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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 : //#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 <float.h>
112 :
113 : #include "math_private.h"
114 :
115 : static const double
116 : one = 1.0,
117 : tiny = 1.0e-300,
118 : o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
119 : ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
120 : ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
121 : invln2 = 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: */
123 : Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
124 : Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
125 : Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
126 : Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
127 : Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
128 :
129 : static volatile double huge = 1.0e+300;
130 :
131 : double
132 0 : expm1(double x)
133 : {
134 : double y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
135 : int32_t k,xsb;
136 : u_int32_t hx;
137 :
138 0 : GET_HIGH_WORD(hx,x);
139 0 : xsb = hx&0x80000000; /* sign bit of x */
140 0 : hx &= 0x7fffffff; /* high word of |x| */
141 :
142 : /* filter out huge and non-finite argument */
143 0 : if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
144 0 : if(hx >= 0x40862E42) { /* if |x|>=709.78... */
145 0 : if(hx>=0x7ff00000) {
146 : u_int32_t low;
147 0 : GET_LOW_WORD(low,x);
148 0 : if(((hx&0xfffff)|low)!=0)
149 0 : return x+x; /* NaN */
150 0 : else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
151 : }
152 0 : if(x > o_threshold) return huge*huge; /* overflow */
153 : }
154 0 : if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
155 0 : if(x+tiny<0.0) /* raise inexact */
156 0 : return tiny-one; /* return -1 */
157 : }
158 : }
159 :
160 : /* argument reduction */
161 0 : if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
162 0 : if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
163 0 : if(xsb==0)
164 0 : {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
165 : else
166 0 : {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
167 : } else {
168 0 : k = invln2*x+((xsb==0)?0.5:-0.5);
169 0 : t = k;
170 0 : hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
171 0 : lo = t*ln2_lo;
172 : }
173 0 : STRICT_ASSIGN(double, x, hi - lo);
174 0 : c = (hi-x)-lo;
175 : }
176 0 : else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
177 0 : t = huge+x; /* return x with inexact flags when x!=0 */
178 0 : return x - (t-(huge+x));
179 : }
180 0 : else k = 0;
181 :
182 : /* x is now in primary range */
183 0 : hfx = 0.5*x;
184 0 : hxs = x*hfx;
185 0 : r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
186 0 : t = 3.0-r1*hfx;
187 0 : e = hxs*((r1-t)/(6.0 - x*t));
188 0 : if(k==0) return x - (x*e-hxs); /* c is 0 */
189 : else {
190 0 : INSERT_WORDS(twopk,0x3ff00000+(k<<20),0); /* 2^k */
191 0 : e = (x*(e-c)-c);
192 0 : e -= hxs;
193 0 : if(k== -1) return 0.5*(x-e)-0.5;
194 0 : if(k==1) {
195 0 : if(x < -0.25) return -2.0*(e-(x+0.5));
196 0 : else return one+2.0*(x-e);
197 : }
198 0 : if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
199 0 : y = one-(e-x);
200 0 : if (k == 1024) {
201 0 : double const_0x1p1023 = pow(2, 1023);
202 0 : y = y*2.0*const_0x1p1023;
203 : }
204 0 : else y = y*twopk;
205 0 : return y-one;
206 : }
207 0 : t = one;
208 0 : if(k<20) {
209 0 : SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
210 0 : y = t-(e-x);
211 0 : y = y*twopk;
212 : } else {
213 0 : SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
214 0 : y = x-(e+t);
215 0 : y += one;
216 0 : y = y*twopk;
217 : }
218 : }
219 0 : return y;
220 : }
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