31 * Since
32 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
33 * we define R1(r*r) by
34 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
35 * That is,
36 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
37 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
38 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
39 * We use a special Reme algorithm on [0,0.347] to generate
40 * a polynomial of degree 5 in r*r to approximate R1. The
41 * maximum error of this polynomial approximation is bounded
42 * by 2**-61. In other words,
43 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
44 * where Q1 = -1.6666666666666567384E-2,
45 * Q2 = 3.9682539681370365873E-4,
46 * Q3 = -9.9206344733435987357E-6,
47 * Q4 = 2.5051361420808517002E-7,
48 * Q5 = -6.2843505682382617102E-9;
49 * (where z=r*r, and the values of Q1 to Q5 are listed below)
50 * with error bounded by
51 * | 5 | -61
52 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
53 * | |
54 *
55 * expm1(r) = exp(r)-1 is then computed by the following
56 * specific way which minimize the accumulation rounding error:
57 * 2 3
58 * r r [ 3 - (R1 + R1*r/2) ]
59 * expm1(r) = r + --- + --- * [--------------------]
60 * 2 2 [ 6 - r*(3 - R1*r/2) ]
61 *
62 * To compensate the error in the argument reduction, we use
63 * expm1(r+c) = expm1(r) + c + expm1(r)*c
64 * ~ expm1(r) + c + r*c
65 * Thus c+r*c will be added in as the correction terms for
66 * expm1(r+c). Now rearrange the term to avoid optimization
67 * screw up:
68 * ( 2 2 )
69 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
70 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
71 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
72 * ( )
73 *
74 * = r - E
75 * 3. Scale back to obtain expm1(x):
76 * From step 1, we have
77 * expm1(x) = either 2^k*[expm1(r)+1] - 1
78 * = or 2^k*[expm1(r) + (1-2^-k)]
79 * 4. Implementation notes:
80 * (A). To save one multiplication, we scale the coefficient Qi
81 * to Qi*2^i, and replace z by (x^2)/2.
82 * (B). To achieve maximum accuracy, we compute expm1(x) by
83 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
84 * (ii) if k=0, return r-E
85 * (iii) if k=-1, return 0.5*(r-E)-0.5
86 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
87 * else return 1.0+2.0*(r-E);
88 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
89 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
90 * (vii) return 2^k(1-((E+2^-k)-r))
91 *
92 * Special cases:
93 * expm1(INF) is INF, expm1(NaN) is NaN;
94 * expm1(-INF) is -1, and
95 * for finite argument, only expm1(0)=0 is exact.
96 *
97 * Accuracy:
98 * according to an error analysis, the error is always less than
99 * 1 ulp (unit in the last place).
100 *
101 * Misc. info.
102 * For IEEE double
103 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
104 *
105 * Constants:
106 * The hexadecimal values are the intended ones for the following
107 * constants. The decimal values may be used, provided that the
108 * compiler will convert from decimal to binary accurately enough
109 * to produce the hexadecimal values shown.
110 */
111
112 #include "math.h"
113 #include "math_private.h"
114
115 static const double
116 one = 1.0,
117 huge = 1.0e+300,
118 tiny = 1.0e-300,
119 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
120 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
121 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
122 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
123 /* scaled coefficients related to expm1 */
124 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
125 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
126 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
127 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
128 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
129
130 double
131 expm1(double x)
132 {
133 double y,hi,lo,c,t,e,hxs,hfx,r1;
134 int32_t k,xsb;
135 u_int32_t hx;
136
137 GET_HIGH_WORD(hx,x);
138 xsb = hx&0x80000000; /* sign bit of x */
139 if(xsb==0) y=x; else y= -x; /* y = |x| */
140 hx &= 0x7fffffff; /* high word of |x| */
141
142 /* filter out huge and non-finite argument */
143 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
144 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
145 if(hx>=0x7ff00000) {
146 u_int32_t low;
147 GET_LOW_WORD(low,x);
148 if(((hx&0xfffff)|low)!=0)
149 return x+x; /* NaN */
150 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
151 }
152 if(x > o_threshold) return huge*huge; /* overflow */
153 }
154 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
155 if(x+tiny<0.0) /* raise inexact */
156 return tiny-one; /* return -1 */
157 }
158 }
159
160 /* argument reduction */
161 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
162 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
163 if(xsb==0)
164 {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
165 else
166 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
167 } else {
168 k = invln2*x+((xsb==0)?0.5:-0.5);
169 t = k;
170 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
171 lo = t*ln2_lo;
172 }
173 x = hi - lo;
174 c = (hi-x)-lo;
175 }
176 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
177 t = huge+x; /* return x with inexact flags when x!=0 */
178 return x - (t-(huge+x));
179 }
180 else k = 0;
181
182 /* x is now in primary range */
183 hfx = 0.5*x;
184 hxs = x*hfx;
185 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
186 t = 3.0-r1*hfx;
187 e = hxs*((r1-t)/(6.0 - x*t));
188 if(k==0) return x - (x*e-hxs); /* c is 0 */
189 else {
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 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
197 u_int32_t high;
198 y = one-(e-x);
199 GET_HIGH_WORD(high,y);
200 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
201 return y-one;
202 }
203 t = one;
204 if(k<20) {
205 u_int32_t high;
206 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
207 y = t-(e-x);
208 GET_HIGH_WORD(high,y);
209 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
210 } else {
211 u_int32_t high;
212 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
213 y = x-(e+t);
214 y += one;
215 GET_HIGH_WORD(high,y);
216 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
217 }
218 }
219 return y;
220 }