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  * 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: */
44 static const volatile long double
45 huge = 0x1p10000L,
46 tiny = 0x1p-10000L;
47 
48 static const long double
49 twom10000 = 0x1p-10000L;
50 
51 static 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: */
54 o_threshold =  11356.523406294143949491931077970763428L,
55 /* log(2**(-16381-64-1)) rounded towards zero: */
56 u_threshold = -11433.462743336297878837243843452621503L;
57 
58 long double
expl(long double x)59 expl(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  */
126 static const double
127 T1 = -0.1659,				/* ~-30.625/128 * log(2) */
128 T2 =  0.1659,				/* ~30.625/128 * log(2) */
129 T3 =  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  */
141 static const long double
142 C3  =  1.66666666666666666666666666666666667e-1L,
143 C4  =  4.16666666666666666666666666666666645e-2L,
144 C5  =  8.33333333333333333333333333333371638e-3L,
145 C6  =  1.38888888888888888888888888891188658e-3L,
146 C7  =  1.98412698412698412698412697235950394e-4L,
147 C8  =  2.48015873015873015873015112487849040e-5L,
148 C9  =  2.75573192239858906525606685484412005e-6L,
149 C10 =  2.75573192239858906612966093057020362e-7L,
150 C11 =  2.50521083854417203619031960151253944e-8L,
151 C12 =  2.08767569878679576457272282566520649e-9L,
152 C13 =  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  */
161 static const double
162 C14 =  1.1470745580491932e-11,		/*  0x1.93974a81dae30p-37 */
163 C15 =  7.6471620181090468e-13,		/*  0x1.ae7f3820adab1p-41 */
164 C16 =  4.7793721460260450e-14,		/*  0x1.ae7cd18a18eacp-45 */
165 C17 =  2.8074757356658877e-15,		/*  0x1.949992a1937d9p-49 */
166 C18 =  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  */
172 static const long double
173 D3  =  1.66666666666666666666666666666682245e-1L,
174 D4  =  4.16666666666666666666666666634228324e-2L,
175 D5  =  8.33333333333333333333333364022244481e-3L,
176 D6  =  1.38888888888888888888887138722762072e-3L,
177 D7  =  1.98412698412698412699085805424661471e-4L,
178 D8  =  2.48015873015873015687993712101479612e-5L,
179 D9  =  2.75573192239858944101036288338208042e-6L,
180 D10 =  2.75573192239853161148064676533754048e-7L,
181 D11 =  2.50521083855084570046480450935267433e-8L,
182 D12 =  2.08767569819738524488686318024854942e-9L,
183 D13 =  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  */
192 static const double
193 D14 =  1.1470726176204336e-11,		/*  0x1.93971dc395d9ep-37 */
194 D15 =  7.6478532249581686e-13,		/*  0x1.ae892e3D16fcep-41 */
195 D16 =  4.7628892832607741e-14,		/*  0x1.ad00Dfe41feccp-45 */
196 D17 =  3.0524857220358650e-15;		/*  0x1.D7e8d886Df921p-49 */
197 
198 long double
expm1l(long double x)199 expm1l(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