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 __FBSDID("$FreeBSD$"); 33 34 /* 35 * ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments. 36 */ 37 38 #include <float.h> 39 40 #include "fpmath.h" 41 #include "math.h" 42 #include "math_private.h" 43 #include "k_expl.h" 44 45 /* XXX Prevent compilers from erroneously constant folding these: */ 46 static const volatile long double 47 huge = 0x1p10000L, 48 tiny = 0x1p-10000L; 49 50 static const long double 51 twom10000 = 0x1p-10000L; 52 53 static const long double 54 /* log(2**16384 - 0.5) rounded towards zero: */ 55 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ 56 o_threshold = 11356.523406294143949491931077970763428L, 57 /* log(2**(-16381-64-1)) rounded towards zero: */ 58 u_threshold = -11433.462743336297878837243843452621503L; 59 60 long double 61 expl(long double x) 62 { 63 union IEEEl2bits u; 64 long double hi, lo, t, twopk; 65 int k; 66 uint16_t hx, ix; 67 68 /* Filter out exceptional cases. */ 69 u.e = x; 70 hx = u.xbits.expsign; 71 ix = hx & 0x7fff; 72 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 73 if (ix == BIAS + LDBL_MAX_EXP) { 74 if (hx & 0x8000) /* x is -Inf or -NaN */ 75 RETURNF(-1 / x); 76 RETURNF(x + x); /* x is +Inf or +NaN */ 77 } 78 if (x > o_threshold) 79 RETURNF(huge * huge); 80 if (x < u_threshold) 81 RETURNF(tiny * tiny); 82 } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */ 83 RETURNF(1 + x); /* 1 with inexact iff x != 0 */ 84 } 85 86 ENTERI(); 87 88 twopk = 1; 89 __k_expl(x, &hi, &lo, &k); 90 t = SUM2P(hi, lo); 91 92 /* Scale by 2**k. */ 93 /* 94 * XXX sparc64 multiplication was so slow that scalbnl() is faster, 95 * but performance on aarch64 and riscv hasn't yet been quantified. 96 */ 97 if (k >= LDBL_MIN_EXP) { 98 if (k == LDBL_MAX_EXP) 99 RETURNI(t * 2 * 0x1p16383L); 100 SET_LDBL_EXPSIGN(twopk, BIAS + k); 101 RETURNI(t * twopk); 102 } else { 103 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); 104 RETURNI(t * twopk * twom10000); 105 } 106 } 107 108 /* 109 * Our T1 and T2 are chosen to be approximately the points where method 110 * A and method B have the same accuracy. Tang's T1 and T2 are the 111 * points where method A's accuracy changes by a full bit. For Tang, 112 * this drop in accuracy makes method A immediately less accurate than 113 * method B, but our larger INTERVALS makes method A 2 bits more 114 * accurate so it remains the most accurate method significantly 115 * closer to the origin despite losing the full bit in our extended 116 * range for it. 117 * 118 * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2]. 119 * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear 120 * in both subintervals, so set T3 = 2**-5, which places the condition 121 * into the [T1, T3] interval. 122 * 123 * XXX we now do this more to (partially) balance the number of terms 124 * in the C and D polys than to avoid checking the condition in both 125 * intervals. 126 * 127 * XXX these micro-optimizations are excessive. 128 */ 129 static const double 130 T1 = -0.1659, /* ~-30.625/128 * log(2) */ 131 T2 = 0.1659, /* ~30.625/128 * log(2) */ 132 T3 = 0.03125; 133 134 /* 135 * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]: 136 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03 137 * 138 * XXX none of the long double C or D coeffs except C10 is correctly printed. 139 * If you re-print their values in %.35Le format, the result is always 140 * different. For example, the last 2 digits in C3 should be 59, not 67. 141 * 67 is apparently from rounding an extra-precision value to 36 decimal 142 * places. 143 */ 144 static const long double 145 C3 = 1.66666666666666666666666666666666667e-1L, 146 C4 = 4.16666666666666666666666666666666645e-2L, 147 C5 = 8.33333333333333333333333333333371638e-3L, 148 C6 = 1.38888888888888888888888888891188658e-3L, 149 C7 = 1.98412698412698412698412697235950394e-4L, 150 C8 = 2.48015873015873015873015112487849040e-5L, 151 C9 = 2.75573192239858906525606685484412005e-6L, 152 C10 = 2.75573192239858906612966093057020362e-7L, 153 C11 = 2.50521083854417203619031960151253944e-8L, 154 C12 = 2.08767569878679576457272282566520649e-9L, 155 C13 = 1.60590438367252471783548748824255707e-10L; 156 157 /* 158 * XXX this has 1 more coeff than needed. 159 * XXX can start the double coeffs but not the double mults at C10. 160 * With my coeffs (C10-C17 double; s = best_s): 161 * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]: 162 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 163 */ 164 static const double 165 C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */ 166 C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */ 167 C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */ 168 C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */ 169 C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */ 170 171 /* 172 * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]: 173 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44 174 */ 175 static const long double 176 D3 = 1.66666666666666666666666666666682245e-1L, 177 D4 = 4.16666666666666666666666666634228324e-2L, 178 D5 = 8.33333333333333333333333364022244481e-3L, 179 D6 = 1.38888888888888888888887138722762072e-3L, 180 D7 = 1.98412698412698412699085805424661471e-4L, 181 D8 = 2.48015873015873015687993712101479612e-5L, 182 D9 = 2.75573192239858944101036288338208042e-6L, 183 D10 = 2.75573192239853161148064676533754048e-7L, 184 D11 = 2.50521083855084570046480450935267433e-8L, 185 D12 = 2.08767569819738524488686318024854942e-9L, 186 D13 = 1.60590442297008495301927448122499313e-10L; 187 188 /* 189 * XXX this has 1 more coeff than needed. 190 * XXX can start the double coeffs but not the double mults at D11. 191 * With my coeffs (D11-D16 double): 192 * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]: 193 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 194 */ 195 static const double 196 D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */ 197 D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */ 198 D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */ 199 D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */ 200 201 long double 202 expm1l(long double x) 203 { 204 union IEEEl2bits u, v; 205 long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi; 206 long double x_lo, x2; 207 double dr, dx, fn, r2; 208 int k, n, n2; 209 uint16_t hx, ix; 210 211 /* Filter out exceptional cases. */ 212 u.e = x; 213 hx = u.xbits.expsign; 214 ix = hx & 0x7fff; 215 if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */ 216 if (ix == BIAS + LDBL_MAX_EXP) { 217 if (hx & 0x8000) /* x is -Inf or -NaN */ 218 RETURNF(-1 / x - 1); 219 RETURNF(x + x); /* x is +Inf or +NaN */ 220 } 221 if (x > o_threshold) 222 RETURNF(huge * huge); 223 /* 224 * expm1l() never underflows, but it must avoid 225 * unrepresentable large negative exponents. We used a 226 * much smaller threshold for large |x| above than in 227 * expl() so as to handle not so large negative exponents 228 * in the same way as large ones here. 229 */ 230 if (hx & 0x8000) /* x <= -128 */ 231 RETURNF(tiny - 1); /* good for x < -114ln2 - eps */ 232 } 233 234 ENTERI(); 235 236 if (T1 < x && x < T2) { 237 x2 = x * x; 238 dx = x; 239 240 if (x < T3) { 241 if (ix < BIAS - 113) { /* |x| < 0x1p-113 */ 242 /* x (rounded) with inexact if x != 0: */ 243 RETURNI(x == 0 ? x : 244 (0x1p200 * x + fabsl(x)) * 0x1p-200); 245 } 246 q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 + 247 x * (C7 + x * (C8 + x * (C9 + x * (C10 + 248 x * (C11 + x * (C12 + x * (C13 + 249 dx * (C14 + dx * (C15 + dx * (C16 + 250 dx * (C17 + dx * C18)))))))))))))); 251 } else { 252 q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 + 253 x * (D7 + x * (D8 + x * (D9 + x * (D10 + 254 x * (D11 + x * (D12 + x * (D13 + 255 dx * (D14 + dx * (D15 + dx * (D16 + 256 dx * D17))))))))))))); 257 } 258 259 x_hi = (float)x; 260 x_lo = x - x_hi; 261 hx2_hi = x_hi * x_hi / 2; 262 hx2_lo = x_lo * (x + x_hi) / 2; 263 if (ix >= BIAS - 7) 264 RETURNI((hx2_hi + x_hi) + (hx2_lo + x_lo + q)); 265 else 266 RETURNI(x + (hx2_lo + q + hx2_hi)); 267 } 268 269 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 270 fn = rnint((double)x * INV_L); 271 n = irint(fn); 272 n2 = (unsigned)n % INTERVALS; 273 k = n >> LOG2_INTERVALS; 274 r1 = x - fn * L1; 275 r2 = fn * -L2; 276 r = r1 + r2; 277 278 /* Prepare scale factor. */ 279 v.e = 1; 280 v.xbits.expsign = BIAS + k; 281 twopk = v.e; 282 283 /* 284 * Evaluate lower terms of 285 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 286 */ 287 dr = r; 288 q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + 289 dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); 290 291 t = tbl[n2].lo + tbl[n2].hi; 292 293 if (k == 0) { 294 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + 295 tbl[n2].hi * r1); 296 RETURNI(t); 297 } 298 if (k == -1) { 299 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + 300 tbl[n2].hi * r1); 301 RETURNI(t / 2); 302 } 303 if (k < -7) { 304 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 305 RETURNI(t * twopk - 1); 306 } 307 if (k > 2 * LDBL_MANT_DIG - 1) { 308 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 309 if (k == LDBL_MAX_EXP) 310 RETURNI(t * 2 * 0x1p16383L - 1); 311 RETURNI(t * twopk - 1); 312 } 313 314 v.xbits.expsign = BIAS - k; 315 twomk = v.e; 316 317 if (k > LDBL_MANT_DIG - 1) 318 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); 319 else 320 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); 321 RETURNI(t * twopk); 322 } 323