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 "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: */ 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 /* 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