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 /** 32 * Compute the exponential of x for Intel 80-bit format. This is based on: 33 * 34 * PTP Tang, "Table-driven implementation of the exponential function 35 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15, 36 * 144-157 (1989). 37 * 38 * where the 32 table entries have been expanded to INTERVALS (see below). 39 */ 40 41 #include <float.h> 42 43 #ifdef __i386__ 44 #include <ieeefp.h> 45 #endif 46 47 #include "fpmath.h" 48 #include "math.h" 49 #include "math_private.h" 50 #include "k_expl.h" 51 52 /* XXX Prevent compilers from erroneously constant folding these: */ 53 static const volatile long double 54 huge = 0x1p10000L, 55 tiny = 0x1p-10000L; 56 57 static const long double 58 twom10000 = 0x1p-10000L; 59 60 static const union IEEEl2bits 61 /* log(2**16384 - 0.5) rounded towards zero: */ 62 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ 63 o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), 64 #define o_threshold (o_thresholdu.e) 65 /* log(2**(-16381-64-1)) rounded towards zero: */ 66 u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); 67 #define u_threshold (u_thresholdu.e) 68 69 long double 70 expl(long double x) 71 { 72 union IEEEl2bits u; 73 long double hi, lo, t, twopk; 74 int k; 75 uint16_t hx, ix; 76 77 /* Filter out exceptional cases. */ 78 u.e = x; 79 hx = u.xbits.expsign; 80 ix = hx & 0x7fff; 81 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 82 if (ix == BIAS + LDBL_MAX_EXP) { 83 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 84 RETURNF(-1 / x); 85 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */ 86 } 87 if (x > o_threshold) 88 RETURNF(huge * huge); 89 if (x < u_threshold) 90 RETURNF(tiny * tiny); 91 } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ 92 RETURNF(1 + x); /* 1 with inexact iff x != 0 */ 93 } 94 95 ENTERI(); 96 97 twopk = 1; 98 __k_expl(x, &hi, &lo, &k); 99 t = SUM2P(hi, lo); 100 101 /* Scale by 2**k. */ 102 if (k >= LDBL_MIN_EXP) { 103 if (k == LDBL_MAX_EXP) 104 RETURNI(t * 2 * 0x1p16383L); 105 SET_LDBL_EXPSIGN(twopk, BIAS + k); 106 RETURNI(t * twopk); 107 } else { 108 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); 109 RETURNI(t * twopk * twom10000); 110 } 111 } 112 113 /** 114 * Compute expm1l(x) for Intel 80-bit format. This is based on: 115 * 116 * PTP Tang, "Table-driven implementation of the Expm1 function 117 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, 118 * 211-222 (1992). 119 */ 120 121 /* 122 * Our T1 and T2 are chosen to be approximately the points where method 123 * A and method B have the same accuracy. Tang's T1 and T2 are the 124 * points where method A's accuracy changes by a full bit. For Tang, 125 * this drop in accuracy makes method A immediately less accurate than 126 * method B, but our larger INTERVALS makes method A 2 bits more 127 * accurate so it remains the most accurate method significantly 128 * closer to the origin despite losing the full bit in our extended 129 * range for it. 130 */ 131 static const double 132 T1 = -0.1659, /* ~-30.625/128 * log(2) */ 133 T2 = 0.1659; /* ~30.625/128 * log(2) */ 134 135 /* 136 * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]: 137 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6 138 * 139 * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits, 140 * but unlike for ld128 we can't drop any terms. 141 */ 142 static const union IEEEl2bits 143 B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), 144 B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); 145 146 static const double 147 B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ 148 B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ 149 B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ 150 B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ 151 B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ 152 B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ 153 B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ 154 B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ 155 156 long double 157 expm1l(long double x) 158 { 159 union IEEEl2bits u, v; 160 long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; 161 long double x_lo, x2, z; 162 long double x4; 163 int k, n, n2; 164 uint16_t hx, ix; 165 166 /* Filter out exceptional cases. */ 167 u.e = x; 168 hx = u.xbits.expsign; 169 ix = hx & 0x7fff; 170 if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ 171 if (ix == BIAS + LDBL_MAX_EXP) { 172 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 173 RETURNF(-1 / x - 1); 174 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */ 175 } 176 if (x > o_threshold) 177 RETURNF(huge * huge); 178 /* 179 * expm1l() never underflows, but it must avoid 180 * unrepresentable large negative exponents. We used a 181 * much smaller threshold for large |x| above than in 182 * expl() so as to handle not so large negative exponents 183 * in the same way as large ones here. 184 */ 185 if (hx & 0x8000) /* x <= -64 */ 186 RETURNF(tiny - 1); /* good for x < -65ln2 - eps */ 187 } 188 189 ENTERI(); 190 191 if (T1 < x && x < T2) { 192 if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */ 193 /* x (rounded) with inexact if x != 0: */ 194 RETURNI(x == 0 ? x : 195 (0x1p100 * x + fabsl(x)) * 0x1p-100); 196 } 197 198 x2 = x * x; 199 x4 = x2 * x2; 200 q = x4 * (x2 * (x4 * 201 /* 202 * XXX the number of terms is no longer good for 203 * pairwise grouping of all except B3, and the 204 * grouping is no longer from highest down. 205 */ 206 (x2 * B12 + (x * B11 + B10)) + 207 (x2 * (x * B9 + B8) + (x * B7 + B6))) + 208 (x * B5 + B4.e)) + x2 * x * B3.e; 209 210 x_hi = (float)x; 211 x_lo = x - x_hi; 212 hx2_hi = x_hi * x_hi / 2; 213 hx2_lo = x_lo * (x + x_hi) / 2; 214 if (ix >= BIAS - 7) 215 RETURNI((hx2_hi + x_hi) + (hx2_lo + x_lo + q)); 216 else 217 RETURNI(x + (hx2_lo + q + hx2_hi)); 218 } 219 220 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 221 fn = rnintl(x * INV_L); 222 n = irint(fn); 223 n2 = (unsigned)n % INTERVALS; 224 k = n >> LOG2_INTERVALS; 225 r1 = x - fn * L1; 226 r2 = fn * -L2; 227 r = r1 + r2; 228 229 /* Prepare scale factor. */ 230 v.e = 1; 231 v.xbits.expsign = BIAS + k; 232 twopk = v.e; 233 234 /* 235 * Evaluate lower terms of 236 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 237 */ 238 z = r * r; 239 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 240 241 t = (long double)tbl[n2].lo + tbl[n2].hi; 242 243 if (k == 0) { 244 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + 245 tbl[n2].hi * r1); 246 RETURNI(t); 247 } 248 if (k == -1) { 249 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + 250 tbl[n2].hi * r1); 251 RETURNI(t / 2); 252 } 253 if (k < -7) { 254 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 255 RETURNI(t * twopk - 1); 256 } 257 if (k > 2 * LDBL_MANT_DIG - 1) { 258 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 259 if (k == LDBL_MAX_EXP) 260 RETURNI(t * 2 * 0x1p16383L - 1); 261 RETURNI(t * twopk - 1); 262 } 263 264 v.xbits.expsign = BIAS - k; 265 twomk = v.e; 266 267 if (k > LDBL_MANT_DIG - 1) 268 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); 269 else 270 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); 271 RETURNI(t * twopk); 272 } 273