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 | /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, |
13 | for performance improvement on pipelined processors. |
14 | */ |
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 | * (where z=r*r, and the values of Q1 to Q5 are listed below) |
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 <errno.h> |
112 | #include <float.h> |
113 | #include <math.h> |
114 | #include <math_private.h> |
115 | #include <libm-alias-double.h> |
116 | #define one Q[0] |
117 | static const double |
118 | huge = 1.0e+300, |
119 | tiny = 1.0e-300, |
120 | o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ |
121 | ln2_hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ |
122 | ln2_lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ |
123 | invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ |
124 | /* scaled coefficients related to expm1 */ |
125 | Q[] = { 1.0, -3.33333333333331316428e-02, /* BFA11111 111110F4 */ |
126 | 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ |
127 | -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ |
128 | 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ |
129 | -2.01099218183624371326e-07 }; /* BE8AFDB7 6E09C32D */ |
130 | |
131 | double |
132 | __expm1 (double x) |
133 | { |
134 | double y, hi, lo, c, t, e, hxs, hfx, r1, h2, h4, R1, R2, R3; |
135 | int32_t k, xsb; |
136 | uint32_t hx; |
137 | |
138 | GET_HIGH_WORD (hx, x); |
139 | xsb = hx & 0x80000000; /* sign bit of x */ |
140 | if (xsb == 0) |
141 | y = x; |
142 | else |
143 | y = -x; /* y = |x| */ |
144 | hx &= 0x7fffffff; /* high word of |x| */ |
145 | |
146 | /* filter out huge and non-finite argument */ |
147 | if (hx >= 0x4043687A) /* if |x|>=56*ln2 */ |
148 | { |
149 | if (hx >= 0x40862E42) /* if |x|>=709.78... */ |
150 | { |
151 | if (hx >= 0x7ff00000) |
152 | { |
153 | uint32_t low; |
154 | GET_LOW_WORD (low, x); |
155 | if (((hx & 0xfffff) | low) != 0) |
156 | return x + x; /* NaN */ |
157 | else |
158 | return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */ |
159 | } |
160 | if (x > o_threshold) |
161 | { |
162 | __set_errno (ERANGE); |
163 | return huge * huge; /* overflow */ |
164 | } |
165 | } |
166 | if (xsb != 0) /* x < -56*ln2, return -1.0 with inexact */ |
167 | { |
168 | math_force_eval (x + tiny); /* raise inexact */ |
169 | return tiny - one; /* return -1 */ |
170 | } |
171 | } |
172 | |
173 | /* argument reduction */ |
174 | if (hx > 0x3fd62e42) /* if |x| > 0.5 ln2 */ |
175 | { |
176 | if (hx < 0x3FF0A2B2) /* and |x| < 1.5 ln2 */ |
177 | { |
178 | if (xsb == 0) |
179 | { |
180 | hi = x - ln2_hi; lo = ln2_lo; k = 1; |
181 | } |
182 | else |
183 | { |
184 | hi = x + ln2_hi; lo = -ln2_lo; k = -1; |
185 | } |
186 | } |
187 | else |
188 | { |
189 | k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5); |
190 | t = k; |
191 | hi = x - t * ln2_hi; /* t*ln2_hi is exact here */ |
192 | lo = t * ln2_lo; |
193 | } |
194 | x = hi - lo; |
195 | c = (hi - x) - lo; |
196 | } |
197 | else if (hx < 0x3c900000) /* when |x|<2**-54, return x */ |
198 | { |
199 | math_check_force_underflow (x); |
200 | t = huge + x; /* return x with inexact flags when x!=0 */ |
201 | return x - (t - (huge + x)); |
202 | } |
203 | else |
204 | k = 0; |
205 | |
206 | /* x is now in primary range */ |
207 | hfx = 0.5 * x; |
208 | hxs = x * hfx; |
209 | R1 = one + hxs * Q[1]; h2 = hxs * hxs; |
210 | R2 = Q[2] + hxs * Q[3]; h4 = h2 * h2; |
211 | R3 = Q[4] + hxs * Q[5]; |
212 | r1 = R1 + h2 * R2 + h4 * R3; |
213 | t = 3.0 - r1 * hfx; |
214 | e = hxs * ((r1 - t) / (6.0 - x * t)); |
215 | if (k == 0) |
216 | return x - (x * e - hxs); /* c is 0 */ |
217 | else |
218 | { |
219 | e = (x * (e - c) - c); |
220 | e -= hxs; |
221 | if (k == -1) |
222 | return 0.5 * (x - e) - 0.5; |
223 | if (k == 1) |
224 | { |
225 | if (x < -0.25) |
226 | return -2.0 * (e - (x + 0.5)); |
227 | else |
228 | return one + 2.0 * (x - e); |
229 | } |
230 | if (k <= -2 || k > 56) /* suffice to return exp(x)-1 */ |
231 | { |
232 | uint32_t high; |
233 | y = one - (e - x); |
234 | GET_HIGH_WORD (high, y); |
235 | SET_HIGH_WORD (y, high + (k << 20)); /* add k to y's exponent */ |
236 | return y - one; |
237 | } |
238 | t = one; |
239 | if (k < 20) |
240 | { |
241 | uint32_t high; |
242 | SET_HIGH_WORD (t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */ |
243 | y = t - (e - x); |
244 | GET_HIGH_WORD (high, y); |
245 | SET_HIGH_WORD (y, high + (k << 20)); /* add k to y's exponent */ |
246 | } |
247 | else |
248 | { |
249 | uint32_t high; |
250 | SET_HIGH_WORD (t, ((0x3ff - k) << 20)); /* 2^-k */ |
251 | y = x - (e + t); |
252 | y += one; |
253 | GET_HIGH_WORD (high, y); |
254 | SET_HIGH_WORD (y, high + (k << 20)); /* add k to y's exponent */ |
255 | } |
256 | } |
257 | return y; |
258 | } |
259 | libm_alias_double (__expm1, expm1) |
260 | |