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b34b60bc 1/* @(#)s_expm1.c 5.1 93/09/24 */
219be92a 2/* \$FreeBSD: head/lib/msun/src/s_expm1.c 251343 2013-06-03 19:51:32Z kargl \$ */
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3/*
4 * ====================================================
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
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12 */
13
14/* expm1(x)
15 * Returns exp(x)-1, the exponential of x minus 1.
16 *
17 * Method
18 * 1. Argument reduction:
19 * Given x, find r and integer k such that
20 *
21 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
22 *
23 * Here a correction term c will be computed to compensate
24 * the error in r when rounded to a floating-point number.
25 *
26 * 2. Approximating expm1(r) by a special rational function on
27 * the interval [0,0.34658]:
28 * Since
29 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
30 * we define R1(r*r) by
31 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
32 * That is,
33 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
34 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
35 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
36 * We use a special Reme algorithm on [0,0.347] to generate
37 * a polynomial of degree 5 in r*r to approximate R1. The
38 * maximum error of this polynomial approximation is bounded
39 * by 2**-61. In other words,
40 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
41 * where Q1 = -1.6666666666666567384E-2,
42 * Q2 = 3.9682539681370365873E-4,
43 * Q3 = -9.9206344733435987357E-6,
44 * Q4 = 2.5051361420808517002E-7,
45 * Q5 = -6.2843505682382617102E-9;
6ff43c94 46 * z = r*r,
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47 * with error bounded by
48 * | 5 | -61
49 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
50 * | |
51 *
52 * expm1(r) = exp(r)-1 is then computed by the following
53 * specific way which minimize the accumulation rounding error:
54 * 2 3
55 * r r [ 3 - (R1 + R1*r/2) ]
56 * expm1(r) = r + --- + --- * [--------------------]
57 * 2 2 [ 6 - r*(3 - R1*r/2) ]
58 *
59 * To compensate the error in the argument reduction, we use
60 * expm1(r+c) = expm1(r) + c + expm1(r)*c
61 * ~ expm1(r) + c + r*c
62 * Thus c+r*c will be added in as the correction terms for
63 * expm1(r+c). Now rearrange the term to avoid optimization
64 * screw up:
65 * ( 2 2 )
66 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
67 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
68 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
69 * ( )
70 *
71 * = r - E
72 * 3. Scale back to obtain expm1(x):
73 * From step 1, we have
74 * expm1(x) = either 2^k*[expm1(r)+1] - 1
75 * = or 2^k*[expm1(r) + (1-2^-k)]
76 * 4. Implementation notes:
77 * (A). To save one multiplication, we scale the coefficient Qi
78 * to Qi*2^i, and replace z by (x^2)/2.
79 * (B). To achieve maximum accuracy, we compute expm1(x) by
80 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
81 * (ii) if k=0, return r-E
82 * (iii) if k=-1, return 0.5*(r-E)-0.5
83 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
84 * else return 1.0+2.0*(r-E);
85 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
86 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
87 * (vii) return 2^k(1-((E+2^-k)-r))
88 *
89 * Special cases:
90 * expm1(INF) is INF, expm1(NaN) is NaN;
91 * expm1(-INF) is -1, and
92 * for finite argument, only expm1(0)=0 is exact.
93 *
94 * Accuracy:
95 * according to an error analysis, the error is always less than
96 * 1 ulp (unit in the last place).
97 *
98 * Misc. info.
99 * For IEEE double
100 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
101 *
102 * Constants:
103 * The hexadecimal values are the intended ones for the following
104 * constants. The decimal values may be used, provided that the
105 * compiler will convert from decimal to binary accurately enough
106 * to produce the hexadecimal values shown.
107 */
108
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109#include <float.h>
110
111#include "math.h"
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112#include "math_private.h"
113
114static const double
115one = 1.0,
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116tiny = 1.0e-300,
117o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
118ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
119ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
120invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
6ff43c94 121/* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
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122Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
123Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
124Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
125Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
126Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
127
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128static volatile double huge = 1.0e+300;
129
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130double
131expm1(double x)
132{
6ff43c94 133 double y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
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134 int32_t k,xsb;
135 u_int32_t hx;
136
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137 GET_HIGH_WORD(hx,x);
138 xsb = hx&0x80000000; /* sign bit of x */
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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 }
6ff43c94 172 STRICT_ASSIGN(double, x, hi - lo);
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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 {
6ff43c94 189 INSERT_WORDS(twopk,0x3ff00000+(k<<20),0); /* 2^k */
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190 e = (x*(e-c)-c);
191 e -= hxs;
192 if(k== -1) return 0.5*(x-e)-0.5;
6ff43c94 193 if(k==1) {
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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 */
b34b60bc 198 y = one-(e-x);
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199 if (k == 1024) y = y*2.0*0x1p1023;
200 else y = y*twopk;
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201 return y-one;
202 }
203 t = one;
204 if(k<20) {
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205 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
206 y = t-(e-x);
6ff43c94 207 y = y*twopk;
b34b60bc 208 } else {
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209 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
210 y = x-(e+t);
211 y += one;
6ff43c94 212 y = y*twopk;
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213 }
214 }
215 return y;
216}
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217
218#if (LDBL_MANT_DIG == 53)
219__weak_reference(expm1, expm1l);
220#endif