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
expm1(double x)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 }
202