newlib/libm/ld80/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  * 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 "../ld/fpmath.h"
  51 #include "math.h"
  52 #include "../ld/math_private.h"
  53 #include "k_expl.h"
  54
  55 /* XXX Prevent compilers from erroneously constant folding these: */
  56 static const volatile long double
  57 huge = 0x1p10000L,
  58 tiny = 0x1p-10000L;
  59
  60 static const long double
  61 twom10000 = 0x1p-10000L;
  62
  63 static 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: */
  66 o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13,  11356.5234062941439488L),
  67 #define o_threshold      (o_thresholdu.e)
  68 /* log(2**(-16381-64-1)) rounded towards zero: */
  69 u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
  70 #define u_threshold      (u_thresholdu.e)
  71
  72 long double
  73 expl(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  */
 136 static const double
 137 T1 = -0.1659,                           /* ~-30.625/128 * log(2) */
 138 T2 =  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  */
 147 static const union IEEEl2bits
 148 B3 = LD80C(0xaaaaaaaaaaaaaaab, -3,  1.66666666666666666671e-1L),
 149 B4 = LD80C(0xaaaaaaaaaaaaaaac, -5,  4.16666666666666666712e-2L);
 150
 151 static const double
 152 B5  =  8.3333333333333245e-3,           /*  0x1.111111111110cp-7 */
 153 B6  =  1.3888888888888861e-3,           /*  0x1.6c16c16c16c0ap-10 */
 154 B7  =  1.9841269841532042e-4,           /*  0x1.a01a01a0319f9p-13 */
 155 B8  =  2.4801587302069236e-5,           /*  0x1.a01a01a03cbbcp-16 */
 156 B9  =  2.7557316558468562e-6,           /*  0x1.71de37fd33d67p-19 */
 157 B10 =  2.7557315829785151e-7,           /*  0x1.27e4f91418144p-22 */
 158 B11 =  2.5063168199779829e-8,           /*  0x1.ae94fabdc6b27p-26 */
 159 B12 =  2.0887164654459567e-9;           /*  0x1.1f122d6413fe1p-29 */
 160
 161 long double
 162 expm1l(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         fn = rnintl(x * INV_L);
 229         n = irint(fn);
 230         n2 = (unsigned)n % INTERVALS;
 231         k = n >> LOG2_INTERVALS;
 232         r1 = x - fn * L1;
 233         r2 = fn * -L2;
 234         r = r1 + r2;
 235
 236         /* Prepare scale factor. */
 237         v.e = 1;
 238         v.xbits.expsign = BIAS + k;
 239         twopk = v.e;
 240
 241         /*
 242          * Evaluate lower terms of
 243          * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
 244          */
 245         z = r * r;
 246         q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
 247
 248         t = (long double)tbl[n2].lo + tbl[n2].hi;
 249
 250         if (k == 0) {
 251                 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
 252                     tbl[n2].hi * r1);
 253                 RETURNI(t);
 254         }
 255         if (k == -1) {
 256                 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
 257                     tbl[n2].hi * r1);
 258                 RETURNI(t / 2);
 259         }
 260         if (k < -7) {
 261                 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
 262                 RETURNI(t * twopk - 1);
 263         }
 264         if (k > 2 * LDBL_MANT_DIG - 1) {
 265                 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
 266                 if (k == LDBL_MAX_EXP)
 267                         RETURNI(t * 2 * 0x1p16383L - 1);
 268                 RETURNI(t * twopk - 1);
 269         }
 270
 271         v.xbits.expsign = BIAS - k;
 272         twomk = v.e;
 273
 274         if (k > LDBL_MANT_DIG - 1)
 275                 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
 276         else
 277                 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
 278         RETURNI(t * twopk);
 279 }