s_expl.c revision 331722 
1/*-
2 * Copyright (c) 2009-2013 Steven G. Kargl
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 *    notice unmodified, this list of conditions, and the following
10 *    disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 *    notice, this list of conditions and the following disclaimer in the
13 *    documentation and/or other materials provided with the distribution.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
25 *
26 * Optimized by Bruce D. Evans.
27 */
28
29#include <sys/cdefs.h>
30__FBSDID("$FreeBSD: stable/11/lib/msun/ld128/s_expl.c 331722 2018-03-29 02:50:57Z eadler $");
31
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: */
44static const volatile long double
45huge = 0x1p10000L,
46tiny = 0x1p-10000L;
47
48static const long double
49twom10000 = 0x1p-10000L;
50
51static 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: */
54o_threshold =  11356.523406294143949491931077970763428L,
55/* log(2**(-16381-64-1)) rounded towards zero: */
56u_threshold = -11433.462743336297878837243843452621503L;
57
58long double
59expl(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	DOPRINT_START(&x);
67
68	/* Filter out exceptional cases. */
69	u.e = x;
70	hx = u.xbits.expsign;
71	ix = hx & 0x7fff;
72	if (ix >= BIAS + 13) {		/* |x| >= 8192 or x is NaN */
73		if (ix == BIAS + LDBL_MAX_EXP) {
74			if (hx & 0x8000)  /* x is -Inf or -NaN */
75				RETURNP(-1 / x);
76			RETURNP(x + x);	/* x is +Inf or +NaN */
77		}
78		if (x > o_threshold)
79			RETURNP(huge * huge);
80		if (x < u_threshold)
81			RETURNP(tiny * tiny);
82	} else if (ix < BIAS - 114) {	/* |x| < 0x1p-114 */
83		RETURN2P(1, x);		/* 1 with inexact iff x != 0 */
84	}
85
86	ENTERI();
87
88	twopk = 1;
89	__k_expl(x, &hi, &lo, &k);
90	t = SUM2P(hi, lo);
91
92	/* Scale by 2**k. */
93	/* XXX sparc64 multiplication is so slow that scalbnl() is faster. */
94	if (k >= LDBL_MIN_EXP) {
95		if (k == LDBL_MAX_EXP)
96			RETURNI(t * 2 * 0x1p16383L);
97		SET_LDBL_EXPSIGN(twopk, BIAS + k);
98		RETURNI(t * twopk);
99	} else {
100		SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
101		RETURNI(t * twopk * twom10000);
102	}
103}
104
105/*
106 * Our T1 and T2 are chosen to be approximately the points where method
107 * A and method B have the same accuracy.  Tang's T1 and T2 are the
108 * points where method A's accuracy changes by a full bit.  For Tang,
109 * this drop in accuracy makes method A immediately less accurate than
110 * method B, but our larger INTERVALS makes method A 2 bits more
111 * accurate so it remains the most accurate method significantly
112 * closer to the origin despite losing the full bit in our extended
113 * range for it.
114 *
115 * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2].
116 * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear
117 * in both subintervals, so set T3 = 2**-5, which places the condition
118 * into the [T1, T3] interval.
119 *
120 * XXX we now do this more to (partially) balance the number of terms
121 * in the C and D polys than to avoid checking the condition in both
122 * intervals.
123 *
124 * XXX these micro-optimizations are excessive.
125 */
126static const double
127T1 = -0.1659,				/* ~-30.625/128 * log(2) */
128T2 =  0.1659,				/* ~30.625/128 * log(2) */
129T3 =  0.03125;
130
131/*
132 * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]:
133 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03
134 *
135 * XXX none of the long double C or D coeffs except C10 is correctly printed.
136 * If you re-print their values in %.35Le format, the result is always
137 * different.  For example, the last 2 digits in C3 should be 59, not 67.
138 * 67 is apparently from rounding an extra-precision value to 36 decimal
139 * places.
140 */
141static const long double
142C3  =  1.66666666666666666666666666666666667e-1L,
143C4  =  4.16666666666666666666666666666666645e-2L,
144C5  =  8.33333333333333333333333333333371638e-3L,
145C6  =  1.38888888888888888888888888891188658e-3L,
146C7  =  1.98412698412698412698412697235950394e-4L,
147C8  =  2.48015873015873015873015112487849040e-5L,
148C9  =  2.75573192239858906525606685484412005e-6L,
149C10 =  2.75573192239858906612966093057020362e-7L,
150C11 =  2.50521083854417203619031960151253944e-8L,
151C12 =  2.08767569878679576457272282566520649e-9L,
152C13 =  1.60590438367252471783548748824255707e-10L;
153
154/*
155 * XXX this has 1 more coeff than needed.
156 * XXX can start the double coeffs but not the double mults at C10.
157 * With my coeffs (C10-C17 double; s = best_s):
158 * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]:
159 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
160 */
161static const double
162C14 =  1.1470745580491932e-11,		/*  0x1.93974a81dae30p-37 */
163C15 =  7.6471620181090468e-13,		/*  0x1.ae7f3820adab1p-41 */
164C16 =  4.7793721460260450e-14,		/*  0x1.ae7cd18a18eacp-45 */
165C17 =  2.8074757356658877e-15,		/*  0x1.949992a1937d9p-49 */
166C18 =  1.4760610323699476e-16;		/*  0x1.545b43aabfbcdp-53 */
167
168/*
169 * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]:
170 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44
171 */
172static const long double
173D3  =  1.66666666666666666666666666666682245e-1L,
174D4  =  4.16666666666666666666666666634228324e-2L,
175D5  =  8.33333333333333333333333364022244481e-3L,
176D6  =  1.38888888888888888888887138722762072e-3L,
177D7  =  1.98412698412698412699085805424661471e-4L,
178D8  =  2.48015873015873015687993712101479612e-5L,
179D9  =  2.75573192239858944101036288338208042e-6L,
180D10 =  2.75573192239853161148064676533754048e-7L,
181D11 =  2.50521083855084570046480450935267433e-8L,
182D12 =  2.08767569819738524488686318024854942e-9L,
183D13 =  1.60590442297008495301927448122499313e-10L;
184
185/*
186 * XXX this has 1 more coeff than needed.
187 * XXX can start the double coeffs but not the double mults at D11.
188 * With my coeffs (D11-D16 double):
189 * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]:
190 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
191 */
192static const double
193D14 =  1.1470726176204336e-11,		/*  0x1.93971dc395d9ep-37 */
194D15 =  7.6478532249581686e-13,		/*  0x1.ae892e3D16fcep-41 */
195D16 =  4.7628892832607741e-14,		/*  0x1.ad00Dfe41feccp-45 */
196D17 =  3.0524857220358650e-15;		/*  0x1.D7e8d886Df921p-49 */
197
198long double
199expm1l(long double x)
200{
201	union IEEEl2bits u, v;
202	long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi;
203	long double x_lo, x2;
204	double dr, dx, fn, r2;
205	int k, n, n2;
206	uint16_t hx, ix;
207
208	DOPRINT_START(&x);
209
210	/* Filter out exceptional cases. */
211	u.e = x;
212	hx = u.xbits.expsign;
213	ix = hx & 0x7fff;
214	if (ix >= BIAS + 7) {		/* |x| >= 128 or x is NaN */
215		if (ix == BIAS + LDBL_MAX_EXP) {
216			if (hx & 0x8000)  /* x is -Inf or -NaN */
217				RETURNP(-1 / x - 1);
218			RETURNP(x + x);	/* x is +Inf or +NaN */
219		}
220		if (x > o_threshold)
221			RETURNP(huge * huge);
222		/*
223		 * expm1l() never underflows, but it must avoid
224		 * unrepresentable large negative exponents.  We used a
225		 * much smaller threshold for large |x| above than in
226		 * expl() so as to handle not so large negative exponents
227		 * in the same way as large ones here.
228		 */
229		if (hx & 0x8000)	/* x <= -128 */
230			RETURN2P(tiny, -1);	/* good for x < -114ln2 - eps */
231	}
232
233	ENTERI();
234
235	if (T1 < x && x < T2) {
236		x2 = x * x;
237		dx = x;
238
239		if (x < T3) {
240			if (ix < BIAS - 113) {	/* |x| < 0x1p-113 */
241				/* x (rounded) with inexact if x != 0: */
242				RETURNPI(x == 0 ? x :
243				    (0x1p200 * x + fabsl(x)) * 0x1p-200);
244			}
245			q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 +
246			    x * (C7 + x * (C8 + x * (C9 + x * (C10 +
247			    x * (C11 + x * (C12 + x * (C13 +
248			    dx * (C14 + dx * (C15 + dx * (C16 +
249			    dx * (C17 + dx * C18))))))))))))));
250		} else {
251			q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 +
252			    x * (D7 + x * (D8 + x * (D9 + x * (D10 +
253			    x * (D11 + x * (D12 + x * (D13 +
254			    dx * (D14 + dx * (D15 + dx * (D16 +
255			    dx * D17)))))))))))));
256		}
257
258		x_hi = (float)x;
259		x_lo = x - x_hi;
260		hx2_hi = x_hi * x_hi / 2;
261		hx2_lo = x_lo * (x + x_hi) / 2;
262		if (ix >= BIAS - 7)
263			RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q);
264		else
265			RETURN2PI(x, hx2_lo + q + hx2_hi);
266	}
267
268	/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
269	/* Use a specialized rint() to get fn.  Assume round-to-nearest. */
270	fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52;
271#if defined(HAVE_EFFICIENT_IRINT)
272	n = irint(fn);
273#else
274	n = (int)fn;
275#endif
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