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