1/* 2 * QEMU float support 3 * 4 * The code in this source file is derived from release 2a of the SoftFloat 5 * IEC/IEEE Floating-point Arithmetic Package. Those parts of the code (and 6 * some later contributions) are provided under that license, as detailed below. 7 * It has subsequently been modified by contributors to the QEMU Project, 8 * so some portions are provided under: 9 * the SoftFloat-2a license 10 * the BSD license 11 * GPL-v2-or-later 12 * 13 * Any future contributions to this file after December 1st 2014 will be 14 * taken to be licensed under the Softfloat-2a license unless specifically 15 * indicated otherwise. 16 */ 17 18static void partsN(return_nan)(FloatPartsN *a, float_status *s) 19{ 20 switch (a->cls) { 21 case float_class_snan: 22 float_raise(float_flag_invalid, s); 23 if (s->default_nan_mode) { 24 parts_default_nan(a, s); 25 } else { 26 parts_silence_nan(a, s); 27 } 28 break; 29 case float_class_qnan: 30 if (s->default_nan_mode) { 31 parts_default_nan(a, s); 32 } 33 break; 34 default: 35 g_assert_not_reached(); 36 } 37} 38 39static FloatPartsN *partsN(pick_nan)(FloatPartsN *a, FloatPartsN *b, 40 float_status *s) 41{ 42 if (is_snan(a->cls) || is_snan(b->cls)) { 43 float_raise(float_flag_invalid, s); 44 } 45 46 if (s->default_nan_mode) { 47 parts_default_nan(a, s); 48 } else { 49 int cmp = frac_cmp(a, b); 50 if (cmp == 0) { 51 cmp = a->sign < b->sign; 52 } 53 54 if (pickNaN(a->cls, b->cls, cmp > 0, s)) { 55 a = b; 56 } 57 if (is_snan(a->cls)) { 58 parts_silence_nan(a, s); 59 } 60 } 61 return a; 62} 63 64static FloatPartsN *partsN(pick_nan_muladd)(FloatPartsN *a, FloatPartsN *b, 65 FloatPartsN *c, float_status *s, 66 int ab_mask, int abc_mask) 67{ 68 int which; 69 70 if (unlikely(abc_mask & float_cmask_snan)) { 71 float_raise(float_flag_invalid, s); 72 } 73 74 which = pickNaNMulAdd(a->cls, b->cls, c->cls, 75 ab_mask == float_cmask_infzero, s); 76 77 if (s->default_nan_mode || which == 3) { 78 /* 79 * Note that this check is after pickNaNMulAdd so that function 80 * has an opportunity to set the Invalid flag for infzero. 81 */ 82 parts_default_nan(a, s); 83 return a; 84 } 85 86 switch (which) { 87 case 0: 88 break; 89 case 1: 90 a = b; 91 break; 92 case 2: 93 a = c; 94 break; 95 default: 96 g_assert_not_reached(); 97 } 98 if (is_snan(a->cls)) { 99 parts_silence_nan(a, s); 100 } 101 return a; 102} 103 104/* 105 * Canonicalize the FloatParts structure. Determine the class, 106 * unbias the exponent, and normalize the fraction. 107 */ 108static void partsN(canonicalize)(FloatPartsN *p, float_status *status, 109 const FloatFmt *fmt) 110{ 111 if (unlikely(p->exp == 0)) { 112 if (likely(frac_eqz(p))) { 113 p->cls = float_class_zero; 114 } else if (status->flush_inputs_to_zero) { 115 float_raise(float_flag_input_denormal, status); 116 p->cls = float_class_zero; 117 frac_clear(p); 118 } else { 119 int shift = frac_normalize(p); 120 p->cls = float_class_normal; 121 p->exp = fmt->frac_shift - fmt->exp_bias - shift + 1; 122 } 123 } else if (likely(p->exp < fmt->exp_max) || fmt->arm_althp) { 124 p->cls = float_class_normal; 125 p->exp -= fmt->exp_bias; 126 frac_shl(p, fmt->frac_shift); 127 p->frac_hi |= DECOMPOSED_IMPLICIT_BIT; 128 } else if (likely(frac_eqz(p))) { 129 p->cls = float_class_inf; 130 } else { 131 frac_shl(p, fmt->frac_shift); 132 p->cls = (parts_is_snan_frac(p->frac_hi, status) 133 ? float_class_snan : float_class_qnan); 134 } 135} 136 137/* 138 * Round and uncanonicalize a floating-point number by parts. There 139 * are FRAC_SHIFT bits that may require rounding at the bottom of the 140 * fraction; these bits will be removed. The exponent will be biased 141 * by EXP_BIAS and must be bounded by [EXP_MAX-1, 0]. 142 */ 143static void partsN(uncanon_normal)(FloatPartsN *p, float_status *s, 144 const FloatFmt *fmt) 145{ 146 const int exp_max = fmt->exp_max; 147 const int frac_shift = fmt->frac_shift; 148 const uint64_t round_mask = fmt->round_mask; 149 const uint64_t frac_lsb = round_mask + 1; 150 const uint64_t frac_lsbm1 = round_mask ^ (round_mask >> 1); 151 const uint64_t roundeven_mask = round_mask | frac_lsb; 152 uint64_t inc; 153 bool overflow_norm = false; 154 int exp, flags = 0; 155 156 switch (s->float_rounding_mode) { 157 case float_round_nearest_even: 158 if (N > 64 && frac_lsb == 0) { 159 inc = ((p->frac_hi & 1) || (p->frac_lo & round_mask) != frac_lsbm1 160 ? frac_lsbm1 : 0); 161 } else { 162 inc = ((p->frac_lo & roundeven_mask) != frac_lsbm1 163 ? frac_lsbm1 : 0); 164 } 165 break; 166 case float_round_ties_away: 167 inc = frac_lsbm1; 168 break; 169 case float_round_to_zero: 170 overflow_norm = true; 171 inc = 0; 172 break; 173 case float_round_up: 174 inc = p->sign ? 0 : round_mask; 175 overflow_norm = p->sign; 176 break; 177 case float_round_down: 178 inc = p->sign ? round_mask : 0; 179 overflow_norm = !p->sign; 180 break; 181 case float_round_to_odd: 182 overflow_norm = true; 183 /* fall through */ 184 case float_round_to_odd_inf: 185 if (N > 64 && frac_lsb == 0) { 186 inc = p->frac_hi & 1 ? 0 : round_mask; 187 } else { 188 inc = p->frac_lo & frac_lsb ? 0 : round_mask; 189 } 190 break; 191 default: 192 g_assert_not_reached(); 193 } 194 195 exp = p->exp + fmt->exp_bias; 196 if (likely(exp > 0)) { 197 if (p->frac_lo & round_mask) { 198 flags |= float_flag_inexact; 199 if (frac_addi(p, p, inc)) { 200 frac_shr(p, 1); 201 p->frac_hi |= DECOMPOSED_IMPLICIT_BIT; 202 exp++; 203 } 204 p->frac_lo &= ~round_mask; 205 } 206 207 if (fmt->arm_althp) { 208 /* ARM Alt HP eschews Inf and NaN for a wider exponent. */ 209 if (unlikely(exp > exp_max)) { 210 /* Overflow. Return the maximum normal. */ 211 flags = float_flag_invalid; 212 exp = exp_max; 213 frac_allones(p); 214 p->frac_lo &= ~round_mask; 215 } 216 } else if (unlikely(exp >= exp_max)) { 217 flags |= float_flag_overflow | float_flag_inexact; 218 if (overflow_norm) { 219 exp = exp_max - 1; 220 frac_allones(p); 221 p->frac_lo &= ~round_mask; 222 } else { 223 p->cls = float_class_inf; 224 exp = exp_max; 225 frac_clear(p); 226 } 227 } 228 frac_shr(p, frac_shift); 229 } else if (s->flush_to_zero) { 230 flags |= float_flag_output_denormal; 231 p->cls = float_class_zero; 232 exp = 0; 233 frac_clear(p); 234 } else { 235 bool is_tiny = s->tininess_before_rounding || exp < 0; 236 237 if (!is_tiny) { 238 FloatPartsN discard; 239 is_tiny = !frac_addi(&discard, p, inc); 240 } 241 242 frac_shrjam(p, 1 - exp); 243 244 if (p->frac_lo & round_mask) { 245 /* Need to recompute round-to-even/round-to-odd. */ 246 switch (s->float_rounding_mode) { 247 case float_round_nearest_even: 248 if (N > 64 && frac_lsb == 0) { 249 inc = ((p->frac_hi & 1) || 250 (p->frac_lo & round_mask) != frac_lsbm1 251 ? frac_lsbm1 : 0); 252 } else { 253 inc = ((p->frac_lo & roundeven_mask) != frac_lsbm1 254 ? frac_lsbm1 : 0); 255 } 256 break; 257 case float_round_to_odd: 258 case float_round_to_odd_inf: 259 if (N > 64 && frac_lsb == 0) { 260 inc = p->frac_hi & 1 ? 0 : round_mask; 261 } else { 262 inc = p->frac_lo & frac_lsb ? 0 : round_mask; 263 } 264 break; 265 default: 266 break; 267 } 268 flags |= float_flag_inexact; 269 frac_addi(p, p, inc); 270 p->frac_lo &= ~round_mask; 271 } 272 273 exp = (p->frac_hi & DECOMPOSED_IMPLICIT_BIT) != 0; 274 frac_shr(p, frac_shift); 275 276 if (is_tiny && (flags & float_flag_inexact)) { 277 flags |= float_flag_underflow; 278 } 279 if (exp == 0 && frac_eqz(p)) { 280 p->cls = float_class_zero; 281 } 282 } 283 p->exp = exp; 284 float_raise(flags, s); 285} 286 287static void partsN(uncanon)(FloatPartsN *p, float_status *s, 288 const FloatFmt *fmt) 289{ 290 if (likely(p->cls == float_class_normal)) { 291 parts_uncanon_normal(p, s, fmt); 292 } else { 293 switch (p->cls) { 294 case float_class_zero: 295 p->exp = 0; 296 frac_clear(p); 297 return; 298 case float_class_inf: 299 g_assert(!fmt->arm_althp); 300 p->exp = fmt->exp_max; 301 frac_clear(p); 302 return; 303 case float_class_qnan: 304 case float_class_snan: 305 g_assert(!fmt->arm_althp); 306 p->exp = fmt->exp_max; 307 frac_shr(p, fmt->frac_shift); 308 return; 309 default: 310 break; 311 } 312 g_assert_not_reached(); 313 } 314} 315 316/* 317 * Returns the result of adding or subtracting the values of the 318 * floating-point values `a' and `b'. The operation is performed 319 * according to the IEC/IEEE Standard for Binary Floating-Point 320 * Arithmetic. 321 */ 322static FloatPartsN *partsN(addsub)(FloatPartsN *a, FloatPartsN *b, 323 float_status *s, bool subtract) 324{ 325 bool b_sign = b->sign ^ subtract; 326 int ab_mask = float_cmask(a->cls) | float_cmask(b->cls); 327 328 if (a->sign != b_sign) { 329 /* Subtraction */ 330 if (likely(ab_mask == float_cmask_normal)) { 331 if (parts_sub_normal(a, b)) { 332 return a; 333 } 334 /* Subtract was exact, fall through to set sign. */ 335 ab_mask = float_cmask_zero; 336 } 337 338 if (ab_mask == float_cmask_zero) { 339 a->sign = s->float_rounding_mode == float_round_down; 340 return a; 341 } 342 343 if (unlikely(ab_mask & float_cmask_anynan)) { 344 goto p_nan; 345 } 346 347 if (ab_mask & float_cmask_inf) { 348 if (a->cls != float_class_inf) { 349 /* N - Inf */ 350 goto return_b; 351 } 352 if (b->cls != float_class_inf) { 353 /* Inf - N */ 354 return a; 355 } 356 /* Inf - Inf */ 357 float_raise(float_flag_invalid, s); 358 parts_default_nan(a, s); 359 return a; 360 } 361 } else { 362 /* Addition */ 363 if (likely(ab_mask == float_cmask_normal)) { 364 parts_add_normal(a, b); 365 return a; 366 } 367 368 if (ab_mask == float_cmask_zero) { 369 return a; 370 } 371 372 if (unlikely(ab_mask & float_cmask_anynan)) { 373 goto p_nan; 374 } 375 376 if (ab_mask & float_cmask_inf) { 377 a->cls = float_class_inf; 378 return a; 379 } 380 } 381 382 if (b->cls == float_class_zero) { 383 g_assert(a->cls == float_class_normal); 384 return a; 385 } 386 387 g_assert(a->cls == float_class_zero); 388 g_assert(b->cls == float_class_normal); 389 return_b: 390 b->sign = b_sign; 391 return b; 392 393 p_nan: 394 return parts_pick_nan(a, b, s); 395} 396 397/* 398 * Returns the result of multiplying the floating-point values `a' and 399 * `b'. The operation is performed according to the IEC/IEEE Standard 400 * for Binary Floating-Point Arithmetic. 401 */ 402static FloatPartsN *partsN(mul)(FloatPartsN *a, FloatPartsN *b, 403 float_status *s) 404{ 405 int ab_mask = float_cmask(a->cls) | float_cmask(b->cls); 406 bool sign = a->sign ^ b->sign; 407 408 if (likely(ab_mask == float_cmask_normal)) { 409 FloatPartsW tmp; 410 411 frac_mulw(&tmp, a, b); 412 frac_truncjam(a, &tmp); 413 414 a->exp += b->exp + 1; 415 if (!(a->frac_hi & DECOMPOSED_IMPLICIT_BIT)) { 416 frac_add(a, a, a); 417 a->exp -= 1; 418 } 419 420 a->sign = sign; 421 return a; 422 } 423 424 /* Inf * Zero == NaN */ 425 if (unlikely(ab_mask == float_cmask_infzero)) { 426 float_raise(float_flag_invalid, s); 427 parts_default_nan(a, s); 428 return a; 429 } 430 431 if (unlikely(ab_mask & float_cmask_anynan)) { 432 return parts_pick_nan(a, b, s); 433 } 434 435 /* Multiply by 0 or Inf */ 436 if (ab_mask & float_cmask_inf) { 437 a->cls = float_class_inf; 438 a->sign = sign; 439 return a; 440 } 441 442 g_assert(ab_mask & float_cmask_zero); 443 a->cls = float_class_zero; 444 a->sign = sign; 445 return a; 446} 447 448/* 449 * Returns the result of multiplying the floating-point values `a' and 450 * `b' then adding 'c', with no intermediate rounding step after the 451 * multiplication. The operation is performed according to the 452 * IEC/IEEE Standard for Binary Floating-Point Arithmetic 754-2008. 453 * The flags argument allows the caller to select negation of the 454 * addend, the intermediate product, or the final result. (The 455 * difference between this and having the caller do a separate 456 * negation is that negating externally will flip the sign bit on NaNs.) 457 * 458 * Requires A and C extracted into a double-sized structure to provide the 459 * extra space for the widening multiply. 460 */ 461static FloatPartsN *partsN(muladd)(FloatPartsN *a, FloatPartsN *b, 462 FloatPartsN *c, int flags, float_status *s) 463{ 464 int ab_mask, abc_mask; 465 FloatPartsW p_widen, c_widen; 466 467 ab_mask = float_cmask(a->cls) | float_cmask(b->cls); 468 abc_mask = float_cmask(c->cls) | ab_mask; 469 470 /* 471 * It is implementation-defined whether the cases of (0,inf,qnan) 472 * and (inf,0,qnan) raise InvalidOperation or not (and what QNaN 473 * they return if they do), so we have to hand this information 474 * off to the target-specific pick-a-NaN routine. 475 */ 476 if (unlikely(abc_mask & float_cmask_anynan)) { 477 return parts_pick_nan_muladd(a, b, c, s, ab_mask, abc_mask); 478 } 479 480 if (flags & float_muladd_negate_c) { 481 c->sign ^= 1; 482 } 483 484 /* Compute the sign of the product into A. */ 485 a->sign ^= b->sign; 486 if (flags & float_muladd_negate_product) { 487 a->sign ^= 1; 488 } 489 490 if (unlikely(ab_mask != float_cmask_normal)) { 491 if (unlikely(ab_mask == float_cmask_infzero)) { 492 goto d_nan; 493 } 494 495 if (ab_mask & float_cmask_inf) { 496 if (c->cls == float_class_inf && a->sign != c->sign) { 497 goto d_nan; 498 } 499 goto return_inf; 500 } 501 502 g_assert(ab_mask & float_cmask_zero); 503 if (c->cls == float_class_normal) { 504 *a = *c; 505 goto return_normal; 506 } 507 if (c->cls == float_class_zero) { 508 if (a->sign != c->sign) { 509 goto return_sub_zero; 510 } 511 goto return_zero; 512 } 513 g_assert(c->cls == float_class_inf); 514 } 515 516 if (unlikely(c->cls == float_class_inf)) { 517 a->sign = c->sign; 518 goto return_inf; 519 } 520 521 /* Perform the multiplication step. */ 522 p_widen.sign = a->sign; 523 p_widen.exp = a->exp + b->exp + 1; 524 frac_mulw(&p_widen, a, b); 525 if (!(p_widen.frac_hi & DECOMPOSED_IMPLICIT_BIT)) { 526 frac_add(&p_widen, &p_widen, &p_widen); 527 p_widen.exp -= 1; 528 } 529 530 /* Perform the addition step. */ 531 if (c->cls != float_class_zero) { 532 /* Zero-extend C to less significant bits. */ 533 frac_widen(&c_widen, c); 534 c_widen.exp = c->exp; 535 536 if (a->sign == c->sign) { 537 parts_add_normal(&p_widen, &c_widen); 538 } else if (!parts_sub_normal(&p_widen, &c_widen)) { 539 goto return_sub_zero; 540 } 541 } 542 543 /* Narrow with sticky bit, for proper rounding later. */ 544 frac_truncjam(a, &p_widen); 545 a->sign = p_widen.sign; 546 a->exp = p_widen.exp; 547 548 return_normal: 549 if (flags & float_muladd_halve_result) { 550 a->exp -= 1; 551 } 552 finish_sign: 553 if (flags & float_muladd_negate_result) { 554 a->sign ^= 1; 555 } 556 return a; 557 558 return_sub_zero: 559 a->sign = s->float_rounding_mode == float_round_down; 560 return_zero: 561 a->cls = float_class_zero; 562 goto finish_sign; 563 564 return_inf: 565 a->cls = float_class_inf; 566 goto finish_sign; 567 568 d_nan: 569 float_raise(float_flag_invalid, s); 570 parts_default_nan(a, s); 571 return a; 572} 573 574/* 575 * Returns the result of dividing the floating-point value `a' by the 576 * corresponding value `b'. The operation is performed according to 577 * the IEC/IEEE Standard for Binary Floating-Point Arithmetic. 578 */ 579static FloatPartsN *partsN(div)(FloatPartsN *a, FloatPartsN *b, 580 float_status *s) 581{ 582 int ab_mask = float_cmask(a->cls) | float_cmask(b->cls); 583 bool sign = a->sign ^ b->sign; 584 585 if (likely(ab_mask == float_cmask_normal)) { 586 a->sign = sign; 587 a->exp -= b->exp + frac_div(a, b); 588 return a; 589 } 590 591 /* 0/0 or Inf/Inf => NaN */ 592 if (unlikely(ab_mask == float_cmask_zero) || 593 unlikely(ab_mask == float_cmask_inf)) { 594 float_raise(float_flag_invalid, s); 595 parts_default_nan(a, s); 596 return a; 597 } 598 599 /* All the NaN cases */ 600 if (unlikely(ab_mask & float_cmask_anynan)) { 601 return parts_pick_nan(a, b, s); 602 } 603 604 a->sign = sign; 605 606 /* Inf / X */ 607 if (a->cls == float_class_inf) { 608 return a; 609 } 610 611 /* 0 / X */ 612 if (a->cls == float_class_zero) { 613 return a; 614 } 615 616 /* X / Inf */ 617 if (b->cls == float_class_inf) { 618 a->cls = float_class_zero; 619 return a; 620 } 621 622 /* X / 0 => Inf */ 623 g_assert(b->cls == float_class_zero); 624 float_raise(float_flag_divbyzero, s); 625 a->cls = float_class_inf; 626 return a; 627} 628 629/* 630 * Floating point remainder, per IEC/IEEE, or modulus. 631 */ 632static FloatPartsN *partsN(modrem)(FloatPartsN *a, FloatPartsN *b, 633 uint64_t *mod_quot, float_status *s) 634{ 635 int ab_mask = float_cmask(a->cls) | float_cmask(b->cls); 636 637 if (likely(ab_mask == float_cmask_normal)) { 638 frac_modrem(a, b, mod_quot); 639 return a; 640 } 641 642 if (mod_quot) { 643 *mod_quot = 0; 644 } 645 646 /* All the NaN cases */ 647 if (unlikely(ab_mask & float_cmask_anynan)) { 648 return parts_pick_nan(a, b, s); 649 } 650 651 /* Inf % N; N % 0 */ 652 if (a->cls == float_class_inf || b->cls == float_class_zero) { 653 float_raise(float_flag_invalid, s); 654 parts_default_nan(a, s); 655 return a; 656 } 657 658 /* N % Inf; 0 % N */ 659 g_assert(b->cls == float_class_inf || a->cls == float_class_zero); 660 return a; 661} 662 663/* 664 * Square Root 665 * 666 * The base algorithm is lifted from 667 * https://git.musl-libc.org/cgit/musl/tree/src/math/sqrtf.c 668 * https://git.musl-libc.org/cgit/musl/tree/src/math/sqrt.c 669 * https://git.musl-libc.org/cgit/musl/tree/src/math/sqrtl.c 670 * and is thus MIT licenced. 671 */ 672static void partsN(sqrt)(FloatPartsN *a, float_status *status, 673 const FloatFmt *fmt) 674{ 675 const uint32_t three32 = 3u << 30; 676 const uint64_t three64 = 3ull << 62; 677 uint32_t d32, m32, r32, s32, u32; /* 32-bit computation */ 678 uint64_t d64, m64, r64, s64, u64; /* 64-bit computation */ 679 uint64_t dh, dl, rh, rl, sh, sl, uh, ul; /* 128-bit computation */ 680 uint64_t d0h, d0l, d1h, d1l, d2h, d2l; 681 uint64_t discard; 682 bool exp_odd; 683 size_t index; 684 685 if (unlikely(a->cls != float_class_normal)) { 686 switch (a->cls) { 687 case float_class_snan: 688 case float_class_qnan: 689 parts_return_nan(a, status); 690 return; 691 case float_class_zero: 692 return; 693 case float_class_inf: 694 if (unlikely(a->sign)) { 695 goto d_nan; 696 } 697 return; 698 default: 699 g_assert_not_reached(); 700 } 701 } 702 703 if (unlikely(a->sign)) { 704 goto d_nan; 705 } 706 707 /* 708 * Argument reduction. 709 * x = 4^e frac; with integer e, and frac in [1, 4) 710 * m = frac fixed point at bit 62, since we're in base 4. 711 * If base-2 exponent is odd, exchange that for multiply by 2, 712 * which results in no shift. 713 */ 714 exp_odd = a->exp & 1; 715 index = extract64(a->frac_hi, 57, 6) | (!exp_odd << 6); 716 if (!exp_odd) { 717 frac_shr(a, 1); 718 } 719 720 /* 721 * Approximate r ~= 1/sqrt(m) and s ~= sqrt(m) when m in [1, 4). 722 * 723 * Initial estimate: 724 * 7-bit lookup table (1-bit exponent and 6-bit significand). 725 * 726 * The relative error (e = r0*sqrt(m)-1) of a linear estimate 727 * (r0 = a*m + b) is |e| < 0.085955 ~ 0x1.6p-4 at best; 728 * a table lookup is faster and needs one less iteration. 729 * The 7-bit table gives |e| < 0x1.fdp-9. 730 * 731 * A Newton-Raphson iteration for r is 732 * s = m*r 733 * d = s*r 734 * u = 3 - d 735 * r = r*u/2 736 * 737 * Fixed point representations: 738 * m, s, d, u, three are all 2.30; r is 0.32 739 */ 740 m64 = a->frac_hi; 741 m32 = m64 >> 32; 742 743 r32 = rsqrt_tab[index] << 16; 744 /* |r*sqrt(m) - 1| < 0x1.FDp-9 */ 745 746 s32 = ((uint64_t)m32 * r32) >> 32; 747 d32 = ((uint64_t)s32 * r32) >> 32; 748 u32 = three32 - d32; 749 750 if (N == 64) { 751 /* float64 or smaller */ 752 753 r32 = ((uint64_t)r32 * u32) >> 31; 754 /* |r*sqrt(m) - 1| < 0x1.7Bp-16 */ 755 756 s32 = ((uint64_t)m32 * r32) >> 32; 757 d32 = ((uint64_t)s32 * r32) >> 32; 758 u32 = three32 - d32; 759 760 if (fmt->frac_size <= 23) { 761 /* float32 or smaller */ 762 763 s32 = ((uint64_t)s32 * u32) >> 32; /* 3.29 */ 764 s32 = (s32 - 1) >> 6; /* 9.23 */ 765 /* s < sqrt(m) < s + 0x1.08p-23 */ 766 767 /* compute nearest rounded result to 2.23 bits */ 768 uint32_t d0 = (m32 << 16) - s32 * s32; 769 uint32_t d1 = s32 - d0; 770 uint32_t d2 = d1 + s32 + 1; 771 s32 += d1 >> 31; 772 a->frac_hi = (uint64_t)s32 << (64 - 25); 773 774 /* increment or decrement for inexact */ 775 if (d2 != 0) { 776 a->frac_hi += ((int32_t)(d1 ^ d2) < 0 ? -1 : 1); 777 } 778 goto done; 779 } 780 781 /* float64 */ 782 783 r64 = (uint64_t)r32 * u32 * 2; 784 /* |r*sqrt(m) - 1| < 0x1.37-p29; convert to 64-bit arithmetic */ 785 mul64To128(m64, r64, &s64, &discard); 786 mul64To128(s64, r64, &d64, &discard); 787 u64 = three64 - d64; 788 789 mul64To128(s64, u64, &s64, &discard); /* 3.61 */ 790 s64 = (s64 - 2) >> 9; /* 12.52 */ 791 792 /* Compute nearest rounded result */ 793 uint64_t d0 = (m64 << 42) - s64 * s64; 794 uint64_t d1 = s64 - d0; 795 uint64_t d2 = d1 + s64 + 1; 796 s64 += d1 >> 63; 797 a->frac_hi = s64 << (64 - 54); 798 799 /* increment or decrement for inexact */ 800 if (d2 != 0) { 801 a->frac_hi += ((int64_t)(d1 ^ d2) < 0 ? -1 : 1); 802 } 803 goto done; 804 } 805 806 r64 = (uint64_t)r32 * u32 * 2; 807 /* |r*sqrt(m) - 1| < 0x1.7Bp-16; convert to 64-bit arithmetic */ 808 809 mul64To128(m64, r64, &s64, &discard); 810 mul64To128(s64, r64, &d64, &discard); 811 u64 = three64 - d64; 812 mul64To128(u64, r64, &r64, &discard); 813 r64 <<= 1; 814 /* |r*sqrt(m) - 1| < 0x1.a5p-31 */ 815 816 mul64To128(m64, r64, &s64, &discard); 817 mul64To128(s64, r64, &d64, &discard); 818 u64 = three64 - d64; 819 mul64To128(u64, r64, &rh, &rl); 820 add128(rh, rl, rh, rl, &rh, &rl); 821 /* |r*sqrt(m) - 1| < 0x1.c001p-59; change to 128-bit arithmetic */ 822 823 mul128To256(a->frac_hi, a->frac_lo, rh, rl, &sh, &sl, &discard, &discard); 824 mul128To256(sh, sl, rh, rl, &dh, &dl, &discard, &discard); 825 sub128(three64, 0, dh, dl, &uh, &ul); 826 mul128To256(uh, ul, sh, sl, &sh, &sl, &discard, &discard); /* 3.125 */ 827 /* -0x1p-116 < s - sqrt(m) < 0x3.8001p-125 */ 828 829 sub128(sh, sl, 0, 4, &sh, &sl); 830 shift128Right(sh, sl, 13, &sh, &sl); /* 16.112 */ 831 /* s < sqrt(m) < s + 1ulp */ 832 833 /* Compute nearest rounded result */ 834 mul64To128(sl, sl, &d0h, &d0l); 835 d0h += 2 * sh * sl; 836 sub128(a->frac_lo << 34, 0, d0h, d0l, &d0h, &d0l); 837 sub128(sh, sl, d0h, d0l, &d1h, &d1l); 838 add128(sh, sl, 0, 1, &d2h, &d2l); 839 add128(d2h, d2l, d1h, d1l, &d2h, &d2l); 840 add128(sh, sl, 0, d1h >> 63, &sh, &sl); 841 shift128Left(sh, sl, 128 - 114, &sh, &sl); 842 843 /* increment or decrement for inexact */ 844 if (d2h | d2l) { 845 if ((int64_t)(d1h ^ d2h) < 0) { 846 sub128(sh, sl, 0, 1, &sh, &sl); 847 } else { 848 add128(sh, sl, 0, 1, &sh, &sl); 849 } 850 } 851 a->frac_lo = sl; 852 a->frac_hi = sh; 853 854 done: 855 /* Convert back from base 4 to base 2. */ 856 a->exp >>= 1; 857 if (!(a->frac_hi & DECOMPOSED_IMPLICIT_BIT)) { 858 frac_add(a, a, a); 859 } else { 860 a->exp += 1; 861 } 862 return; 863 864 d_nan: 865 float_raise(float_flag_invalid, status); 866 parts_default_nan(a, status); 867} 868 869/* 870 * Rounds the floating-point value `a' to an integer, and returns the 871 * result as a floating-point value. The operation is performed 872 * according to the IEC/IEEE Standard for Binary Floating-Point 873 * Arithmetic. 874 * 875 * parts_round_to_int_normal is an internal helper function for 876 * normal numbers only, returning true for inexact but not directly 877 * raising float_flag_inexact. 878 */ 879static bool partsN(round_to_int_normal)(FloatPartsN *a, FloatRoundMode rmode, 880 int scale, int frac_size) 881{ 882 uint64_t frac_lsb, frac_lsbm1, rnd_even_mask, rnd_mask, inc; 883 int shift_adj; 884 885 scale = MIN(MAX(scale, -0x10000), 0x10000); 886 a->exp += scale; 887 888 if (a->exp < 0) { 889 bool one; 890 891 /* All fractional */ 892 switch (rmode) { 893 case float_round_nearest_even: 894 one = false; 895 if (a->exp == -1) { 896 FloatPartsN tmp; 897 /* Shift left one, discarding DECOMPOSED_IMPLICIT_BIT */ 898 frac_add(&tmp, a, a); 899 /* Anything remaining means frac > 0.5. */ 900 one = !frac_eqz(&tmp); 901 } 902 break; 903 case float_round_ties_away: 904 one = a->exp == -1; 905 break; 906 case float_round_to_zero: 907 one = false; 908 break; 909 case float_round_up: 910 one = !a->sign; 911 break; 912 case float_round_down: 913 one = a->sign; 914 break; 915 case float_round_to_odd: 916 one = true; 917 break; 918 default: 919 g_assert_not_reached(); 920 } 921 922 frac_clear(a); 923 a->exp = 0; 924 if (one) { 925 a->frac_hi = DECOMPOSED_IMPLICIT_BIT; 926 } else { 927 a->cls = float_class_zero; 928 } 929 return true; 930 } 931 932 if (a->exp >= frac_size) { 933 /* All integral */ 934 return false; 935 } 936 937 if (N > 64 && a->exp < N - 64) { 938 /* 939 * Rounding is not in the low word -- shift lsb to bit 2, 940 * which leaves room for sticky and rounding bit. 941 */ 942 shift_adj = (N - 1) - (a->exp + 2); 943 frac_shrjam(a, shift_adj); 944 frac_lsb = 1 << 2; 945 } else { 946 shift_adj = 0; 947 frac_lsb = DECOMPOSED_IMPLICIT_BIT >> (a->exp & 63); 948 } 949 950 frac_lsbm1 = frac_lsb >> 1; 951 rnd_mask = frac_lsb - 1; 952 rnd_even_mask = rnd_mask | frac_lsb; 953 954 if (!(a->frac_lo & rnd_mask)) { 955 /* Fractional bits already clear, undo the shift above. */ 956 frac_shl(a, shift_adj); 957 return false; 958 } 959 960 switch (rmode) { 961 case float_round_nearest_even: 962 inc = ((a->frac_lo & rnd_even_mask) != frac_lsbm1 ? frac_lsbm1 : 0); 963 break; 964 case float_round_ties_away: 965 inc = frac_lsbm1; 966 break; 967 case float_round_to_zero: 968 inc = 0; 969 break; 970 case float_round_up: 971 inc = a->sign ? 0 : rnd_mask; 972 break; 973 case float_round_down: 974 inc = a->sign ? rnd_mask : 0; 975 break; 976 case float_round_to_odd: 977 inc = a->frac_lo & frac_lsb ? 0 : rnd_mask; 978 break; 979 default: 980 g_assert_not_reached(); 981 } 982 983 if (shift_adj == 0) { 984 if (frac_addi(a, a, inc)) { 985 frac_shr(a, 1); 986 a->frac_hi |= DECOMPOSED_IMPLICIT_BIT; 987 a->exp++; 988 } 989 a->frac_lo &= ~rnd_mask; 990 } else { 991 frac_addi(a, a, inc); 992 a->frac_lo &= ~rnd_mask; 993 /* Be careful shifting back, not to overflow */ 994 frac_shl(a, shift_adj - 1); 995 if (a->frac_hi & DECOMPOSED_IMPLICIT_BIT) { 996 a->exp++; 997 } else { 998 frac_add(a, a, a); 999 } 1000 } 1001 return true; 1002} 1003 1004static void partsN(round_to_int)(FloatPartsN *a, FloatRoundMode rmode, 1005 int scale, float_status *s, 1006 const FloatFmt *fmt) 1007{ 1008 switch (a->cls) { 1009 case float_class_qnan: 1010 case float_class_snan: 1011 parts_return_nan(a, s); 1012 break; 1013 case float_class_zero: 1014 case float_class_inf: 1015 break; 1016 case float_class_normal: 1017 if (parts_round_to_int_normal(a, rmode, scale, fmt->frac_size)) { 1018 float_raise(float_flag_inexact, s); 1019 } 1020 break; 1021 default: 1022 g_assert_not_reached(); 1023 } 1024} 1025 1026/* 1027 * Returns the result of converting the floating-point value `a' to 1028 * the two's complement integer format. The conversion is performed 1029 * according to the IEC/IEEE Standard for Binary Floating-Point 1030 * Arithmetic---which means in particular that the conversion is 1031 * rounded according to the current rounding mode. If `a' is a NaN, 1032 * the largest positive integer is returned. Otherwise, if the 1033 * conversion overflows, the largest integer with the same sign as `a' 1034 * is returned. 1035 */ 1036static int64_t partsN(float_to_sint)(FloatPartsN *p, FloatRoundMode rmode, 1037 int scale, int64_t min, int64_t max, 1038 float_status *s) 1039{ 1040 int flags = 0; 1041 uint64_t r; 1042 1043 switch (p->cls) { 1044 case float_class_snan: 1045 case float_class_qnan: 1046 flags = float_flag_invalid; 1047 r = max; 1048 break; 1049 1050 case float_class_inf: 1051 flags = float_flag_invalid; 1052 r = p->sign ? min : max; 1053 break; 1054 1055 case float_class_zero: 1056 return 0; 1057 1058 case float_class_normal: 1059 /* TODO: N - 2 is frac_size for rounding; could use input fmt. */ 1060 if (parts_round_to_int_normal(p, rmode, scale, N - 2)) { 1061 flags = float_flag_inexact; 1062 } 1063 1064 if (p->exp <= DECOMPOSED_BINARY_POINT) { 1065 r = p->frac_hi >> (DECOMPOSED_BINARY_POINT - p->exp); 1066 } else { 1067 r = UINT64_MAX; 1068 } 1069 if (p->sign) { 1070 if (r <= -(uint64_t)min) { 1071 r = -r; 1072 } else { 1073 flags = float_flag_invalid; 1074 r = min; 1075 } 1076 } else if (r > max) { 1077 flags = float_flag_invalid; 1078 r = max; 1079 } 1080 break; 1081 1082 default: 1083 g_assert_not_reached(); 1084 } 1085 1086 float_raise(flags, s); 1087 return r; 1088} 1089 1090/* 1091 * Returns the result of converting the floating-point value `a' to 1092 * the unsigned integer format. The conversion is performed according 1093 * to the IEC/IEEE Standard for Binary Floating-Point 1094 * Arithmetic---which means in particular that the conversion is 1095 * rounded according to the current rounding mode. If `a' is a NaN, 1096 * the largest unsigned integer is returned. Otherwise, if the 1097 * conversion overflows, the largest unsigned integer is returned. If 1098 * the 'a' is negative, the result is rounded and zero is returned; 1099 * values that do not round to zero will raise the inexact exception 1100 * flag. 1101 */ 1102static uint64_t partsN(float_to_uint)(FloatPartsN *p, FloatRoundMode rmode, 1103 int scale, uint64_t max, float_status *s) 1104{ 1105 int flags = 0; 1106 uint64_t r; 1107 1108 switch (p->cls) { 1109 case float_class_snan: 1110 case float_class_qnan: 1111 flags = float_flag_invalid; 1112 r = max; 1113 break; 1114 1115 case float_class_inf: 1116 flags = float_flag_invalid; 1117 r = p->sign ? 0 : max; 1118 break; 1119 1120 case float_class_zero: 1121 return 0; 1122 1123 case float_class_normal: 1124 /* TODO: N - 2 is frac_size for rounding; could use input fmt. */ 1125 if (parts_round_to_int_normal(p, rmode, scale, N - 2)) { 1126 flags = float_flag_inexact; 1127 if (p->cls == float_class_zero) { 1128 r = 0; 1129 break; 1130 } 1131 } 1132 1133 if (p->sign) { 1134 flags = float_flag_invalid; 1135 r = 0; 1136 } else if (p->exp > DECOMPOSED_BINARY_POINT) { 1137 flags = float_flag_invalid; 1138 r = max; 1139 } else { 1140 r = p->frac_hi >> (DECOMPOSED_BINARY_POINT - p->exp); 1141 if (r > max) { 1142 flags = float_flag_invalid; 1143 r = max; 1144 } 1145 } 1146 break; 1147 1148 default: 1149 g_assert_not_reached(); 1150 } 1151 1152 float_raise(flags, s); 1153 return r; 1154} 1155 1156/* 1157 * Integer to float conversions 1158 * 1159 * Returns the result of converting the two's complement integer `a' 1160 * to the floating-point format. The conversion is performed according 1161 * to the IEC/IEEE Standard for Binary Floating-Point Arithmetic. 1162 */ 1163static void partsN(sint_to_float)(FloatPartsN *p, int64_t a, 1164 int scale, float_status *s) 1165{ 1166 uint64_t f = a; 1167 int shift; 1168 1169 memset(p, 0, sizeof(*p)); 1170 1171 if (a == 0) { 1172 p->cls = float_class_zero; 1173 return; 1174 } 1175 1176 p->cls = float_class_normal; 1177 if (a < 0) { 1178 f = -f; 1179 p->sign = true; 1180 } 1181 shift = clz64(f); 1182 scale = MIN(MAX(scale, -0x10000), 0x10000); 1183 1184 p->exp = DECOMPOSED_BINARY_POINT - shift + scale; 1185 p->frac_hi = f << shift; 1186} 1187 1188/* 1189 * Unsigned Integer to float conversions 1190 * 1191 * Returns the result of converting the unsigned integer `a' to the 1192 * floating-point format. The conversion is performed according to the 1193 * IEC/IEEE Standard for Binary Floating-Point Arithmetic. 1194 */ 1195static void partsN(uint_to_float)(FloatPartsN *p, uint64_t a, 1196 int scale, float_status *status) 1197{ 1198 memset(p, 0, sizeof(*p)); 1199 1200 if (a == 0) { 1201 p->cls = float_class_zero; 1202 } else { 1203 int shift = clz64(a); 1204 scale = MIN(MAX(scale, -0x10000), 0x10000); 1205 p->cls = float_class_normal; 1206 p->exp = DECOMPOSED_BINARY_POINT - shift + scale; 1207 p->frac_hi = a << shift; 1208 } 1209} 1210 1211/* 1212 * Float min/max. 1213 */ 1214static FloatPartsN *partsN(minmax)(FloatPartsN *a, FloatPartsN *b, 1215 float_status *s, int flags) 1216{ 1217 int ab_mask = float_cmask(a->cls) | float_cmask(b->cls); 1218 int a_exp, b_exp, cmp; 1219 1220 if (unlikely(ab_mask & float_cmask_anynan)) { 1221 /* 1222 * For minnum/maxnum, if one operand is a QNaN, and the other 1223 * operand is numerical, then return numerical argument. 1224 */ 1225 if ((flags & minmax_isnum) 1226 && !(ab_mask & float_cmask_snan) 1227 && (ab_mask & ~float_cmask_qnan)) { 1228 return is_nan(a->cls) ? b : a; 1229 } 1230 return parts_pick_nan(a, b, s); 1231 } 1232 1233 a_exp = a->exp; 1234 b_exp = b->exp; 1235 1236 if (unlikely(ab_mask != float_cmask_normal)) { 1237 switch (a->cls) { 1238 case float_class_normal: 1239 break; 1240 case float_class_inf: 1241 a_exp = INT16_MAX; 1242 break; 1243 case float_class_zero: 1244 a_exp = INT16_MIN; 1245 break; 1246 default: 1247 g_assert_not_reached(); 1248 break; 1249 } 1250 switch (b->cls) { 1251 case float_class_normal: 1252 break; 1253 case float_class_inf: 1254 b_exp = INT16_MAX; 1255 break; 1256 case float_class_zero: 1257 b_exp = INT16_MIN; 1258 break; 1259 default: 1260 g_assert_not_reached(); 1261 break; 1262 } 1263 } 1264 1265 /* Compare magnitudes. */ 1266 cmp = a_exp - b_exp; 1267 if (cmp == 0) { 1268 cmp = frac_cmp(a, b); 1269 } 1270 1271 /* 1272 * Take the sign into account. 1273 * For ismag, only do this if the magnitudes are equal. 1274 */ 1275 if (!(flags & minmax_ismag) || cmp == 0) { 1276 if (a->sign != b->sign) { 1277 /* For differing signs, the negative operand is less. */ 1278 cmp = a->sign ? -1 : 1; 1279 } else if (a->sign) { 1280 /* For two negative operands, invert the magnitude comparison. */ 1281 cmp = -cmp; 1282 } 1283 } 1284 1285 if (flags & minmax_ismin) { 1286 cmp = -cmp; 1287 } 1288 return cmp < 0 ? b : a; 1289} 1290 1291/* 1292 * Floating point compare 1293 */ 1294static FloatRelation partsN(compare)(FloatPartsN *a, FloatPartsN *b, 1295 float_status *s, bool is_quiet) 1296{ 1297 int ab_mask = float_cmask(a->cls) | float_cmask(b->cls); 1298 int cmp; 1299 1300 if (likely(ab_mask == float_cmask_normal)) { 1301 if (a->sign != b->sign) { 1302 goto a_sign; 1303 } 1304 if (a->exp != b->exp) { 1305 cmp = a->exp < b->exp ? -1 : 1; 1306 } else { 1307 cmp = frac_cmp(a, b); 1308 } 1309 if (a->sign) { 1310 cmp = -cmp; 1311 } 1312 return cmp; 1313 } 1314 1315 if (unlikely(ab_mask & float_cmask_anynan)) { 1316 if (!is_quiet || (ab_mask & float_cmask_snan)) { 1317 float_raise(float_flag_invalid, s); 1318 } 1319 return float_relation_unordered; 1320 } 1321 1322 if (ab_mask & float_cmask_zero) { 1323 if (ab_mask == float_cmask_zero) { 1324 return float_relation_equal; 1325 } else if (a->cls == float_class_zero) { 1326 goto b_sign; 1327 } else { 1328 goto a_sign; 1329 } 1330 } 1331 1332 if (ab_mask == float_cmask_inf) { 1333 if (a->sign == b->sign) { 1334 return float_relation_equal; 1335 } 1336 } else if (b->cls == float_class_inf) { 1337 goto b_sign; 1338 } else { 1339 g_assert(a->cls == float_class_inf); 1340 } 1341 1342 a_sign: 1343 return a->sign ? float_relation_less : float_relation_greater; 1344 b_sign: 1345 return b->sign ? float_relation_greater : float_relation_less; 1346} 1347 1348/* 1349 * Multiply A by 2 raised to the power N. 1350 */ 1351static void partsN(scalbn)(FloatPartsN *a, int n, float_status *s) 1352{ 1353 switch (a->cls) { 1354 case float_class_snan: 1355 case float_class_qnan: 1356 parts_return_nan(a, s); 1357 break; 1358 case float_class_zero: 1359 case float_class_inf: 1360 break; 1361 case float_class_normal: 1362 a->exp += MIN(MAX(n, -0x10000), 0x10000); 1363 break; 1364 default: 1365 g_assert_not_reached(); 1366 } 1367} 1368 1369/* 1370 * Return log2(A) 1371 */ 1372static void partsN(log2)(FloatPartsN *a, float_status *s, const FloatFmt *fmt) 1373{ 1374 uint64_t a0, a1, r, t, ign; 1375 FloatPartsN f; 1376 int i, n, a_exp, f_exp; 1377 1378 if (unlikely(a->cls != float_class_normal)) { 1379 switch (a->cls) { 1380 case float_class_snan: 1381 case float_class_qnan: 1382 parts_return_nan(a, s); 1383 return; 1384 case float_class_zero: 1385 /* log2(0) = -inf */ 1386 a->cls = float_class_inf; 1387 a->sign = 1; 1388 return; 1389 case float_class_inf: 1390 if (unlikely(a->sign)) { 1391 goto d_nan; 1392 } 1393 return; 1394 default: 1395 break; 1396 } 1397 g_assert_not_reached(); 1398 } 1399 if (unlikely(a->sign)) { 1400 goto d_nan; 1401 } 1402 1403 /* TODO: This algorithm looses bits too quickly for float128. */ 1404 g_assert(N == 64); 1405 1406 a_exp = a->exp; 1407 f_exp = -1; 1408 1409 r = 0; 1410 t = DECOMPOSED_IMPLICIT_BIT; 1411 a0 = a->frac_hi; 1412 a1 = 0; 1413 1414 n = fmt->frac_size + 2; 1415 if (unlikely(a_exp == -1)) { 1416 /* 1417 * When a_exp == -1, we're computing the log2 of a value [0.5,1.0). 1418 * When the value is very close to 1.0, there are lots of 1's in 1419 * the msb parts of the fraction. At the end, when we subtract 1420 * this value from -1.0, we can see a catastrophic loss of precision, 1421 * as 0x800..000 - 0x7ff..ffx becomes 0x000..00y, leaving only the 1422 * bits of y in the final result. To minimize this, compute as many 1423 * digits as we can. 1424 * ??? This case needs another algorithm to avoid this. 1425 */ 1426 n = fmt->frac_size * 2 + 2; 1427 /* Don't compute a value overlapping the sticky bit */ 1428 n = MIN(n, 62); 1429 } 1430 1431 for (i = 0; i < n; i++) { 1432 if (a1) { 1433 mul128To256(a0, a1, a0, a1, &a0, &a1, &ign, &ign); 1434 } else if (a0 & 0xffffffffull) { 1435 mul64To128(a0, a0, &a0, &a1); 1436 } else if (a0 & ~DECOMPOSED_IMPLICIT_BIT) { 1437 a0 >>= 32; 1438 a0 *= a0; 1439 } else { 1440 goto exact; 1441 } 1442 1443 if (a0 & DECOMPOSED_IMPLICIT_BIT) { 1444 if (unlikely(a_exp == 0 && r == 0)) { 1445 /* 1446 * When a_exp == 0, we're computing the log2 of a value 1447 * [1.0,2.0). When the value is very close to 1.0, there 1448 * are lots of 0's in the msb parts of the fraction. 1449 * We need to compute more digits to produce a correct 1450 * result -- restart at the top of the fraction. 1451 * ??? This is likely to lose precision quickly, as for 1452 * float128; we may need another method. 1453 */ 1454 f_exp -= i; 1455 t = r = DECOMPOSED_IMPLICIT_BIT; 1456 i = 0; 1457 } else { 1458 r |= t; 1459 } 1460 } else { 1461 add128(a0, a1, a0, a1, &a0, &a1); 1462 } 1463 t >>= 1; 1464 } 1465 1466 /* Set sticky for inexact. */ 1467 r |= (a1 || a0 & ~DECOMPOSED_IMPLICIT_BIT); 1468 1469 exact: 1470 parts_sint_to_float(a, a_exp, 0, s); 1471 if (r == 0) { 1472 return; 1473 } 1474 1475 memset(&f, 0, sizeof(f)); 1476 f.cls = float_class_normal; 1477 f.frac_hi = r; 1478 f.exp = f_exp - frac_normalize(&f); 1479 1480 if (a_exp < 0) { 1481 parts_sub_normal(a, &f); 1482 } else if (a_exp > 0) { 1483 parts_add_normal(a, &f); 1484 } else { 1485 *a = f; 1486 } 1487 return; 1488 1489 d_nan: 1490 float_raise(float_flag_invalid, s); 1491 parts_default_nan(a, s); 1492} 1493