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 __i386__ 45 #include <ieeefp.h> 46 #endif 47 48 #include "fpmath.h" 49 #include "math.h" 50 #include "math_private.h" 51 #include "k_expl.h" 52 53 /* XXX Prevent compilers from erroneously constant folding these: */ 54 static const volatile long double 55 huge = 0x1p10000L, 56 tiny = 0x1p-10000L; 57 58 static const long double 59 twom10000 = 0x1p-10000L; 60 61 static const union IEEEl2bits 62 /* log(2**16384 - 0.5) rounded towards zero: */ 63 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ 64 o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), 65 #define o_threshold (o_thresholdu.e) 66 /* log(2**(-16381-64-1)) rounded towards zero: */ 67 u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); 68 #define u_threshold (u_thresholdu.e) 69 70 long double 71 expl(long double x) 72 { 73 union IEEEl2bits u; 74 long double hi, lo, t, twopk; 75 int k; 76 uint16_t hx, ix; 77 78 /* Filter out exceptional cases. */ 79 u.e = x; 80 hx = u.xbits.expsign; 81 ix = hx & 0x7fff; 82 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 83 if (ix == BIAS + LDBL_MAX_EXP) { 84 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 85 RETURNF(-1 / x); 86 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */ 87 } 88 if (x > o_threshold) 89 RETURNF(huge * huge); 90 if (x < u_threshold) 91 RETURNF(tiny * tiny); 92 } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ 93 RETURNF(1 + x); /* 1 with inexact iff x != 0 */ 94 } 95 96 ENTERI(); 97 98 twopk = 1; 99 __k_expl(x, &hi, &lo, &k); 100 t = SUM2P(hi, lo); 101 102 /* Scale by 2**k. */ 103 if (k >= LDBL_MIN_EXP) { 104 if (k == LDBL_MAX_EXP) 105 RETURNI(t * 2 * 0x1p16383L); 106 SET_LDBL_EXPSIGN(twopk, BIAS + k); 107 RETURNI(t * twopk); 108 } else { 109 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); 110 RETURNI(t * twopk * twom10000); 111 } 112 } 113 114 /** 115 * Compute expm1l(x) for Intel 80-bit format. This is based on: 116 * 117 * PTP Tang, "Table-driven implementation of the Expm1 function 118 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, 119 * 211-222 (1992). 120 */ 121 122 /* 123 * Our T1 and T2 are chosen to be approximately the points where method 124 * A and method B have the same accuracy. Tang's T1 and T2 are the 125 * points where method A's accuracy changes by a full bit. For Tang, 126 * this drop in accuracy makes method A immediately less accurate than 127 * method B, but our larger INTERVALS makes method A 2 bits more 128 * accurate so it remains the most accurate method significantly 129 * closer to the origin despite losing the full bit in our extended 130 * range for it. 131 */ 132 static const double 133 T1 = -0.1659, /* ~-30.625/128 * log(2) */ 134 T2 = 0.1659; /* ~30.625/128 * log(2) */ 135 136 /* 137 * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]: 138 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6 139 * 140 * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits, 141 * but unlike for ld128 we can't drop any terms. 142 */ 143 static const union IEEEl2bits 144 B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), 145 B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); 146 147 static const double 148 B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ 149 B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ 150 B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ 151 B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ 152 B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ 153 B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ 154 B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ 155 B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ 156 157 long double 158 expm1l(long double x) 159 { 160 union IEEEl2bits u, v; 161 long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; 162 long double x_lo, x2, z; 163 long double x4; 164 int k, n, n2; 165 uint16_t hx, ix; 166 167 /* Filter out exceptional cases. */ 168 u.e = x; 169 hx = u.xbits.expsign; 170 ix = hx & 0x7fff; 171 if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ 172 if (ix == BIAS + LDBL_MAX_EXP) { 173 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 174 RETURNF(-1 / x - 1); 175 RETURNF(x + x); /* x is +Inf, +NaN or unsupported */ 176 } 177 if (x > o_threshold) 178 RETURNF(huge * huge); 179 /* 180 * expm1l() never underflows, but it must avoid 181 * unrepresentable large negative exponents. We used a 182 * much smaller threshold for large |x| above than in 183 * expl() so as to handle not so large negative exponents 184 * in the same way as large ones here. 185 */ 186 if (hx & 0x8000) /* x <= -64 */ 187 RETURNF(tiny - 1); /* good for x < -65ln2 - eps */ 188 } 189 190 ENTERI(); 191 192 if (T1 < x && x < T2) { 193 if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */ 194 /* x (rounded) with inexact if x != 0: */ 195 RETURNI(x == 0 ? x : 196 (0x1p100 * x + fabsl(x)) * 0x1p-100); 197 } 198 199 x2 = x * x; 200 x4 = x2 * x2; 201 q = x4 * (x2 * (x4 * 202 /* 203 * XXX the number of terms is no longer good for 204 * pairwise grouping of all except B3, and the 205 * grouping is no longer from highest down. 206 */ 207 (x2 * B12 + (x * B11 + B10)) + 208 (x2 * (x * B9 + B8) + (x * B7 + B6))) + 209 (x * B5 + B4.e)) + x2 * x * B3.e; 210 211 x_hi = (float)x; 212 x_lo = x - x_hi; 213 hx2_hi = x_hi * x_hi / 2; 214 hx2_lo = x_lo * (x + x_hi) / 2; 215 if (ix >= BIAS - 7) 216 RETURNI((hx2_hi + x_hi) + (hx2_lo + x_lo + q)); 217 else 218 RETURNI(x + (hx2_lo + q + hx2_hi)); 219 } 220 221 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 222 fn = rnintl(x * INV_L); 223 n = irint(fn); 224 n2 = (unsigned)n % INTERVALS; 225 k = n >> LOG2_INTERVALS; 226 r1 = x - fn * L1; 227 r2 = fn * -L2; 228 r = r1 + r2; 229 230 /* Prepare scale factor. */ 231 v.e = 1; 232 v.xbits.expsign = BIAS + k; 233 twopk = v.e; 234 235 /* 236 * Evaluate lower terms of 237 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 238 */ 239 z = r * r; 240 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 241 242 t = (long double)tbl[n2].lo + tbl[n2].hi; 243 244 if (k == 0) { 245 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + 246 tbl[n2].hi * r1); 247 RETURNI(t); 248 } 249 if (k == -1) { 250 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + 251 tbl[n2].hi * r1); 252 RETURNI(t / 2); 253 } 254 if (k < -7) { 255 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 256 RETURNI(t * twopk - 1); 257 } 258 if (k > 2 * LDBL_MANT_DIG - 1) { 259 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 260 if (k == LDBL_MAX_EXP) 261 RETURNI(t * 2 * 0x1p16383L - 1); 262 RETURNI(t * twopk - 1); 263 } 264 265 v.xbits.expsign = BIAS - k; 266 twomk = v.e; 267 268 if (k > LDBL_MANT_DIG - 1) 269 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); 270 else 271 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); 272 RETURNI(t * twopk); 273 } 274