1 /*- 2 * Copyright (c) 2009-2013 Steven G. Kargl 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice unmodified, this list of conditions, and the following 10 * disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 25 * 26 * Optimized by Bruce D. Evans. 27 */ 28 29 #include <sys/cdefs.h> 30 __FBSDID("$FreeBSD$"); 31 32 /* 33 * ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments. 34 */ 35 36 #include <float.h> 37 38 #include "fpmath.h" 39 #include "math.h" 40 #include "math_private.h" 41 #include "k_expl.h" 42 43 /* XXX Prevent compilers from erroneously constant folding these: */ 44 static const volatile long double 45 huge = 0x1p10000L, 46 tiny = 0x1p-10000L; 47 48 static const long double 49 twom10000 = 0x1p-10000L; 50 51 static const long double 52 /* log(2**16384 - 0.5) rounded towards zero: */ 53 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ 54 o_threshold = 11356.523406294143949491931077970763428L, 55 /* log(2**(-16381-64-1)) rounded towards zero: */ 56 u_threshold = -11433.462743336297878837243843452621503L; 57 58 long double 59 expl(long double x) 60 { 61 union IEEEl2bits u; 62 long double hi, lo, t, twopk; 63 int k; 64 uint16_t hx, ix; 65 66 DOPRINT_START(&x); 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 RETURNP(-1 / x); 76 RETURNP(x + x); /* x is +Inf or +NaN */ 77 } 78 if (x > o_threshold) 79 RETURNP(huge * huge); 80 if (x < u_threshold) 81 RETURNP(tiny * tiny); 82 } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */ 83 RETURN2P(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 /* XXX sparc64 multiplication is so slow that scalbnl() is faster. */ 94 if (k >= LDBL_MIN_EXP) { 95 if (k == LDBL_MAX_EXP) 96 RETURNI(t * 2 * 0x1p16383L); 97 SET_LDBL_EXPSIGN(twopk, BIAS + k); 98 RETURNI(t * twopk); 99 } else { 100 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); 101 RETURNI(t * twopk * twom10000); 102 } 103 } 104 105 /* 106 * Our T1 and T2 are chosen to be approximately the points where method 107 * A and method B have the same accuracy. Tang's T1 and T2 are the 108 * points where method A's accuracy changes by a full bit. For Tang, 109 * this drop in accuracy makes method A immediately less accurate than 110 * method B, but our larger INTERVALS makes method A 2 bits more 111 * accurate so it remains the most accurate method significantly 112 * closer to the origin despite losing the full bit in our extended 113 * range for it. 114 * 115 * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2]. 116 * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear 117 * in both subintervals, so set T3 = 2**-5, which places the condition 118 * into the [T1, T3] interval. 119 * 120 * XXX we now do this more to (partially) balance the number of terms 121 * in the C and D polys than to avoid checking the condition in both 122 * intervals. 123 * 124 * XXX these micro-optimizations are excessive. 125 */ 126 static const double 127 T1 = -0.1659, /* ~-30.625/128 * log(2) */ 128 T2 = 0.1659, /* ~30.625/128 * log(2) */ 129 T3 = 0.03125; 130 131 /* 132 * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]: 133 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03 134 * 135 * XXX none of the long double C or D coeffs except C10 is correctly printed. 136 * If you re-print their values in %.35Le format, the result is always 137 * different. For example, the last 2 digits in C3 should be 59, not 67. 138 * 67 is apparently from rounding an extra-precision value to 36 decimal 139 * places. 140 */ 141 static const long double 142 C3 = 1.66666666666666666666666666666666667e-1L, 143 C4 = 4.16666666666666666666666666666666645e-2L, 144 C5 = 8.33333333333333333333333333333371638e-3L, 145 C6 = 1.38888888888888888888888888891188658e-3L, 146 C7 = 1.98412698412698412698412697235950394e-4L, 147 C8 = 2.48015873015873015873015112487849040e-5L, 148 C9 = 2.75573192239858906525606685484412005e-6L, 149 C10 = 2.75573192239858906612966093057020362e-7L, 150 C11 = 2.50521083854417203619031960151253944e-8L, 151 C12 = 2.08767569878679576457272282566520649e-9L, 152 C13 = 1.60590438367252471783548748824255707e-10L; 153 154 /* 155 * XXX this has 1 more coeff than needed. 156 * XXX can start the double coeffs but not the double mults at C10. 157 * With my coeffs (C10-C17 double; s = best_s): 158 * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]: 159 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 160 */ 161 static const double 162 C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */ 163 C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */ 164 C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */ 165 C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */ 166 C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */ 167 168 /* 169 * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]: 170 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44 171 */ 172 static const long double 173 D3 = 1.66666666666666666666666666666682245e-1L, 174 D4 = 4.16666666666666666666666666634228324e-2L, 175 D5 = 8.33333333333333333333333364022244481e-3L, 176 D6 = 1.38888888888888888888887138722762072e-3L, 177 D7 = 1.98412698412698412699085805424661471e-4L, 178 D8 = 2.48015873015873015687993712101479612e-5L, 179 D9 = 2.75573192239858944101036288338208042e-6L, 180 D10 = 2.75573192239853161148064676533754048e-7L, 181 D11 = 2.50521083855084570046480450935267433e-8L, 182 D12 = 2.08767569819738524488686318024854942e-9L, 183 D13 = 1.60590442297008495301927448122499313e-10L; 184 185 /* 186 * XXX this has 1 more coeff than needed. 187 * XXX can start the double coeffs but not the double mults at D11. 188 * With my coeffs (D11-D16 double): 189 * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]: 190 * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 191 */ 192 static const double 193 D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */ 194 D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */ 195 D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */ 196 D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */ 197 198 long double 199 expm1l(long double x) 200 { 201 union IEEEl2bits u, v; 202 long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi; 203 long double x_lo, x2; 204 double dr, dx, fn, r2; 205 int k, n, n2; 206 uint16_t hx, ix; 207 208 DOPRINT_START(&x); 209 210 /* Filter out exceptional cases. */ 211 u.e = x; 212 hx = u.xbits.expsign; 213 ix = hx & 0x7fff; 214 if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */ 215 if (ix == BIAS + LDBL_MAX_EXP) { 216 if (hx & 0x8000) /* x is -Inf or -NaN */ 217 RETURNP(-1 / x - 1); 218 RETURNP(x + x); /* x is +Inf or +NaN */ 219 } 220 if (x > o_threshold) 221 RETURNP(huge * huge); 222 /* 223 * expm1l() never underflows, but it must avoid 224 * unrepresentable large negative exponents. We used a 225 * much smaller threshold for large |x| above than in 226 * expl() so as to handle not so large negative exponents 227 * in the same way as large ones here. 228 */ 229 if (hx & 0x8000) /* x <= -128 */ 230 RETURN2P(tiny, -1); /* good for x < -114ln2 - eps */ 231 } 232 233 ENTERI(); 234 235 if (T1 < x && x < T2) { 236 x2 = x * x; 237 dx = x; 238 239 if (x < T3) { 240 if (ix < BIAS - 113) { /* |x| < 0x1p-113 */ 241 /* x (rounded) with inexact if x != 0: */ 242 RETURNPI(x == 0 ? x : 243 (0x1p200 * x + fabsl(x)) * 0x1p-200); 244 } 245 q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 + 246 x * (C7 + x * (C8 + x * (C9 + x * (C10 + 247 x * (C11 + x * (C12 + x * (C13 + 248 dx * (C14 + dx * (C15 + dx * (C16 + 249 dx * (C17 + dx * C18)))))))))))))); 250 } else { 251 q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 + 252 x * (D7 + x * (D8 + x * (D9 + x * (D10 + 253 x * (D11 + x * (D12 + x * (D13 + 254 dx * (D14 + dx * (D15 + dx * (D16 + 255 dx * D17))))))))))))); 256 } 257 258 x_hi = (float)x; 259 x_lo = x - x_hi; 260 hx2_hi = x_hi * x_hi / 2; 261 hx2_lo = x_lo * (x + x_hi) / 2; 262 if (ix >= BIAS - 7) 263 RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); 264 else 265 RETURN2PI(x, hx2_lo + q + hx2_hi); 266 } 267 268 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 269 /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 270 fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52; 271 #if defined(HAVE_EFFICIENT_IRINT) 272 n = irint(fn); 273 #else 274 n = (int)fn; 275 #endif 276 n2 = (unsigned)n % INTERVALS; 277 k = n >> LOG2_INTERVALS; 278 r1 = x - fn * L1; 279 r2 = fn * -L2; 280 r = r1 + r2; 281 282 /* Prepare scale factor. */ 283 v.e = 1; 284 v.xbits.expsign = BIAS + k; 285 twopk = v.e; 286 287 /* 288 * Evaluate lower terms of 289 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 290 */ 291 dr = r; 292 q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + 293 dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); 294 295 t = tbl[n2].lo + tbl[n2].hi; 296 297 if (k == 0) { 298 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + 299 tbl[n2].hi * r1); 300 RETURNI(t); 301 } 302 if (k == -1) { 303 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + 304 tbl[n2].hi * r1); 305 RETURNI(t / 2); 306 } 307 if (k < -7) { 308 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 309 RETURNI(t * twopk - 1); 310 } 311 if (k > 2 * LDBL_MANT_DIG - 1) { 312 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 313 if (k == LDBL_MAX_EXP) 314 RETURNI(t * 2 * 0x1p16383L - 1); 315 RETURNI(t * twopk - 1); 316 } 317 318 v.xbits.expsign = BIAS - k; 319 twomk = v.e; 320 321 if (k > LDBL_MANT_DIG - 1) 322 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); 323 else 324 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); 325 RETURNI(t * twopk); 326 } 327