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 #include <sys/cdefs.h>
32 /**
33 * Compute the exponential of x for Intel 80-bit format. This is based on:
34 *
35 * PTP Tang, "Table-driven implementation of the exponential function
36 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15,
37 * 144-157 (1989).
38 *
39 * where the 32 table entries have been expanded to INTERVALS (see below).
40 */
41
42 #include <float.h>
43
44 #ifdef __FreeBSD__
45 #include "fpmath.h"
46 #endif
47 #include "math.h"
48 #include "math_private.h"
49 #include "k_expl.h"
50
51 /* XXX Prevent compilers from erroneously constant folding these: */
52 static const volatile long double
53 huge = 0x1p10000L,
54 tiny = 0x1p-10000L;
55
56 static const long double
57 twom10000 = 0x1p-10000L;
58
59 static const union ieee_ext_u
60 /* log(2**16384 - 0.5) rounded towards zero: */
61 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
62 o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L),
63 #define o_threshold (o_thresholdu.extu_ld)
64 /* log(2**(-16381-64-1)) rounded towards zero: */
65 u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
66 #define u_threshold (u_thresholdu.extu_ld)
67
68 long double
expl(long double x)69 expl(long double x)
70 {
71 union ieee_ext_u u;
72 long double hi, lo, t, twopk;
73 int k;
74 uint16_t hx, ix;
75
76 /* Filter out exceptional cases. */
77 u.extu_ld = x;
78 hx = GET_EXPSIGN(&u);
79 ix = hx & 0x7fff;
80 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
81 if (ix == BIAS + LDBL_MAX_EXP) {
82 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
83 RETURNF(-1 / x);
84 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */
85 }
86 if (x > o_threshold)
87 RETURNF(huge * huge);
88 if (x < u_threshold)
89 RETURNF(tiny * tiny);
90 } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */
91 RETURNF(1 + x); /* 1 with inexact iff x != 0 */
92 }
93
94 ENTERI();
95
96 twopk = 1;
97 __k_expl(x, &hi, &lo, &k);
98 t = SUM2P(hi, lo);
99
100 /* Scale by 2**k. */
101 if (k >= LDBL_MIN_EXP) {
102 if (k == LDBL_MAX_EXP)
103 RETURNI(t * 2 * 0x1p16383L);
104 SET_LDBL_EXPSIGN(twopk, BIAS + k);
105 RETURNI(t * twopk);
106 } else {
107 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
108 RETURNI(t * twopk * twom10000);
109 }
110 }
111
112 /**
113 * Compute expm1l(x) for Intel 80-bit format. This is based on:
114 *
115 * PTP Tang, "Table-driven implementation of the Expm1 function
116 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18,
117 * 211-222 (1992).
118 */
119
120 /*
121 * Our T1 and T2 are chosen to be approximately the points where method
122 * A and method B have the same accuracy. Tang's T1 and T2 are the
123 * points where method A's accuracy changes by a full bit. For Tang,
124 * this drop in accuracy makes method A immediately less accurate than
125 * method B, but our larger INTERVALS makes method A 2 bits more
126 * accurate so it remains the most accurate method significantly
127 * closer to the origin despite losing the full bit in our extended
128 * range for it.
129 */
130 static const double
131 T1 = -0.1659, /* ~-30.625/128 * log(2) */
132 T2 = 0.1659; /* ~30.625/128 * log(2) */
133
134 /*
135 * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]:
136 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6
137 *
138 * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits,
139 * but unlike for ld128 we can't drop any terms.
140 */
141 static const union ieee_ext_u
142 B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L),
143 B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L);
144
145 static const double
146 B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */
147 B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */
148 B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */
149 B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */
150 B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */
151 B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */
152 B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */
153 B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */
154
155 long double
expm1l(long double x)156 expm1l(long double x)
157 {
158 union ieee_ext_u u, v;
159 long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi;
160 long double x_lo, x2, z;
161 long double x4;
162 int k, n, n2;
163 uint16_t hx, ix;
164
165 /* Filter out exceptional cases. */
166 u.extu_ld = x;
167 hx = GET_EXPSIGN(&u);
168 ix = hx & 0x7fff;
169 if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */
170 if (ix == BIAS + LDBL_MAX_EXP) {
171 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
172 RETURNF(-1 / x - 1);
173 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */
174 }
175 if (x > o_threshold)
176 RETURNF(huge * huge);
177 /*
178 * expm1l() never underflows, but it must avoid
179 * unrepresentable large negative exponents. We used a
180 * much smaller threshold for large |x| above than in
181 * expl() so as to handle not so large negative exponents
182 * in the same way as large ones here.
183 */
184 if (hx & 0x8000) /* x <= -64 */
185 RETURNF(tiny - 1); /* good for x < -65ln2 - eps */
186 }
187
188 ENTERI();
189
190 if (T1 < x && x < T2) {
191 if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */
192 /* x (rounded) with inexact if x != 0: */
193 RETURNI(x == 0 ? x :
194 (0x1p100 * x + fabsl(x)) * 0x1p-100);
195 }
196
197 x2 = x * x;
198 x4 = x2 * x2;
199 q = x4 * (x2 * (x4 *
200 /*
201 * XXX the number of terms is no longer good for
202 * pairwise grouping of all except B3, and the
203 * grouping is no longer from highest down.
204 */
205 (x2 * B12 + (x * B11 + B10)) +
206 (x2 * (x * B9 + B8) + (x * B7 + B6))) +
207 (x * B5 + B4.extu_ld)) + x2 * x * B3.extu_ld;
208
209 x_hi = (float)x;
210 x_lo = x - x_hi;
211 hx2_hi = x_hi * x_hi / 2;
212 hx2_lo = x_lo * (x + x_hi) / 2;
213 if (ix >= BIAS - 7)
214 RETURNI((hx2_hi + x_hi) + (hx2_lo + x_lo + q));
215 else
216 RETURNI(x + (hx2_lo + q + hx2_hi));
217 }
218
219 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
220 fn = rnintl(x * INV_L);
221 n = irint(fn);
222 n2 = (unsigned)n % INTERVALS;
223 k = n >> LOG2_INTERVALS;
224 r1 = x - fn * L1;
225 r2 = fn * -L2;
226 r = r1 + r2;
227
228 /* Prepare scale factor. */
229 v.extu_ld = 1;
230 SET_EXPSIGN(&v, BIAS + k);
231 twopk = v.extu_ld;
232
233 /*
234 * Evaluate lower terms of
235 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
236 */
237 z = r * r;
238 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
239
240 t = (long double)tbl[n2].lo + tbl[n2].hi;
241
242 if (k == 0) {
243 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
244 tbl[n2].hi * r1);
245 RETURNI(t);
246 }
247 if (k == -1) {
248 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
249 tbl[n2].hi * r1);
250 RETURNI(t / 2);
251 }
252 if (k < -7) {
253 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
254 RETURNI(t * twopk - 1);
255 }
256 if (k > 2 * LDBL_MANT_DIG - 1) {
257 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
258 if (k == LDBL_MAX_EXP)
259 RETURNI(t * 2 * 0x1p16383L - 1);
260 RETURNI(t * twopk - 1);
261 }
262
263 SET_EXPSIGN(&v, BIAS - k);
264 twomk = v.extu_ld;
265
266 if (k > LDBL_MANT_DIG - 1)
267 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
268 else
269 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
270 RETURNI(t * twopk);
271 }
272