1/*-
2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
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__FBSDID("$FreeBSD$");
33
34/*
35 * ld128 version of s_expl.c.  See ../ld80/s_expl.c for most comments.
36 */
37
38#include <float.h>
39
40#include "fpmath.h"
41#include "math.h"
42#include "math_private.h"
43#include "k_expl.h"
44
45/* XXX Prevent compilers from erroneously constant folding these: */
46static const volatile long double
47huge = 0x1p10000L,
48tiny = 0x1p-10000L;
49
50static const long double
51twom10000 = 0x1p-10000L;
52
53static const long double
54/* log(2**16384 - 0.5) rounded towards zero: */
55/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
56o_threshold =  11356.523406294143949491931077970763428L,
57/* log(2**(-16381-64-1)) rounded towards zero: */
58u_threshold = -11433.462743336297878837243843452621503L;
59
60long double
61expl(long double x)
62{
63	union IEEEl2bits u;
64	long double hi, lo, t, twopk;
65	int k;
66	uint16_t hx, ix;
67
68	DOPRINT_START(&x);
69
70	/* Filter out exceptional cases. */
71	u.e = x;
72	hx = u.xbits.expsign;
73	ix = hx & 0x7fff;
74	if (ix >= BIAS + 13) {		/* |x| >= 8192 or x is NaN */
75		if (ix == BIAS + LDBL_MAX_EXP) {
76			if (hx & 0x8000)  /* x is -Inf or -NaN */
77				RETURNP(-1 / x);
78			RETURNP(x + x);	/* x is +Inf or +NaN */
79		}
80		if (x > o_threshold)
81			RETURNP(huge * huge);
82		if (x < u_threshold)
83			RETURNP(tiny * tiny);
84	} else if (ix < BIAS - 114) {	/* |x| < 0x1p-114 */
85		RETURN2P(1, x);		/* 1 with inexact iff x != 0 */
86	}
87
88	ENTERI();
89
90	twopk = 1;
91	__k_expl(x, &hi, &lo, &k);
92	t = SUM2P(hi, lo);
93
94	/* Scale by 2**k. */
95	/*
96	 * XXX sparc64 multiplication was so slow that scalbnl() is faster,
97	 * but performance on aarch64 and riscv hasn't yet been quantified.
98	 */
99	if (k >= LDBL_MIN_EXP) {
100		if (k == LDBL_MAX_EXP)
101			RETURNI(t * 2 * 0x1p16383L);
102		SET_LDBL_EXPSIGN(twopk, BIAS + k);
103		RETURNI(t * twopk);
104	} else {
105		SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
106		RETURNI(t * twopk * twom10000);
107	}
108}
109
110/*
111 * Our T1 and T2 are chosen to be approximately the points where method
112 * A and method B have the same accuracy.  Tang's T1 and T2 are the
113 * points where method A's accuracy changes by a full bit.  For Tang,
114 * this drop in accuracy makes method A immediately less accurate than
115 * method B, but our larger INTERVALS makes method A 2 bits more
116 * accurate so it remains the most accurate method significantly
117 * closer to the origin despite losing the full bit in our extended
118 * range for it.
119 *
120 * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2].
121 * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear
122 * in both subintervals, so set T3 = 2**-5, which places the condition
123 * into the [T1, T3] interval.
124 *
125 * XXX we now do this more to (partially) balance the number of terms
126 * in the C and D polys than to avoid checking the condition in both
127 * intervals.
128 *
129 * XXX these micro-optimizations are excessive.
130 */
131static const double
132T1 = -0.1659,				/* ~-30.625/128 * log(2) */
133T2 =  0.1659,				/* ~30.625/128 * log(2) */
134T3 =  0.03125;
135
136/*
137 * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]:
138 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03
139 *
140 * XXX none of the long double C or D coeffs except C10 is correctly printed.
141 * If you re-print their values in %.35Le format, the result is always
142 * different.  For example, the last 2 digits in C3 should be 59, not 67.
143 * 67 is apparently from rounding an extra-precision value to 36 decimal
144 * places.
145 */
146static const long double
147C3  =  1.66666666666666666666666666666666667e-1L,
148C4  =  4.16666666666666666666666666666666645e-2L,
149C5  =  8.33333333333333333333333333333371638e-3L,
150C6  =  1.38888888888888888888888888891188658e-3L,
151C7  =  1.98412698412698412698412697235950394e-4L,
152C8  =  2.48015873015873015873015112487849040e-5L,
153C9  =  2.75573192239858906525606685484412005e-6L,
154C10 =  2.75573192239858906612966093057020362e-7L,
155C11 =  2.50521083854417203619031960151253944e-8L,
156C12 =  2.08767569878679576457272282566520649e-9L,
157C13 =  1.60590438367252471783548748824255707e-10L;
158
159/*
160 * XXX this has 1 more coeff than needed.
161 * XXX can start the double coeffs but not the double mults at C10.
162 * With my coeffs (C10-C17 double; s = best_s):
163 * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]:
164 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
165 */
166static const double
167C14 =  1.1470745580491932e-11,		/*  0x1.93974a81dae30p-37 */
168C15 =  7.6471620181090468e-13,		/*  0x1.ae7f3820adab1p-41 */
169C16 =  4.7793721460260450e-14,		/*  0x1.ae7cd18a18eacp-45 */
170C17 =  2.8074757356658877e-15,		/*  0x1.949992a1937d9p-49 */
171C18 =  1.4760610323699476e-16;		/*  0x1.545b43aabfbcdp-53 */
172
173/*
174 * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]:
175 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44
176 */
177static const long double
178D3  =  1.66666666666666666666666666666682245e-1L,
179D4  =  4.16666666666666666666666666634228324e-2L,
180D5  =  8.33333333333333333333333364022244481e-3L,
181D6  =  1.38888888888888888888887138722762072e-3L,
182D7  =  1.98412698412698412699085805424661471e-4L,
183D8  =  2.48015873015873015687993712101479612e-5L,
184D9  =  2.75573192239858944101036288338208042e-6L,
185D10 =  2.75573192239853161148064676533754048e-7L,
186D11 =  2.50521083855084570046480450935267433e-8L,
187D12 =  2.08767569819738524488686318024854942e-9L,
188D13 =  1.60590442297008495301927448122499313e-10L;
189
190/*
191 * XXX this has 1 more coeff than needed.
192 * XXX can start the double coeffs but not the double mults at D11.
193 * With my coeffs (D11-D16 double):
194 * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]:
195 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
196 */
197static const double
198D14 =  1.1470726176204336e-11,		/*  0x1.93971dc395d9ep-37 */
199D15 =  7.6478532249581686e-13,		/*  0x1.ae892e3D16fcep-41 */
200D16 =  4.7628892832607741e-14,		/*  0x1.ad00Dfe41feccp-45 */
201D17 =  3.0524857220358650e-15;		/*  0x1.D7e8d886Df921p-49 */
202
203long double
204expm1l(long double x)
205{
206	union IEEEl2bits u, v;
207	long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi;
208	long double x_lo, x2;
209	double dr, dx, fn, r2;
210	int k, n, n2;
211	uint16_t hx, ix;
212
213	DOPRINT_START(&x);
214
215	/* Filter out exceptional cases. */
216	u.e = x;
217	hx = u.xbits.expsign;
218	ix = hx & 0x7fff;
219	if (ix >= BIAS + 7) {		/* |x| >= 128 or x is NaN */
220		if (ix == BIAS + LDBL_MAX_EXP) {
221			if (hx & 0x8000)  /* x is -Inf or -NaN */
222				RETURNP(-1 / x - 1);
223			RETURNP(x + x);	/* x is +Inf or +NaN */
224		}
225		if (x > o_threshold)
226			RETURNP(huge * huge);
227		/*
228		 * expm1l() never underflows, but it must avoid
229		 * unrepresentable large negative exponents.  We used a
230		 * much smaller threshold for large |x| above than in
231		 * expl() so as to handle not so large negative exponents
232		 * in the same way as large ones here.
233		 */
234		if (hx & 0x8000)	/* x <= -128 */
235			RETURN2P(tiny, -1);	/* good for x < -114ln2 - eps */
236	}
237
238	ENTERI();
239
240	if (T1 < x && x < T2) {
241		x2 = x * x;
242		dx = x;
243
244		if (x < T3) {
245			if (ix < BIAS - 113) {	/* |x| < 0x1p-113 */
246				/* x (rounded) with inexact if x != 0: */
247				RETURNPI(x == 0 ? x :
248				    (0x1p200 * x + fabsl(x)) * 0x1p-200);
249			}
250			q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 +
251			    x * (C7 + x * (C8 + x * (C9 + x * (C10 +
252			    x * (C11 + x * (C12 + x * (C13 +
253			    dx * (C14 + dx * (C15 + dx * (C16 +
254			    dx * (C17 + dx * C18))))))))))))));
255		} else {
256			q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 +
257			    x * (D7 + x * (D8 + x * (D9 + x * (D10 +
258			    x * (D11 + x * (D12 + x * (D13 +
259			    dx * (D14 + dx * (D15 + dx * (D16 +
260			    dx * D17)))))))))))));
261		}
262
263		x_hi = (float)x;
264		x_lo = x - x_hi;
265		hx2_hi = x_hi * x_hi / 2;
266		hx2_lo = x_lo * (x + x_hi) / 2;
267		if (ix >= BIAS - 7)
268			RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q);
269		else
270			RETURN2PI(x, hx2_lo + q + hx2_hi);
271	}
272
273	/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
274	fn = rnint((double)x * INV_L);
275	n = irint(fn);
276	n2 = (unsigned)n % INTERVALS;
277	k = n >> LOG2_INTERVALS;
278	r1 = x - fn * L1;
279	r2 = fn * -L2;
280	r = r1 + r2;
281
282	/* Prepare scale factor. */
283	v.e = 1;
284	v.xbits.expsign = BIAS + k;
285	twopk = v.e;
286
287	/*
288	 * Evaluate lower terms of
289	 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
290	 */
291	dr = r;
292	q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 +
293	    dr * (A7 + dr * (A8 + dr * (A9 + dr * A10))))))));
294
295	t = tbl[n2].lo + tbl[n2].hi;
296
297	if (k == 0) {
298		t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
299		    tbl[n2].hi * r1);
300		RETURNI(t);
301	}
302	if (k == -1) {
303		t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
304		    tbl[n2].hi * r1);
305		RETURNI(t / 2);
306	}
307	if (k < -7) {
308		t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
309		RETURNI(t * twopk - 1);
310	}
311	if (k > 2 * LDBL_MANT_DIG - 1) {
312		t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
313		if (k == LDBL_MAX_EXP)
314			RETURNI(t * 2 * 0x1p16383L - 1);
315		RETURNI(t * twopk - 1);
316	}
317
318	v.xbits.expsign = BIAS - k;
319	twomk = v.e;
320
321	if (k > LDBL_MANT_DIG - 1)
322		t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
323	else
324		t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
325	RETURNI(t * twopk);
326}
327