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 * ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments.
34 */
35
36 #include <float.h>
37
38 #include "fpmath.h"
39 #include "math.h"
40 #include "math_private.h"
41 #include "k_expl.h"
42
43 /* XXX Prevent compilers from erroneously constant folding these: */
44 static const volatile long double
45 huge = 0x1p10000L,
46 tiny = 0x1p-10000L;
47
48 static const long double
49 twom10000 = 0x1p-10000L;
50
51 static const long double
52 /* log(2**16384 - 0.5) rounded towards zero: */
53 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
54 o_threshold = 11356.523406294143949491931077970763428L,
55 /* log(2**(-16381-64-1)) rounded towards zero: */
56 u_threshold = -11433.462743336297878837243843452621503L;
57
58 long double
expl(long double x)59 expl(long double x)
60 {
61 union IEEEl2bits u;
62 long double hi, lo, t, twopk;
63 int k;
64 uint16_t hx, ix;
65
66 /* Filter out exceptional cases. */
67 u.e = x;
68 hx = u.xbits.expsign;
69 ix = hx & 0x7fff;
70 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
71 if (ix == BIAS + LDBL_MAX_EXP) {
72 if (hx & 0x8000) /* x is -Inf or -NaN */
73 RETURNF(-1 / x);
74 RETURNF(x + x); /* x is +Inf or +NaN */
75 }
76 if (x > o_threshold)
77 RETURNF(huge * huge);
78 if (x < u_threshold)
79 RETURNF(tiny * tiny);
80 } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */
81 RETURNF(1 + x); /* 1 with inexact iff x != 0 */
82 }
83
84 ENTERI();
85
86 twopk = 1;
87 __k_expl(x, &hi, &lo, &k);
88 t = SUM2P(hi, lo);
89
90 /* Scale by 2**k. */
91 /*
92 * XXX sparc64 multiplication was so slow that scalbnl() is faster,
93 * but performance on aarch64 and riscv hasn't yet been quantified.
94 */
95 if (k >= LDBL_MIN_EXP) {
96 if (k == LDBL_MAX_EXP)
97 RETURNI(t * 2 * 0x1p16383L);
98 SET_LDBL_EXPSIGN(twopk, BIAS + k);
99 RETURNI(t * twopk);
100 } else {
101 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
102 RETURNI(t * twopk * twom10000);
103 }
104 }
105
106 /*
107 * Our T1 and T2 are chosen to be approximately the points where method
108 * A and method B have the same accuracy. Tang's T1 and T2 are the
109 * points where method A's accuracy changes by a full bit. For Tang,
110 * this drop in accuracy makes method A immediately less accurate than
111 * method B, but our larger INTERVALS makes method A 2 bits more
112 * accurate so it remains the most accurate method significantly
113 * closer to the origin despite losing the full bit in our extended
114 * range for it.
115 *
116 * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2].
117 * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear
118 * in both subintervals, so set T3 = 2**-5, which places the condition
119 * into the [T1, T3] interval.
120 *
121 * XXX we now do this more to (partially) balance the number of terms
122 * in the C and D polys than to avoid checking the condition in both
123 * intervals.
124 *
125 * XXX these micro-optimizations are excessive.
126 */
127 static const double
128 T1 = -0.1659, /* ~-30.625/128 * log(2) */
129 T2 = 0.1659, /* ~30.625/128 * log(2) */
130 T3 = 0.03125;
131
132 /*
133 * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]:
134 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03
135 *
136 * XXX none of the long double C or D coeffs except C10 is correctly printed.
137 * If you re-print their values in %.35Le format, the result is always
138 * different. For example, the last 2 digits in C3 should be 59, not 67.
139 * 67 is apparently from rounding an extra-precision value to 36 decimal
140 * places.
141 */
142 static const long double
143 C3 = 1.66666666666666666666666666666666667e-1L,
144 C4 = 4.16666666666666666666666666666666645e-2L,
145 C5 = 8.33333333333333333333333333333371638e-3L,
146 C6 = 1.38888888888888888888888888891188658e-3L,
147 C7 = 1.98412698412698412698412697235950394e-4L,
148 C8 = 2.48015873015873015873015112487849040e-5L,
149 C9 = 2.75573192239858906525606685484412005e-6L,
150 C10 = 2.75573192239858906612966093057020362e-7L,
151 C11 = 2.50521083854417203619031960151253944e-8L,
152 C12 = 2.08767569878679576457272282566520649e-9L,
153 C13 = 1.60590438367252471783548748824255707e-10L;
154
155 /*
156 * XXX this has 1 more coeff than needed.
157 * XXX can start the double coeffs but not the double mults at C10.
158 * With my coeffs (C10-C17 double; s = best_s):
159 * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]:
160 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
161 */
162 static const double
163 C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */
164 C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */
165 C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */
166 C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */
167 C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */
168
169 /*
170 * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]:
171 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44
172 */
173 static const long double
174 D3 = 1.66666666666666666666666666666682245e-1L,
175 D4 = 4.16666666666666666666666666634228324e-2L,
176 D5 = 8.33333333333333333333333364022244481e-3L,
177 D6 = 1.38888888888888888888887138722762072e-3L,
178 D7 = 1.98412698412698412699085805424661471e-4L,
179 D8 = 2.48015873015873015687993712101479612e-5L,
180 D9 = 2.75573192239858944101036288338208042e-6L,
181 D10 = 2.75573192239853161148064676533754048e-7L,
182 D11 = 2.50521083855084570046480450935267433e-8L,
183 D12 = 2.08767569819738524488686318024854942e-9L,
184 D13 = 1.60590442297008495301927448122499313e-10L;
185
186 /*
187 * XXX this has 1 more coeff than needed.
188 * XXX can start the double coeffs but not the double mults at D11.
189 * With my coeffs (D11-D16 double):
190 * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]:
191 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
192 */
193 static const double
194 D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */
195 D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */
196 D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */
197 D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */
198
199 long double
expm1l(long double x)200 expm1l(long double x)
201 {
202 union IEEEl2bits u, v;
203 long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi;
204 long double x_lo, x2;
205 double dr, dx, fn, r2;
206 int k, n, n2;
207 uint16_t hx, ix;
208
209 /* Filter out exceptional cases. */
210 u.e = x;
211 hx = u.xbits.expsign;
212 ix = hx & 0x7fff;
213 if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */
214 if (ix == BIAS + LDBL_MAX_EXP) {
215 if (hx & 0x8000) /* x is -Inf or -NaN */
216 RETURNF(-1 / x - 1);
217 RETURNF(x + x); /* x is +Inf or +NaN */
218 }
219 if (x > o_threshold)
220 RETURNF(huge * huge);
221 /*
222 * expm1l() never underflows, but it must avoid
223 * unrepresentable large negative exponents. We used a
224 * much smaller threshold for large |x| above than in
225 * expl() so as to handle not so large negative exponents
226 * in the same way as large ones here.
227 */
228 if (hx & 0x8000) /* x <= -128 */
229 RETURNF(tiny - 1); /* good for x < -114ln2 - eps */
230 }
231
232 ENTERI();
233
234 if (T1 < x && x < T2) {
235 x2 = x * x;
236 dx = x;
237
238 if (x < T3) {
239 if (ix < BIAS - 113) { /* |x| < 0x1p-113 */
240 /* x (rounded) with inexact if x != 0: */
241 RETURNI(x == 0 ? x :
242 (0x1p200 * x + fabsl(x)) * 0x1p-200);
243 }
244 q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 +
245 x * (C7 + x * (C8 + x * (C9 + x * (C10 +
246 x * (C11 + x * (C12 + x * (C13 +
247 dx * (C14 + dx * (C15 + dx * (C16 +
248 dx * (C17 + dx * C18))))))))))))));
249 } else {
250 q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 +
251 x * (D7 + x * (D8 + x * (D9 + x * (D10 +
252 x * (D11 + x * (D12 + x * (D13 +
253 dx * (D14 + dx * (D15 + dx * (D16 +
254 dx * D17)))))))))))));
255 }
256
257 x_hi = (float)x;
258 x_lo = x - x_hi;
259 hx2_hi = x_hi * x_hi / 2;
260 hx2_lo = x_lo * (x + x_hi) / 2;
261 if (ix >= BIAS - 7)
262 RETURNI((hx2_hi + x_hi) + (hx2_lo + x_lo + q));
263 else
264 RETURNI(x + (hx2_lo + q + hx2_hi));
265 }
266
267 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
268 fn = rnint((double)x * INV_L);
269 n = irint(fn);
270 n2 = (unsigned)n % INTERVALS;
271 k = n >> LOG2_INTERVALS;
272 r1 = x - fn * L1;
273 r2 = fn * -L2;
274 r = r1 + r2;
275
276 /* Prepare scale factor. */
277 v.e = 1;
278 v.xbits.expsign = BIAS + k;
279 twopk = v.e;
280
281 /*
282 * Evaluate lower terms of
283 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
284 */
285 dr = r;
286 q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 +
287 dr * (A7 + dr * (A8 + dr * (A9 + dr * A10))))))));
288
289 t = tbl[n2].lo + tbl[n2].hi;
290
291 if (k == 0) {
292 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
293 tbl[n2].hi * r1);
294 RETURNI(t);
295 }
296 if (k == -1) {
297 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
298 tbl[n2].hi * r1);
299 RETURNI(t / 2);
300 }
301 if (k < -7) {
302 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
303 RETURNI(t * twopk - 1);
304 }
305 if (k > 2 * LDBL_MANT_DIG - 1) {
306 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
307 if (k == LDBL_MAX_EXP)
308 RETURNI(t * 2 * 0x1p16383L - 1);
309 RETURNI(t * twopk - 1);
310 }
311
312 v.xbits.expsign = BIAS - k;
313 twomk = v.e;
314
315 if (k > LDBL_MANT_DIG - 1)
316 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
317 else
318 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
319 RETURNI(t * twopk);
320 }
321