1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_expm1.c */
|
---|
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 | /* expm1(x)
|
---|
13 | * Returns exp(x)-1, the exponential of x minus 1.
|
---|
14 | *
|
---|
15 | * Method
|
---|
16 | * 1. Argument reduction:
|
---|
17 | * Given x, find r and integer k such that
|
---|
18 | *
|
---|
19 | * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
|
---|
20 | *
|
---|
21 | * Here a correction term c will be computed to compensate
|
---|
22 | * the error in r when rounded to a floating-point number.
|
---|
23 | *
|
---|
24 | * 2. Approximating expm1(r) by a special rational function on
|
---|
25 | * the interval [0,0.34658]:
|
---|
26 | * Since
|
---|
27 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
|
---|
28 | * we define R1(r*r) by
|
---|
29 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
|
---|
30 | * That is,
|
---|
31 | * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
|
---|
32 | * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
|
---|
33 | * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
|
---|
34 | * We use a special Remez algorithm on [0,0.347] to generate
|
---|
35 | * a polynomial of degree 5 in r*r to approximate R1. The
|
---|
36 | * maximum error of this polynomial approximation is bounded
|
---|
37 | * by 2**-61. In other words,
|
---|
38 | * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
|
---|
39 | * where Q1 = -1.6666666666666567384E-2,
|
---|
40 | * Q2 = 3.9682539681370365873E-4,
|
---|
41 | * Q3 = -9.9206344733435987357E-6,
|
---|
42 | * Q4 = 2.5051361420808517002E-7,
|
---|
43 | * Q5 = -6.2843505682382617102E-9;
|
---|
44 | * z = r*r,
|
---|
45 | * with error bounded by
|
---|
46 | * | 5 | -61
|
---|
47 | * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
|
---|
48 | * | |
|
---|
49 | *
|
---|
50 | * expm1(r) = exp(r)-1 is then computed by the following
|
---|
51 | * specific way which minimize the accumulation rounding error:
|
---|
52 | * 2 3
|
---|
53 | * r r [ 3 - (R1 + R1*r/2) ]
|
---|
54 | * expm1(r) = r + --- + --- * [--------------------]
|
---|
55 | * 2 2 [ 6 - r*(3 - R1*r/2) ]
|
---|
56 | *
|
---|
57 | * To compensate the error in the argument reduction, we use
|
---|
58 | * expm1(r+c) = expm1(r) + c + expm1(r)*c
|
---|
59 | * ~ expm1(r) + c + r*c
|
---|
60 | * Thus c+r*c will be added in as the correction terms for
|
---|
61 | * expm1(r+c). Now rearrange the term to avoid optimization
|
---|
62 | * screw up:
|
---|
63 | * ( 2 2 )
|
---|
64 | * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
|
---|
65 | * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
|
---|
66 | * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
|
---|
67 | * ( )
|
---|
68 | *
|
---|
69 | * = r - E
|
---|
70 | * 3. Scale back to obtain expm1(x):
|
---|
71 | * From step 1, we have
|
---|
72 | * expm1(x) = either 2^k*[expm1(r)+1] - 1
|
---|
73 | * = or 2^k*[expm1(r) + (1-2^-k)]
|
---|
74 | * 4. Implementation notes:
|
---|
75 | * (A). To save one multiplication, we scale the coefficient Qi
|
---|
76 | * to Qi*2^i, and replace z by (x^2)/2.
|
---|
77 | * (B). To achieve maximum accuracy, we compute expm1(x) by
|
---|
78 | * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
|
---|
79 | * (ii) if k=0, return r-E
|
---|
80 | * (iii) if k=-1, return 0.5*(r-E)-0.5
|
---|
81 | * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
|
---|
82 | * else return 1.0+2.0*(r-E);
|
---|
83 | * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
|
---|
84 | * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
|
---|
85 | * (vii) return 2^k(1-((E+2^-k)-r))
|
---|
86 | *
|
---|
87 | * Special cases:
|
---|
88 | * expm1(INF) is INF, expm1(NaN) is NaN;
|
---|
89 | * expm1(-INF) is -1, and
|
---|
90 | * for finite argument, only expm1(0)=0 is exact.
|
---|
91 | *
|
---|
92 | * Accuracy:
|
---|
93 | * according to an error analysis, the error is always less than
|
---|
94 | * 1 ulp (unit in the last place).
|
---|
95 | *
|
---|
96 | * Misc. info.
|
---|
97 | * For IEEE double
|
---|
98 | * if x > 7.09782712893383973096e+02 then expm1(x) overflow
|
---|
99 | *
|
---|
100 | * Constants:
|
---|
101 | * The hexadecimal values are the intended ones for the following
|
---|
102 | * constants. The decimal values may be used, provided that the
|
---|
103 | * compiler will convert from decimal to binary accurately enough
|
---|
104 | * to produce the hexadecimal values shown.
|
---|
105 | */
|
---|
106 |
|
---|
107 | #include "libm.h"
|
---|
108 |
|
---|
109 | static const double
|
---|
110 | o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
|
---|
111 | ln2_hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
|
---|
112 | ln2_lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
|
---|
113 | invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
|
---|
114 | /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
|
---|
115 | Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
|
---|
116 | Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
|
---|
117 | Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
|
---|
118 | Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
|
---|
119 | Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
|
---|
120 |
|
---|
121 | double expm1(double x)
|
---|
122 | {
|
---|
123 | double_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
|
---|
124 | union {double f; uint64_t i;} u = {x};
|
---|
125 | uint32_t hx = u.i>>32 & 0x7fffffff;
|
---|
126 | int k, sign = u.i>>63;
|
---|
127 |
|
---|
128 | /* filter out huge and non-finite argument */
|
---|
129 | if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */
|
---|
130 | if (isnan(x))
|
---|
131 | return x;
|
---|
132 | if (sign)
|
---|
133 | return -1;
|
---|
134 | if (x > o_threshold) {
|
---|
135 | x *= 0x1p1023;
|
---|
136 | return x;
|
---|
137 | }
|
---|
138 | }
|
---|
139 |
|
---|
140 | /* argument reduction */
|
---|
141 | if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
---|
142 | if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
---|
143 | if (!sign) {
|
---|
144 | hi = x - ln2_hi;
|
---|
145 | lo = ln2_lo;
|
---|
146 | k = 1;
|
---|
147 | } else {
|
---|
148 | hi = x + ln2_hi;
|
---|
149 | lo = -ln2_lo;
|
---|
150 | k = -1;
|
---|
151 | }
|
---|
152 | } else {
|
---|
153 | k = invln2*x + (sign ? -0.5 : 0.5);
|
---|
154 | t = k;
|
---|
155 | hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
|
---|
156 | lo = t*ln2_lo;
|
---|
157 | }
|
---|
158 | x = hi-lo;
|
---|
159 | c = (hi-x)-lo;
|
---|
160 | } else if (hx < 0x3c900000) { /* |x| < 2**-54, return x */
|
---|
161 | if (hx < 0x00100000)
|
---|
162 | FORCE_EVAL((float)x);
|
---|
163 | return x;
|
---|
164 | } else
|
---|
165 | k = 0;
|
---|
166 |
|
---|
167 | /* x is now in primary range */
|
---|
168 | hfx = 0.5*x;
|
---|
169 | hxs = x*hfx;
|
---|
170 | r1 = 1.0+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
|
---|
171 | t = 3.0-r1*hfx;
|
---|
172 | e = hxs*((r1-t)/(6.0 - x*t));
|
---|
173 | if (k == 0) /* c is 0 */
|
---|
174 | return x - (x*e-hxs);
|
---|
175 | e = x*(e-c) - c;
|
---|
176 | e -= hxs;
|
---|
177 | /* exp(x) ~ 2^k (x_reduced - e + 1) */
|
---|
178 | if (k == -1)
|
---|
179 | return 0.5*(x-e) - 0.5;
|
---|
180 | if (k == 1) {
|
---|
181 | if (x < -0.25)
|
---|
182 | return -2.0*(e-(x+0.5));
|
---|
183 | return 1.0+2.0*(x-e);
|
---|
184 | }
|
---|
185 | u.i = (uint64_t)(0x3ff + k)<<52; /* 2^k */
|
---|
186 | twopk = u.f;
|
---|
187 | if (k < 0 || k > 56) { /* suffice to return exp(x)-1 */
|
---|
188 | y = x - e + 1.0;
|
---|
189 | if (k == 1024)
|
---|
190 | y = y*2.0*0x1p1023;
|
---|
191 | else
|
---|
192 | y = y*twopk;
|
---|
193 | return y - 1.0;
|
---|
194 | }
|
---|
195 | u.i = (uint64_t)(0x3ff - k)<<52; /* 2^-k */
|
---|
196 | if (k < 20)
|
---|
197 | y = (x-e+(1-u.f))*twopk;
|
---|
198 | else
|
---|
199 | y = (x-(e+u.f)+1)*twopk;
|
---|
200 | return y;
|
---|
201 | }
|
---|