| s_expl.c (251339) | s_expl.c (251343) |
|---|---|
| 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 --- 13 unchanged lines hidden (view full) --- 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> | 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 --- 13 unchanged lines hidden (view full) --- 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: head/lib/msun/ld80/s_expl.c 251339 2013-06-03 19:13:44Z kargl $"); | 30__FBSDID("$FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl $"); |
| 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 * --- 258 unchanged lines hidden (view full) --- 297 if (k >= LDBL_MIN_EXP) { 298 if (k == LDBL_MAX_EXP) 299 RETURNI(t * 2 * 0x1p16383L); 300 RETURNI(t * twopk); 301 } else { 302 RETURNI(t * twopkp10000 * twom10000); 303 } 304} | 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 * --- 258 unchanged lines hidden (view full) --- 297 if (k >= LDBL_MIN_EXP) { 298 if (k == LDBL_MAX_EXP) 299 RETURNI(t * 2 * 0x1p16383L); 300 RETURNI(t * twopk); 301 } else { 302 RETURNI(t * twopkp10000 * twom10000); 303 } 304} |
| 305 306/** 307 * Compute expm1l(x) for Intel 80-bit format. This is based on: 308 * 309 * PTP Tang, "Table-driven implementation of the Expm1 function 310 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, 311 * 211-222 (1992). 312 */ 313 314/* 315 * Our T1 and T2 are chosen to be approximately the points where method 316 * A and method B have the same accuracy. Tang's T1 and T2 are the 317 * points where method A's accuracy changes by a full bit. For Tang, 318 * this drop in accuracy makes method A immediately less accurate than 319 * method B, but our larger INTERVALS makes method A 2 bits more 320 * accurate so it remains the most accurate method significantly 321 * closer to the origin despite losing the full bit in our extended 322 * range for it. 323 */ 324static const double 325T1 = -0.1659, /* ~-30.625/128 * log(2) */ 326T2 = 0.1659; /* ~30.625/128 * log(2) */ 327 328/* 329 * Domain [-0.1659, 0.1659], range ~[-1.2027e-22, 3.4417e-22]: 330 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.2 331 */ 332static const union IEEEl2bits 333B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), 334B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); 335 336static const double 337B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ 338B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ 339B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ 340B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ 341B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ 342B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ 343B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ 344B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ 345 346long double 347expm1l(long double x) 348{ 349 union IEEEl2bits u, v; 350 long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; 351 long double x_lo, x2, z; 352 long double x4; 353 int k, n, n2; 354 uint16_t hx, ix; 355 356 /* Filter out exceptional cases. */ 357 u.e = x; 358 hx = u.xbits.expsign; 359 ix = hx & 0x7fff; 360 if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ 361 if (ix == BIAS + LDBL_MAX_EXP) { 362 if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 363 return (-1 / x - 1); 364 return (x + x); /* x is +Inf, +NaN or unsupported */ 365 } 366 if (x > o_threshold) 367 return (huge * huge); 368 /* 369 * expm1l() never underflows, but it must avoid 370 * unrepresentable large negative exponents. We used a 371 * much smaller threshold for large |x| above than in 372 * expl() so as to handle not so large negative exponents 373 * in the same way as large ones here. 374 */ 375 if (hx & 0x8000) /* x <= -64 */ 376 return (tiny - 1); /* good for x < -65ln2 - eps */ 377 } 378 379 ENTERI(); 380 381 if (T1 < x && x < T2) { 382 if (ix < BIAS - 64) { /* |x| < 0x1p-64 (includes pseudos) */ 383 /* x (rounded) with inexact if x != 0: */ 384 RETURNI(x == 0 ? x : 385 (0x1p100 * x + fabsl(x)) * 0x1p-100); 386 } 387 388 x2 = x * x; 389 x4 = x2 * x2; 390 q = x4 * (x2 * (x4 * 391 /* 392 * XXX the number of terms is no longer good for 393 * pairwise grouping of all except B3, and the 394 * grouping is no longer from highest down. 395 */ 396 (x2 * B12 + (x * B11 + B10)) + 397 (x2 * (x * B9 + B8) + (x * B7 + B6))) + 398 (x * B5 + B4.e)) + x2 * x * B3.e; 399 400 x_hi = (float)x; 401 x_lo = x - x_hi; 402 hx2_hi = x_hi * x_hi / 2; 403 hx2_lo = x_lo * (x + x_hi) / 2; 404 if (ix >= BIAS - 7) 405 RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi)); 406 else 407 RETURNI(hx2_lo + q + hx2_hi + x); 408 } 409 410 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 411 /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 412 fn = x * INV_L + 0x1.8p63 - 0x1.8p63; 413#if defined(HAVE_EFFICIENT_IRINTL) 414 n = irintl(fn); 415#elif defined(HAVE_EFFICIENT_IRINT) 416 n = irint(fn); 417#else 418 n = (int)fn; 419#endif 420 n2 = (unsigned)n % INTERVALS; 421 k = n >> LOG2_INTERVALS; 422 r1 = x - fn * L1; 423 r2 = fn * -L2; 424 r = r1 + r2; 425 426 /* Prepare scale factor. */ 427 v.e = 1; 428 v.xbits.expsign = BIAS + k; 429 twopk = v.e; 430 431 /* 432 * Evaluate lower terms of 433 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 434 */ 435 z = r * r; 436 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 437 438 t = (long double)tbl[n2].lo + tbl[n2].hi; 439 440 if (k == 0) { 441 t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + 442 (tbl[n2].hi - 1); 443 RETURNI(t); 444 } 445 if (k == -1) { 446 t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + 447 (tbl[n2].hi - 2); 448 RETURNI(t / 2); 449 } 450 if (k < -7) { 451 t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; 452 RETURNI(t * twopk - 1); 453 } 454 if (k > 2 * LDBL_MANT_DIG - 1) { 455 t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; 456 if (k == LDBL_MAX_EXP) 457 RETURNI(t * 2 * 0x1p16383L - 1); 458 RETURNI(t * twopk - 1); 459 } 460 461 v.xbits.expsign = BIAS - k; 462 twomk = v.e; 463 464 if (k > LDBL_MANT_DIG - 1) 465 t = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi; 466 else 467 t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk); 468 RETURNI(t * twopk); 469} |
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