1 /*- 2 * SPDX-License-Identifier: BSD-2-Clause 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 /** 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 __FreeBSD__ 45 #include "fpmath.h" 46 #endif 47 #include "math.h" 48 #include "math_private.h" 49 #include "k_expl.h" 50 51 /* XXX Prevent compilers from erroneously constant folding these: */ 52 static const volatile long double 53 huge = 0x1p10000L, 54 tiny = 0x1p-10000L; 55 56 static const long double 57 twom10000 = 0x1p-10000L; 58 59 static const union ieee_ext_u 60 /* log(2**16384 - 0.5) rounded towards zero: */ 61 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ 62 o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), 63 #define o_threshold (o_thresholdu.extu_ld) 64 /* log(2**(-16381-64-1)) rounded towards zero: */ 65 u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); 66 #define u_threshold (u_thresholdu.extu_ld) 67 68 long double 69 expl(long double x) 70 { 71 union ieee_ext_u u; 72 long double hi, lo, t, twopk; 73 int k; 74 uint16_t hx, ix; 75 76 /* Filter out exceptional cases. */ 77 u.extu_ld = x; 78 hx = GET_EXPSIGN(&u); 79 ix = hx & 0x7fff; 80 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 81 if (ix == BIAS + LDBL_MAX_EXP) { 82 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 83 RETURNF(-1 / x); 84 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */ 85 } 86 if (x > o_threshold) 87 RETURNF(huge * huge); 88 if (x < u_threshold) 89 RETURNF(tiny * tiny); 90 } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ 91 RETURNF(1 + x); /* 1 with inexact iff x != 0 */ 92 } 93 94 ENTERI(); 95 96 twopk = 1; 97 __k_expl(x, &hi, &lo, &k); 98 t = SUM2P(hi, lo); 99 100 /* Scale by 2**k. */ 101 if (k >= LDBL_MIN_EXP) { 102 if (k == LDBL_MAX_EXP) 103 RETURNI(t * 2 * 0x1p16383L); 104 SET_LDBL_EXPSIGN(twopk, BIAS + k); 105 RETURNI(t * twopk); 106 } else { 107 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); 108 RETURNI(t * twopk * twom10000); 109 } 110 } 111 112 /** 113 * Compute expm1l(x) for Intel 80-bit format. This is based on: 114 * 115 * PTP Tang, "Table-driven implementation of the Expm1 function 116 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, 117 * 211-222 (1992). 118 */ 119 120 /* 121 * Our T1 and T2 are chosen to be approximately the points where method 122 * A and method B have the same accuracy. Tang's T1 and T2 are the 123 * points where method A's accuracy changes by a full bit. For Tang, 124 * this drop in accuracy makes method A immediately less accurate than 125 * method B, but our larger INTERVALS makes method A 2 bits more 126 * accurate so it remains the most accurate method significantly 127 * closer to the origin despite losing the full bit in our extended 128 * range for it. 129 */ 130 static const double 131 T1 = -0.1659, /* ~-30.625/128 * log(2) */ 132 T2 = 0.1659; /* ~30.625/128 * log(2) */ 133 134 /* 135 * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]: 136 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6 137 * 138 * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits, 139 * but unlike for ld128 we can't drop any terms. 140 */ 141 static const union ieee_ext_u 142 B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), 143 B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); 144 145 static const double 146 B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ 147 B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ 148 B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ 149 B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ 150 B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ 151 B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ 152 B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ 153 B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ 154 155 long double 156 expm1l(long double x) 157 { 158 union ieee_ext_u u, v; 159 long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; 160 long double x_lo, x2, z; 161 long double x4; 162 int k, n, n2; 163 uint16_t hx, ix; 164 165 /* Filter out exceptional cases. */ 166 u.extu_ld = x; 167 hx = GET_EXPSIGN(&u); 168 ix = hx & 0x7fff; 169 if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ 170 if (ix == BIAS + LDBL_MAX_EXP) { 171 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 172 RETURNF(-1 / x - 1); 173 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */ 174 } 175 if (x > o_threshold) 176 RETURNF(huge * huge); 177 /* 178 * expm1l() never underflows, but it must avoid 179 * unrepresentable large negative exponents. We used a 180 * much smaller threshold for large |x| above than in 181 * expl() so as to handle not so large negative exponents 182 * in the same way as large ones here. 183 */ 184 if (hx & 0x8000) /* x <= -64 */ 185 RETURNF(tiny - 1); /* good for x < -65ln2 - eps */ 186 } 187 188 ENTERI(); 189 190 if (T1 < x && x < T2) { 191 if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */ 192 /* x (rounded) with inexact if x != 0: */ 193 RETURNI(x == 0 ? x : 194 (0x1p100 * x + fabsl(x)) * 0x1p-100); 195 } 196 197 x2 = x * x; 198 x4 = x2 * x2; 199 q = x4 * (x2 * (x4 * 200 /* 201 * XXX the number of terms is no longer good for 202 * pairwise grouping of all except B3, and the 203 * grouping is no longer from highest down. 204 */ 205 (x2 * B12 + (x * B11 + B10)) + 206 (x2 * (x * B9 + B8) + (x * B7 + B6))) + 207 (x * B5 + B4.extu_ld)) + x2 * x * B3.extu_ld; 208 209 x_hi = (float)x; 210 x_lo = x - x_hi; 211 hx2_hi = x_hi * x_hi / 2; 212 hx2_lo = x_lo * (x + x_hi) / 2; 213 if (ix >= BIAS - 7) 214 RETURNI((hx2_hi + x_hi) + (hx2_lo + x_lo + q)); 215 else 216 RETURNI(x + (hx2_lo + q + hx2_hi)); 217 } 218 219 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 220 fn = rnintl(x * INV_L); 221 n = irint(fn); 222 n2 = (unsigned)n % INTERVALS; 223 k = n >> LOG2_INTERVALS; 224 r1 = x - fn * L1; 225 r2 = fn * -L2; 226 r = r1 + r2; 227 228 /* Prepare scale factor. */ 229 v.extu_ld = 1; 230 SET_EXPSIGN(&v, BIAS + k); 231 twopk = v.extu_ld; 232 233 /* 234 * Evaluate lower terms of 235 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 236 */ 237 z = r * r; 238 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 239 240 t = (long double)tbl[n2].lo + tbl[n2].hi; 241 242 if (k == 0) { 243 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + 244 tbl[n2].hi * r1); 245 RETURNI(t); 246 } 247 if (k == -1) { 248 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + 249 tbl[n2].hi * r1); 250 RETURNI(t / 2); 251 } 252 if (k < -7) { 253 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 254 RETURNI(t * twopk - 1); 255 } 256 if (k > 2 * LDBL_MANT_DIG - 1) { 257 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 258 if (k == LDBL_MAX_EXP) 259 RETURNI(t * 2 * 0x1p16383L - 1); 260 RETURNI(t * twopk - 1); 261 } 262 263 SET_EXPSIGN(&v, BIAS - k); 264 twomk = v.extu_ld; 265 266 if (k > LDBL_MANT_DIG - 1) 267 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); 268 else 269 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); 270 RETURNI(t * twopk); 271 } 272
Indexes created Sun Sep 21 16:09:51 GMT 2025