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 * 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 DOPRINT_START(&x); 79 80 /* Filter out exceptional cases. */ 81 u.e = x; 82 hx = u.xbits.expsign; 83 ix = hx & 0x7fff; 84 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 85 if (ix == BIAS + LDBL_MAX_EXP) { 86 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 87 RETURNP(-1 / x); 88 RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ 89 } 90 if (x > o_threshold) 91 RETURNP(huge * huge); 92 if (x < u_threshold) 93 RETURNP(tiny * tiny); 94 } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ 95 RETURN2P(1, x); /* 1 with inexact iff x != 0 */ 96 } 97 98 ENTERI(); 99 100 twopk = 1; 101 __k_expl(x, &hi, &lo, &k); 102 t = SUM2P(hi, lo); 103 104 /* Scale by 2**k. */ 105 if (k >= LDBL_MIN_EXP) { 106 if (k == LDBL_MAX_EXP) 107 RETURNI(t * 2 * 0x1p16383L); 108 SET_LDBL_EXPSIGN(twopk, BIAS + k); 109 RETURNI(t * twopk); 110 } else { 111 SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); 112 RETURNI(t * twopk * twom10000); 113 } 114 } 115 116 /** 117 * Compute expm1l(x) for Intel 80-bit format. This is based on: 118 * 119 * PTP Tang, "Table-driven implementation of the Expm1 function 120 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, 121 * 211-222 (1992). 122 */ 123 124 /* 125 * Our T1 and T2 are chosen to be approximately the points where method 126 * A and method B have the same accuracy. Tang's T1 and T2 are the 127 * points where method A's accuracy changes by a full bit. For Tang, 128 * this drop in accuracy makes method A immediately less accurate than 129 * method B, but our larger INTERVALS makes method A 2 bits more 130 * accurate so it remains the most accurate method significantly 131 * closer to the origin despite losing the full bit in our extended 132 * range for it. 133 */ 134 static const double 135 T1 = -0.1659, /* ~-30.625/128 * log(2) */ 136 T2 = 0.1659; /* ~30.625/128 * log(2) */ 137 138 /* 139 * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]: 140 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6 141 * 142 * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits, 143 * but unlike for ld128 we can't drop any terms. 144 */ 145 static const union IEEEl2bits 146 B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), 147 B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); 148 149 static const double 150 B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ 151 B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ 152 B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ 153 B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ 154 B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ 155 B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ 156 B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ 157 B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ 158 159 long double 160 expm1l(long double x) 161 { 162 union IEEEl2bits u, v; 163 long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; 164 long double x_lo, x2, z; 165 long double x4; 166 int k, n, n2; 167 uint16_t hx, ix; 168 169 DOPRINT_START(&x); 170 171 /* Filter out exceptional cases. */ 172 u.e = x; 173 hx = u.xbits.expsign; 174 ix = hx & 0x7fff; 175 if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ 176 if (ix == BIAS + LDBL_MAX_EXP) { 177 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 178 RETURNP(-1 / x - 1); 179 RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ 180 } 181 if (x > o_threshold) 182 RETURNP(huge * huge); 183 /* 184 * expm1l() never underflows, but it must avoid 185 * unrepresentable large negative exponents. We used a 186 * much smaller threshold for large |x| above than in 187 * expl() so as to handle not so large negative exponents 188 * in the same way as large ones here. 189 */ 190 if (hx & 0x8000) /* x <= -64 */ 191 RETURN2P(tiny, -1); /* good for x < -65ln2 - eps */ 192 } 193 194 ENTERI(); 195 196 if (T1 < x && x < T2) { 197 if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */ 198 /* x (rounded) with inexact if x != 0: */ 199 RETURNPI(x == 0 ? x : 200 (0x1p100 * x + fabsl(x)) * 0x1p-100); 201 } 202 203 x2 = x * x; 204 x4 = x2 * x2; 205 q = x4 * (x2 * (x4 * 206 /* 207 * XXX the number of terms is no longer good for 208 * pairwise grouping of all except B3, and the 209 * grouping is no longer from highest down. 210 */ 211 (x2 * B12 + (x * B11 + B10)) + 212 (x2 * (x * B9 + B8) + (x * B7 + B6))) + 213 (x * B5 + B4.e)) + x2 * x * B3.e; 214 215 x_hi = (float)x; 216 x_lo = x - x_hi; 217 hx2_hi = x_hi * x_hi / 2; 218 hx2_lo = x_lo * (x + x_hi) / 2; 219 if (ix >= BIAS - 7) 220 RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); 221 else 222 RETURN2PI(x, hx2_lo + q + hx2_hi); 223 } 224 225 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 226 /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 227 fn = x * INV_L + 0x1.8p63 - 0x1.8p63; 228 #if defined(HAVE_EFFICIENT_IRINTL) 229 n = irintl(fn); 230 #elif defined(HAVE_EFFICIENT_IRINT) 231 n = irint(fn); 232 #else 233 n = (int)fn; 234 #endif 235 n2 = (unsigned)n % INTERVALS; 236 k = n >> LOG2_INTERVALS; 237 r1 = x - fn * L1; 238 r2 = fn * -L2; 239 r = r1 + r2; 240 241 /* Prepare scale factor. */ 242 v.e = 1; 243 v.xbits.expsign = BIAS + k; 244 twopk = v.e; 245 246 /* 247 * Evaluate lower terms of 248 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 249 */ 250 z = r * r; 251 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 252 253 t = (long double)tbl[n2].lo + tbl[n2].hi; 254 255 if (k == 0) { 256 t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + 257 tbl[n2].hi * r1); 258 RETURNI(t); 259 } 260 if (k == -1) { 261 t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + 262 tbl[n2].hi * r1); 263 RETURNI(t / 2); 264 } 265 if (k < -7) { 266 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 267 RETURNI(t * twopk - 1); 268 } 269 if (k > 2 * LDBL_MANT_DIG - 1) { 270 t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 271 if (k == LDBL_MAX_EXP) 272 RETURNI(t * 2 * 0x1p16383L - 1); 273 RETURNI(t * twopk - 1); 274 } 275 276 v.xbits.expsign = BIAS - k; 277 twomk = v.e; 278 279 if (k > LDBL_MANT_DIG - 1) 280 t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); 281 else 282 t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); 283 RETURNI(t * twopk); 284 } 285