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("$FreeBSD$");
31
32/**
33 * Compute the exponential of x for Intel 80-bit format.  This is based on:
34 *
35 *   PTP Tang, "Table-driven implementation of the exponential function
36 *   in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15,
37 *   144-157 (1989).
38 *
39 * where the 32 table entries have been expanded to INTERVALS (see below).
40 */
41
42#include <float.h>
43
44#ifdef __i386__
45#include <ieeefp.h>
46#endif
47
48#include "fpmath.h"
49#include "math.h"
50#include "math_private.h"
51#include "k_expl.h"
52
53/* XXX Prevent compilers from erroneously constant folding these: */
54static const volatile long double
55huge = 0x1p10000L,
56tiny = 0x1p-10000L;
57
58static const long double
59twom10000 = 0x1p-10000L;
60
61static const union IEEEl2bits
62/* log(2**16384 - 0.5) rounded towards zero: */
63/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
64o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13,  11356.5234062941439488L),
65#define o_threshold	 (o_thresholdu.e)
66/* log(2**(-16381-64-1)) rounded towards zero: */
67u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
68#define u_threshold	 (u_thresholdu.e)
69
70long double
71expl(long double x)
72{
73	union IEEEl2bits u;
74	long double hi, lo, t, twopk;
75	int k;
76	uint16_t hx, ix;
77
78	DOPRINT_START(&x);
79
80	/* Filter out exceptional cases. */
81	u.e = x;
82	hx = u.xbits.expsign;
83	ix = hx & 0x7fff;
84	if (ix >= BIAS + 13) {		/* |x| >= 8192 or x is NaN */
85		if (ix == BIAS + LDBL_MAX_EXP) {
86			if (hx & 0x8000)  /* x is -Inf, -NaN or unsupported */
87				RETURNP(-1 / x);
88			RETURNP(x + x);	/* x is +Inf, +NaN or unsupported */
89		}
90		if (x > o_threshold)
91			RETURNP(huge * huge);
92		if (x < u_threshold)
93			RETURNP(tiny * tiny);
94	} else if (ix < BIAS - 75) {	/* |x| < 0x1p-75 (includes pseudos) */
95		RETURN2P(1, x);		/* 1 with inexact iff x != 0 */
96	}
97
98	ENTERI();
99
100	twopk = 1;
101	__k_expl(x, &hi, &lo, &k);
102	t = SUM2P(hi, lo);
103
104	/* Scale by 2**k. */
105	if (k >= LDBL_MIN_EXP) {
106		if (k == LDBL_MAX_EXP)
107			RETURNI(t * 2 * 0x1p16383L);
108		SET_LDBL_EXPSIGN(twopk, BIAS + k);
109		RETURNI(t * twopk);
110	} else {
111		SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
112		RETURNI(t * twopk * twom10000);
113	}
114}
115
116/**
117 * Compute expm1l(x) for Intel 80-bit format.  This is based on:
118 *
119 *   PTP Tang, "Table-driven implementation of the Expm1 function
120 *   in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18,
121 *   211-222 (1992).
122 */
123
124/*
125 * Our T1 and T2 are chosen to be approximately the points where method
126 * A and method B have the same accuracy.  Tang's T1 and T2 are the
127 * points where method A's accuracy changes by a full bit.  For Tang,
128 * this drop in accuracy makes method A immediately less accurate than
129 * method B, but our larger INTERVALS makes method A 2 bits more
130 * accurate so it remains the most accurate method significantly
131 * closer to the origin despite losing the full bit in our extended
132 * range for it.
133 */
134static const double
135T1 = -0.1659,				/* ~-30.625/128 * log(2) */
136T2 =  0.1659;				/* ~30.625/128 * log(2) */
137
138/*
139 * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]:
140 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6
141 *
142 * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits,
143 * but unlike for ld128 we can't drop any terms.
144 */
145static const union IEEEl2bits
146B3 = LD80C(0xaaaaaaaaaaaaaaab, -3,  1.66666666666666666671e-1L),
147B4 = LD80C(0xaaaaaaaaaaaaaaac, -5,  4.16666666666666666712e-2L);
148
149static const double
150B5  =  8.3333333333333245e-3,		/*  0x1.111111111110cp-7 */
151B6  =  1.3888888888888861e-3,		/*  0x1.6c16c16c16c0ap-10 */
152B7  =  1.9841269841532042e-4,		/*  0x1.a01a01a0319f9p-13 */
153B8  =  2.4801587302069236e-5,		/*  0x1.a01a01a03cbbcp-16 */
154B9  =  2.7557316558468562e-6,		/*  0x1.71de37fd33d67p-19 */
155B10 =  2.7557315829785151e-7,		/*  0x1.27e4f91418144p-22 */
156B11 =  2.5063168199779829e-8,		/*  0x1.ae94fabdc6b27p-26 */
157B12 =  2.0887164654459567e-9;		/*  0x1.1f122d6413fe1p-29 */
158
159long double
160expm1l(long double x)
161{
162	union IEEEl2bits u, v;
163	long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi;
164	long double x_lo, x2, z;
165	long double x4;
166	int k, n, n2;
167	uint16_t hx, ix;
168
169	DOPRINT_START(&x);
170
171	/* Filter out exceptional cases. */
172	u.e = x;
173	hx = u.xbits.expsign;
174	ix = hx & 0x7fff;
175	if (ix >= BIAS + 6) {		/* |x| >= 64 or x is NaN */
176		if (ix == BIAS + LDBL_MAX_EXP) {
177			if (hx & 0x8000)  /* x is -Inf, -NaN or unsupported */
178				RETURNP(-1 / x - 1);
179			RETURNP(x + x);	/* x is +Inf, +NaN or unsupported */
180		}
181		if (x > o_threshold)
182			RETURNP(huge * huge);
183		/*
184		 * expm1l() never underflows, but it must avoid
185		 * unrepresentable large negative exponents.  We used a
186		 * much smaller threshold for large |x| above than in
187		 * expl() so as to handle not so large negative exponents
188		 * in the same way as large ones here.
189		 */
190		if (hx & 0x8000)	/* x <= -64 */
191			RETURN2P(tiny, -1);	/* good for x < -65ln2 - eps */
192	}
193
194	ENTERI();
195
196	if (T1 < x && x < T2) {
197		if (ix < BIAS - 74) {	/* |x| < 0x1p-74 (includes pseudos) */
198			/* x (rounded) with inexact if x != 0: */
199			RETURNPI(x == 0 ? x :
200			    (0x1p100 * x + fabsl(x)) * 0x1p-100);
201		}
202
203		x2 = x * x;
204		x4 = x2 * x2;
205		q = x4 * (x2 * (x4 *
206		    /*
207		     * XXX the number of terms is no longer good for
208		     * pairwise grouping of all except B3, and the
209		     * grouping is no longer from highest down.
210		     */
211		    (x2 *            B12  + (x * B11 + B10)) +
212		    (x2 * (x * B9 +  B8) +  (x * B7 +  B6))) +
213			  (x * B5 +  B4.e)) + x2 * x * B3.e;
214
215		x_hi = (float)x;
216		x_lo = x - x_hi;
217		hx2_hi = x_hi * x_hi / 2;
218		hx2_lo = x_lo * (x + x_hi) / 2;
219		if (ix >= BIAS - 7)
220			RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q);
221		else
222			RETURN2PI(x, hx2_lo + q + hx2_hi);
223	}
224
225	/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
226	/* Use a specialized rint() to get fn.  Assume round-to-nearest. */
227	fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
228#if defined(HAVE_EFFICIENT_IRINTL)
229	n = irintl(fn);
230#elif defined(HAVE_EFFICIENT_IRINT)
231	n = irint(fn);
232#else
233	n = (int)fn;
234#endif
235	n2 = (unsigned)n % INTERVALS;
236	k = n >> LOG2_INTERVALS;
237	r1 = x - fn * L1;
238	r2 = fn * -L2;
239	r = r1 + r2;
240
241	/* Prepare scale factor. */
242	v.e = 1;
243	v.xbits.expsign = BIAS + k;
244	twopk = v.e;
245
246	/*
247	 * Evaluate lower terms of
248	 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
249	 */
250	z = r * r;
251	q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
252
253	t = (long double)tbl[n2].lo + tbl[n2].hi;
254
255	if (k == 0) {
256		t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
257		    tbl[n2].hi * r1);
258		RETURNI(t);
259	}
260	if (k == -1) {
261		t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
262		    tbl[n2].hi * r1);
263		RETURNI(t / 2);
264	}
265	if (k < -7) {
266		t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
267		RETURNI(t * twopk - 1);
268	}
269	if (k > 2 * LDBL_MANT_DIG - 1) {
270		t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
271		if (k == LDBL_MAX_EXP)
272			RETURNI(t * 2 * 0x1p16383L - 1);
273		RETURNI(t * twopk - 1);
274	}
275
276	v.xbits.expsign = BIAS - k;
277	twomk = v.e;
278
279	if (k > LDBL_MANT_DIG - 1)
280		t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
281	else
282		t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
283	RETURNI(t * twopk);
284}
285