newlib/libm/ld128/s_expl.c

   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 "../ld/fpmath.h"
  41 #include "math.h"
  42 #include "../ld/math_private.h"
  43 #include "k_expl.h"
  44
  45 /* XXX Prevent compilers from erroneously constant folding these: */
  46 static const volatile long double
  47 huge = 0x1p10000L,
  48 tiny = 0x1p-10000L;
  49
  50 static const long double
  51 twom10000 = 0x1p-10000L;
  52
  53 static 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: */
  56 o_threshold =  11356.523406294143949491931077970763428L,
  57 /* log(2**(-16381-64-1)) rounded towards zero: */
  58 u_threshold = -11433.462743336297878837243843452621503L;
  59
  60 long double
  61 expl(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         /*
  96          * XXX sparc64 multiplication was so slow that scalbnl() is faster,
  97          * but performance on aarch64 and riscv hasn't yet been quantified.
  98          */
  99         if (k >= LDBL_MIN_EXP) {
 100                 if (k == LDBL_MAX_EXP)
 101                         RETURNI(t * 2 * 0x1p16383L);
 102                 SET_LDBL_EXPSIGN(twopk, BIAS + k);
 103                 RETURNI(t * twopk);
 104         } else {
 105                 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
 106                 RETURNI(t * twopk * twom10000);
 107         }
 108 }
 109
 110 /*
 111  * Our T1 and T2 are chosen to be approximately the points where method
 112  * A and method B have the same accuracy.  Tang's T1 and T2 are the
 113  * points where method A's accuracy changes by a full bit.  For Tang,
 114  * this drop in accuracy makes method A immediately less accurate than
 115  * method B, but our larger INTERVALS makes method A 2 bits more
 116  * accurate so it remains the most accurate method significantly
 117  * closer to the origin despite losing the full bit in our extended
 118  * range for it.
 119  *
 120  * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2].
 121  * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear
 122  * in both subintervals, so set T3 = 2**-5, which places the condition
 123  * into the [T1, T3] interval.
 124  *
 125  * XXX we now do this more to (partially) balance the number of terms
 126  * in the C and D polys than to avoid checking the condition in both
 127  * intervals.
 128  *
 129  * XXX these micro-optimizations are excessive.
 130  */
 131 static const double
 132 T1 = -0.1659,                           /* ~-30.625/128 * log(2) */
 133 T2 =  0.1659,                           /* ~30.625/128 * log(2) */
 134 T3 =  0.03125;
 135
 136 /*
 137  * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]:
 138  * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03
 139  *
 140  * XXX none of the long double C or D coeffs except C10 is correctly printed.
 141  * If you re-print their values in %.35Le format, the result is always
 142  * different.  For example, the last 2 digits in C3 should be 59, not 67.
 143  * 67 is apparently from rounding an extra-precision value to 36 decimal
 144  * places.
 145  */
 146 static const long double
 147 C3  =  1.66666666666666666666666666666666667e-1L,
 148 C4  =  4.16666666666666666666666666666666645e-2L,
 149 C5  =  8.33333333333333333333333333333371638e-3L,
 150 C6  =  1.38888888888888888888888888891188658e-3L,
 151 C7  =  1.98412698412698412698412697235950394e-4L,
 152 C8  =  2.48015873015873015873015112487849040e-5L,
 153 C9  =  2.75573192239858906525606685484412005e-6L,
 154 C10 =  2.75573192239858906612966093057020362e-7L,
 155 C11 =  2.50521083854417203619031960151253944e-8L,
 156 C12 =  2.08767569878679576457272282566520649e-9L,
 157 C13 =  1.60590438367252471783548748824255707e-10L;
 158
 159 /*
 160  * XXX this has 1 more coeff than needed.
 161  * XXX can start the double coeffs but not the double mults at C10.
 162  * With my coeffs (C10-C17 double; s = best_s):
 163  * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]:
 164  * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
 165  */
 166 static const double
 167 C14 =  1.1470745580491932e-11,          /*  0x1.93974a81dae30p-37 */
 168 C15 =  7.6471620181090468e-13,          /*  0x1.ae7f3820adab1p-41 */
 169 C16 =  4.7793721460260450e-14,          /*  0x1.ae7cd18a18eacp-45 */
 170 C17 =  2.8074757356658877e-15,          /*  0x1.949992a1937d9p-49 */
 171 C18 =  1.4760610323699476e-16;          /*  0x1.545b43aabfbcdp-53 */
 172
 173 /*
 174  * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]:
 175  * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44
 176  */
 177 static const long double
 178 D3  =  1.66666666666666666666666666666682245e-1L,
 179 D4  =  4.16666666666666666666666666634228324e-2L,
 180 D5  =  8.33333333333333333333333364022244481e-3L,
 181 D6  =  1.38888888888888888888887138722762072e-3L,
 182 D7  =  1.98412698412698412699085805424661471e-4L,
 183 D8  =  2.48015873015873015687993712101479612e-5L,
 184 D9  =  2.75573192239858944101036288338208042e-6L,
 185 D10 =  2.75573192239853161148064676533754048e-7L,
 186 D11 =  2.50521083855084570046480450935267433e-8L,
 187 D12 =  2.08767569819738524488686318024854942e-9L,
 188 D13 =  1.60590442297008495301927448122499313e-10L;
 189
 190 /*
 191  * XXX this has 1 more coeff than needed.
 192  * XXX can start the double coeffs but not the double mults at D11.
 193  * With my coeffs (D11-D16 double):
 194  * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]:
 195  * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
 196  */
 197 static const double
 198 D14 =  1.1470726176204336e-11,          /*  0x1.93971dc395d9ep-37 */
 199 D15 =  7.6478532249581686e-13,          /*  0x1.ae892e3D16fcep-41 */
 200 D16 =  4.7628892832607741e-14,          /*  0x1.ad00Dfe41feccp-45 */
 201 D17 =  3.0524857220358650e-15;          /*  0x1.D7e8d886Df921p-49 */
 202
 203 long double
 204 expm1l(long double x)
 205 {
 206         union IEEEl2bits u, v;
 207         long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi;
 208         long double x_lo, x2;
 209         double dr, dx, fn, r2;
 210         int k, n, n2;
 211         uint16_t hx, ix;
 212
 213         DOPRINT_START(&x);
 214
 215         /* Filter out exceptional cases. */
 216         u.e = x;
 217         hx = u.xbits.expsign;
 218         ix = hx & 0x7fff;
 219         if (ix >= BIAS + 7) {           /* |x| >= 128 or x is NaN */
 220                 if (ix == BIAS + LDBL_MAX_EXP) {
 221                         if (hx & 0x8000)  /* x is -Inf or -NaN */
 222                                 RETURNP(-1 / x - 1);
 223                         RETURNP(x + x); /* x is +Inf or +NaN */
 224                 }
 225                 if (x > o_threshold)
 226                         RETURNP(huge * huge);
 227                 /*
 228                  * expm1l() never underflows, but it must avoid
 229                  * unrepresentable large negative exponents.  We used a
 230                  * much smaller threshold for large |x| above than in
 231                  * expl() so as to handle not so large negative exponents
 232                  * in the same way as large ones here.
 233                  */
 234                 if (hx & 0x8000)        /* x <= -128 */
 235                         RETURN2P(tiny, -1);     /* good for x < -114ln2 - eps */
 236         }
 237
 238         ENTERI();
 239
 240         if (T1 < x && x < T2) {
 241                 x2 = x * x;
 242                 dx = x;
 243
 244                 if (x < T3) {
 245                         if (ix < BIAS - 113) {  /* |x| < 0x1p-113 */
 246                                 /* x (rounded) with inexact if x != 0: */
 247                                 RETURNPI(x == 0 ? x :
 248                                     (0x1p200 * x + fabsl(x)) * 0x1p-200);
 249                         }
 250                         q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 +
 251                             x * (C7 + x * (C8 + x * (C9 + x * (C10 +
 252                             x * (C11 + x * (C12 + x * (C13 +
 253                             dx * (C14 + dx * (C15 + dx * (C16 +
 254                             dx * (C17 + dx * C18))))))))))))));
 255                 } else {
 256                         q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 +
 257                             x * (D7 + x * (D8 + x * (D9 + x * (D10 +
 258                             x * (D11 + x * (D12 + x * (D13 +
 259                             dx * (D14 + dx * (D15 + dx * (D16 +
 260                             dx * D17)))))))))))));
 261                 }
 262
 263                 x_hi = (float)x;
 264                 x_lo = x - x_hi;
 265                 hx2_hi = x_hi * x_hi / 2;
 266                 hx2_lo = x_lo * (x + x_hi) / 2;
 267                 if (ix >= BIAS - 7)
 268                         RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q);
 269                 else
 270                         RETURN2PI(x, hx2_lo + q + hx2_hi);
 271         }
 272
 273         /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
 274         fn = rnint((double)x * INV_L);
 275         n = irint(fn);
 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 }