s_expl.c revision 330897 
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: stable/11/lib/msun/ld80/s_expl.c 330897 2018-03-14 03:19:51Z eadler $");
33
34/**
35 * Compute the exponential of x for Intel 80-bit format.  This is based on:
36 *
37 *   PTP Tang, "Table-driven implementation of the exponential function
38 *   in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15,
39 *   144-157 (1989).
40 *
41 * where the 32 table entries have been expanded to INTERVALS (see below).
42 */
43
44#include <float.h>
45
46#ifdef __i386__
47#include <ieeefp.h>
48#endif
49
50#include "fpmath.h"
51#include "math.h"
52#include "math_private.h"
53#include "k_expl.h"
54
55/* XXX Prevent compilers from erroneously constant folding these: */
56static const volatile long double
57huge = 0x1p10000L,
58tiny = 0x1p-10000L;
59
60static const long double
61twom10000 = 0x1p-10000L;
62
63static const union IEEEl2bits
64/* log(2**16384 - 0.5) rounded towards zero: */
65/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
66o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13,  11356.5234062941439488L),
67#define o_threshold	 (o_thresholdu.e)
68/* log(2**(-16381-64-1)) rounded towards zero: */
69u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
70#define u_threshold	 (u_thresholdu.e)
71
72long double
73expl(long double x)
74{
75	union IEEEl2bits u;
76	long double hi, lo, t, twopk;
77	int k;
78	uint16_t hx, ix;
79
80	DOPRINT_START(&x);
81
82	/* Filter out exceptional cases. */
83	u.e = x;
84	hx = u.xbits.expsign;
85	ix = hx & 0x7fff;
86	if (ix >= BIAS + 13) {		/* |x| >= 8192 or x is NaN */
87		if (ix == BIAS + LDBL_MAX_EXP) {
88			if (hx & 0x8000)  /* x is -Inf, -NaN or unsupported */
89				RETURNP(-1 / x);
90			RETURNP(x + x);	/* x is +Inf, +NaN or unsupported */
91		}
92		if (x > o_threshold)
93			RETURNP(huge * huge);
94		if (x < u_threshold)
95			RETURNP(tiny * tiny);
96	} else if (ix < BIAS - 75) {	/* |x| < 0x1p-75 (includes pseudos) */
97		RETURN2P(1, x);		/* 1 with inexact iff x != 0 */
98	}
99
100	ENTERI();
101
102	twopk = 1;
103	__k_expl(x, &hi, &lo, &k);
104	t = SUM2P(hi, lo);
105
106	/* Scale by 2**k. */
107	if (k >= LDBL_MIN_EXP) {
108		if (k == LDBL_MAX_EXP)
109			RETURNI(t * 2 * 0x1p16383L);
110		SET_LDBL_EXPSIGN(twopk, BIAS + k);
111		RETURNI(t * twopk);
112	} else {
113		SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
114		RETURNI(t * twopk * twom10000);
115	}
116}
117
118/**
119 * Compute expm1l(x) for Intel 80-bit format.  This is based on:
120 *
121 *   PTP Tang, "Table-driven implementation of the Expm1 function
122 *   in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18,
123 *   211-222 (1992).
124 */
125
126/*
127 * Our T1 and T2 are chosen to be approximately the points where method
128 * A and method B have the same accuracy.  Tang's T1 and T2 are the
129 * points where method A's accuracy changes by a full bit.  For Tang,
130 * this drop in accuracy makes method A immediately less accurate than
131 * method B, but our larger INTERVALS makes method A 2 bits more
132 * accurate so it remains the most accurate method significantly
133 * closer to the origin despite losing the full bit in our extended
134 * range for it.
135 */
136static const double
137T1 = -0.1659,				/* ~-30.625/128 * log(2) */
138T2 =  0.1659;				/* ~30.625/128 * log(2) */
139
140/*
141 * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]:
142 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6
143 *
144 * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits,
145 * but unlike for ld128 we can't drop any terms.
146 */
147static const union IEEEl2bits
148B3 = LD80C(0xaaaaaaaaaaaaaaab, -3,  1.66666666666666666671e-1L),
149B4 = LD80C(0xaaaaaaaaaaaaaaac, -5,  4.16666666666666666712e-2L);
150
151static const double
152B5  =  8.3333333333333245e-3,		/*  0x1.111111111110cp-7 */
153B6  =  1.3888888888888861e-3,		/*  0x1.6c16c16c16c0ap-10 */
154B7  =  1.9841269841532042e-4,		/*  0x1.a01a01a0319f9p-13 */
155B8  =  2.4801587302069236e-5,		/*  0x1.a01a01a03cbbcp-16 */
156B9  =  2.7557316558468562e-6,		/*  0x1.71de37fd33d67p-19 */
157B10 =  2.7557315829785151e-7,		/*  0x1.27e4f91418144p-22 */
158B11 =  2.5063168199779829e-8,		/*  0x1.ae94fabdc6b27p-26 */
159B12 =  2.0887164654459567e-9;		/*  0x1.1f122d6413fe1p-29 */
160
161long double
162expm1l(long double x)
163{
164	union IEEEl2bits u, v;
165	long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi;
166	long double x_lo, x2, z;
167	long double x4;
168	int k, n, n2;
169	uint16_t hx, ix;
170
171	DOPRINT_START(&x);
172
173	/* Filter out exceptional cases. */
174	u.e = x;
175	hx = u.xbits.expsign;
176	ix = hx & 0x7fff;
177	if (ix >= BIAS + 6) {		/* |x| >= 64 or x is NaN */
178		if (ix == BIAS + LDBL_MAX_EXP) {
179			if (hx & 0x8000)  /* x is -Inf, -NaN or unsupported */
180				RETURNP(-1 / x - 1);
181			RETURNP(x + x);	/* x is +Inf, +NaN or unsupported */
182		}
183		if (x > o_threshold)
184			RETURNP(huge * huge);
185		/*
186		 * expm1l() never underflows, but it must avoid
187		 * unrepresentable large negative exponents.  We used a
188		 * much smaller threshold for large |x| above than in
189		 * expl() so as to handle not so large negative exponents
190		 * in the same way as large ones here.
191		 */
192		if (hx & 0x8000)	/* x <= -64 */
193			RETURN2P(tiny, -1);	/* good for x < -65ln2 - eps */
194	}
195
196	ENTERI();
197
198	if (T1 < x && x < T2) {
199		if (ix < BIAS - 74) {	/* |x| < 0x1p-74 (includes pseudos) */
200			/* x (rounded) with inexact if x != 0: */
201			RETURNPI(x == 0 ? x :
202			    (0x1p100 * x + fabsl(x)) * 0x1p-100);
203		}
204
205		x2 = x * x;
206		x4 = x2 * x2;
207		q = x4 * (x2 * (x4 *
208		    /*
209		     * XXX the number of terms is no longer good for
210		     * pairwise grouping of all except B3, and the
211		     * grouping is no longer from highest down.
212		     */
213		    (x2 *            B12  + (x * B11 + B10)) +
214		    (x2 * (x * B9 +  B8) +  (x * B7 +  B6))) +
215			  (x * B5 +  B4.e)) + x2 * x * B3.e;
216
217		x_hi = (float)x;
218		x_lo = x - x_hi;
219		hx2_hi = x_hi * x_hi / 2;
220		hx2_lo = x_lo * (x + x_hi) / 2;
221		if (ix >= BIAS - 7)
222			RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q);
223		else
224			RETURN2PI(x, hx2_lo + q + hx2_hi);
225	}
226
227	/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
228	/* Use a specialized rint() to get fn.  Assume round-to-nearest. */
229	fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
230#if defined(HAVE_EFFICIENT_IRINTL)
231	n = irintl(fn);
232#elif defined(HAVE_EFFICIENT_IRINT)
233	n = irint(fn);
234#else
235	n = (int)fn;
236#endif
237	n2 = (unsigned)n % INTERVALS;
238	k = n >> LOG2_INTERVALS;
239	r1 = x - fn * L1;
240	r2 = fn * -L2;
241	r = r1 + r2;
242
243	/* Prepare scale factor. */
244	v.e = 1;
245	v.xbits.expsign = BIAS + k;
246	twopk = v.e;
247
248	/*
249	 * Evaluate lower terms of
250	 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
251	 */
252	z = r * r;
253	q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
254
255	t = (long double)tbl[n2].lo + tbl[n2].hi;
256
257	if (k == 0) {
258		t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
259		    tbl[n2].hi * r1);
260		RETURNI(t);
261	}
262	if (k == -1) {
263		t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
264		    tbl[n2].hi * r1);
265		RETURNI(t / 2);
266	}
267	if (k < -7) {
268		t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
269		RETURNI(t * twopk - 1);
270	}
271	if (k > 2 * LDBL_MANT_DIG - 1) {
272		t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
273		if (k == LDBL_MAX_EXP)
274			RETURNI(t * 2 * 0x1p16383L - 1);
275		RETURNI(t * twopk - 1);
276	}
277
278	v.xbits.expsign = BIAS - k;
279	twomk = v.e;
280
281	if (k > LDBL_MANT_DIG - 1)
282		t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
283	else
284		t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
285	RETURNI(t * twopk);
286}
287