1 /*-
2 * SPDX-License-Identifier: BSD-2-Clause
3 *
4 * Copyright (c) 2009-2013 Steven G. Kargl
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice unmodified, this list of conditions, and the following
12 * disclaimer.
13 * 2. Redistributions in binary form must reproduce the above copyright
14 * notice, this list of conditions and the following disclaimer in the
15 * documentation and/or other materials provided with the distribution.
16 *
17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 *
28 * Optimized by Bruce D. Evans.
29 */
30
31 /**
32 * Compute the exponential of x for Intel 80-bit format. This is based on:
33 *
34 * PTP Tang, "Table-driven implementation of the exponential function
35 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15,
36 * 144-157 (1989).
37 *
38 * where the 32 table entries have been expanded to INTERVALS (see below).
39 */
40
41 #include <float.h>
42
43 #ifdef __i386__
44 #include <ieeefp.h>
45 #endif
46
47 #include "fpmath.h"
48 #include "math.h"
49 #include "math_private.h"
50 #include "k_expl.h"
51
52 /* XXX Prevent compilers from erroneously constant folding these: */
53 static const volatile long double
54 huge = 0x1p10000L,
55 tiny = 0x1p-10000L;
56
57 static const long double
58 twom10000 = 0x1p-10000L;
59
60 static const union IEEEl2bits
61 /* log(2**16384 - 0.5) rounded towards zero: */
62 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
63 o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L),
64 #define o_threshold (o_thresholdu.e)
65 /* log(2**(-16381-64-1)) rounded towards zero: */
66 u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
67 #define u_threshold (u_thresholdu.e)
68
69 long double
expl(long double x)70 expl(long double x)
71 {
72 union IEEEl2bits u;
73 long double hi, lo, t, twopk;
74 int k;
75 uint16_t hx, ix;
76
77 /* Filter out exceptional cases. */
78 u.e = x;
79 hx = u.xbits.expsign;
80 ix = hx & 0x7fff;
81 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
82 if (ix == BIAS + LDBL_MAX_EXP) {
83 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
84 RETURNF(-1 / x);
85 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */
86 }
87 if (x > o_threshold)
88 RETURNF(huge * huge);
89 if (x < u_threshold)
90 RETURNF(tiny * tiny);
91 } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */
92 RETURNF(1 + x); /* 1 with inexact iff x != 0 */
93 }
94
95 ENTERI();
96
97 twopk = 1;
98 __k_expl(x, &hi, &lo, &k);
99 t = SUM2P(hi, lo);
100
101 /* Scale by 2**k. */
102 if (k >= LDBL_MIN_EXP) {
103 if (k == LDBL_MAX_EXP)
104 RETURNI(t * 2 * 0x1p16383L);
105 SET_LDBL_EXPSIGN(twopk, BIAS + k);
106 RETURNI(t * twopk);
107 } else {
108 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
109 RETURNI(t * twopk * twom10000);
110 }
111 }
112
113 /**
114 * Compute expm1l(x) for Intel 80-bit format. This is based on:
115 *
116 * PTP Tang, "Table-driven implementation of the Expm1 function
117 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18,
118 * 211-222 (1992).
119 */
120
121 /*
122 * Our T1 and T2 are chosen to be approximately the points where method
123 * A and method B have the same accuracy. Tang's T1 and T2 are the
124 * points where method A's accuracy changes by a full bit. For Tang,
125 * this drop in accuracy makes method A immediately less accurate than
126 * method B, but our larger INTERVALS makes method A 2 bits more
127 * accurate so it remains the most accurate method significantly
128 * closer to the origin despite losing the full bit in our extended
129 * range for it.
130 */
131 static const double
132 T1 = -0.1659, /* ~-30.625/128 * log(2) */
133 T2 = 0.1659; /* ~30.625/128 * log(2) */
134
135 /*
136 * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]:
137 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6
138 *
139 * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits,
140 * but unlike for ld128 we can't drop any terms.
141 */
142 static const union IEEEl2bits
143 B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L),
144 B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L);
145
146 static const double
147 B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */
148 B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */
149 B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */
150 B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */
151 B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */
152 B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */
153 B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */
154 B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */
155
156 long double
expm1l(long double x)157 expm1l(long double x)
158 {
159 union IEEEl2bits u, v;
160 long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi;
161 long double x_lo, x2, z;
162 long double x4;
163 int k, n, n2;
164 uint16_t hx, ix;
165
166 /* Filter out exceptional cases. */
167 u.e = x;
168 hx = u.xbits.expsign;
169 ix = hx & 0x7fff;
170 if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */
171 if (ix == BIAS + LDBL_MAX_EXP) {
172 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
173 RETURNF(-1 / x - 1);
174 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */
175 }
176 if (x > o_threshold)
177 RETURNF(huge * huge);
178 /*
179 * expm1l() never underflows, but it must avoid
180 * unrepresentable large negative exponents. We used a
181 * much smaller threshold for large |x| above than in
182 * expl() so as to handle not so large negative exponents
183 * in the same way as large ones here.
184 */
185 if (hx & 0x8000) /* x <= -64 */
186 RETURNF(tiny - 1); /* good for x < -65ln2 - eps */
187 }
188
189 ENTERI();
190
191 if (T1 < x && x < T2) {
192 if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */
193 /* x (rounded) with inexact if x != 0: */
194 RETURNI(x == 0 ? x :
195 (0x1p100 * x + fabsl(x)) * 0x1p-100);
196 }
197
198 x2 = x * x;
199 x4 = x2 * x2;
200 q = x4 * (x2 * (x4 *
201 /*
202 * XXX the number of terms is no longer good for
203 * pairwise grouping of all except B3, and the
204 * grouping is no longer from highest down.
205 */
206 (x2 * B12 + (x * B11 + B10)) +
207 (x2 * (x * B9 + B8) + (x * B7 + B6))) +
208 (x * B5 + B4.e)) + x2 * x * B3.e;
209
210 x_hi = (float)x;
211 x_lo = x - x_hi;
212 hx2_hi = x_hi * x_hi / 2;
213 hx2_lo = x_lo * (x + x_hi) / 2;
214 if (ix >= BIAS - 7)
215 RETURNI((hx2_hi + x_hi) + (hx2_lo + x_lo + q));
216 else
217 RETURNI(x + (hx2_lo + q + hx2_hi));
218 }
219
220 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
221 fn = rnintl(x * INV_L);
222 n = irint(fn);
223 n2 = (unsigned)n % INTERVALS;
224 k = n >> LOG2_INTERVALS;
225 r1 = x - fn * L1;
226 r2 = fn * -L2;
227 r = r1 + r2;
228
229 /* Prepare scale factor. */
230 v.e = 1;
231 v.xbits.expsign = BIAS + k;
232 twopk = v.e;
233
234 /*
235 * Evaluate lower terms of
236 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
237 */
238 z = r * r;
239 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
240
241 t = (long double)tbl[n2].lo + tbl[n2].hi;
242
243 if (k == 0) {
244 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
245 tbl[n2].hi * r1);
246 RETURNI(t);
247 }
248 if (k == -1) {
249 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
250 tbl[n2].hi * r1);
251 RETURNI(t / 2);
252 }
253 if (k < -7) {
254 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
255 RETURNI(t * twopk - 1);
256 }
257 if (k > 2 * LDBL_MANT_DIG - 1) {
258 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
259 if (k == LDBL_MAX_EXP)
260 RETURNI(t * 2 * 0x1p16383L - 1);
261 RETURNI(t * twopk - 1);
262 }
263
264 v.xbits.expsign = BIAS - k;
265 twomk = v.e;
266
267 if (k > LDBL_MANT_DIG - 1)
268 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
269 else
270 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
271 RETURNI(t * twopk);
272 }
273