1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623
use crate::natural::arithmetic::add::limbs_slice_add_greater_in_place_left;
use crate::natural::arithmetic::add_mul::limbs_slice_add_mul_limb_same_length_in_place_left;
use crate::natural::arithmetic::mul::fft::limbs_mul_greater_to_out_fft;
use crate::natural::arithmetic::mul::limb::limbs_mul_limb_to_out;
use crate::natural::arithmetic::mul::toom::MUL_TOOM33_THRESHOLD_LIMIT;
use crate::natural::arithmetic::mul::toom::{
limbs_mul_greater_to_out_toom_22, limbs_mul_greater_to_out_toom_22_scratch_len,
limbs_mul_greater_to_out_toom_32, limbs_mul_greater_to_out_toom_33,
limbs_mul_greater_to_out_toom_33_scratch_len, limbs_mul_greater_to_out_toom_42,
limbs_mul_greater_to_out_toom_43, limbs_mul_greater_to_out_toom_44,
limbs_mul_greater_to_out_toom_44_scratch_len, limbs_mul_greater_to_out_toom_53,
limbs_mul_greater_to_out_toom_63, limbs_mul_greater_to_out_toom_6h,
limbs_mul_greater_to_out_toom_6h_scratch_len, limbs_mul_greater_to_out_toom_8h,
limbs_mul_greater_to_out_toom_8h_scratch_len, limbs_mul_same_length_to_out_toom_6h_scratch_len,
limbs_mul_same_length_to_out_toom_8h_scratch_len,
};
use crate::natural::arithmetic::square::limbs_square_to_out;
use crate::natural::InnerNatural::{Large, Small};
use crate::natural::Natural;
use crate::platform::{
Limb, MUL_FFT_THRESHOLD, MUL_TOOM22_THRESHOLD, MUL_TOOM32_TO_TOOM43_THRESHOLD,
MUL_TOOM32_TO_TOOM53_THRESHOLD, MUL_TOOM33_THRESHOLD, MUL_TOOM42_TO_TOOM53_THRESHOLD,
MUL_TOOM42_TO_TOOM63_THRESHOLD, MUL_TOOM44_THRESHOLD, MUL_TOOM6H_THRESHOLD,
MUL_TOOM8H_THRESHOLD,
};
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::num::conversion::traits::WrappingFrom;
use std::ops::{Mul, MulAssign};
// Interpreting two slices of `Limb`s as the limbs (in ascending order) of two `Natural`s, returns
// the limbs of the product of the `Natural`s. `xs` must be as least as long as `ys` and `ys`
// cannot be empty.
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `xs.len()`.
//
// # Panics
// Panics if `xs` is shorter than `ys` or `ys` is empty.
//
// This is equivalent to `mpn_mul` from `mpn/generic/mul.c`, GMP 6.2.1, where `prodp` is returned.
pub_test! {limbs_mul_greater(xs: &[Limb], ys: &[Limb]) -> Vec<Limb> {
let mut out = vec![0; xs.len() + ys.len()];
limbs_mul_greater_to_out(&mut out, xs, ys);
out
}}
// Interpreting two slices of `Limb`s as the limbs (in ascending order) of two `Natural`s, returns
// the limbs of the product of the `Natural`s. Neither slice can be empty. The length of the
// resulting slice is always the sum of the lengths of the input slices, so it may have trailing
// zeros.
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `xs.len()`.
//
// # Panics
// Panics if either slice is empty.
//
// This is equivalent to `mpn_mul` from mpn/generic/mul.c, GMP 6.2.1, where `un` may be less than
// `vn` and `prodp` is returned.
pub_crate_test! {limbs_mul(xs: &[Limb], ys: &[Limb]) -> Vec<Limb> {
if xs.len() >= ys.len() {
limbs_mul_greater(xs, ys)
} else {
limbs_mul_greater(ys, xs)
}
}}
// Interpreting two equal-length slices of `Limb`s as the limbs (in ascending order) of two
// `Natural`s, writes the `2 * xs.len()` least-significant limbs of the product of the `Natural`s
// to an output slice. The output must be at least as long as `2 * xs.len()`, `xs` must be as long
// as `ys`, and neither slice can be empty. Returns the result limb at index `2 * xs.len() - 1`
// (which may be zero).
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `xs.len()`.
//
// # Panics
// Panics if `out` is too short, `xs` and `ys` have different lengths, or either slice is empty.
//
// This is equivalent to `mpn_mul_n` from `mpn/generic/mul_n.c`, GMP 6.2.1.
pub_crate_test! {limbs_mul_same_length_to_out(out: &mut [Limb], xs: &[Limb], ys: &[Limb]) {
let len = xs.len();
assert_eq!(ys.len(), len);
assert_ne!(len, 0);
if std::ptr::eq(xs, ys) {
limbs_square_to_out(out, xs);
} else if len < MUL_TOOM22_THRESHOLD {
limbs_mul_greater_to_out_basecase(out, xs, ys);
} else if len < MUL_TOOM33_THRESHOLD {
// Allocate workspace of fixed size on stack: fast!
let scratch =
&mut [0; limbs_mul_greater_to_out_toom_22_scratch_len(MUL_TOOM33_THRESHOLD_LIMIT - 1)];
assert!(MUL_TOOM33_THRESHOLD <= MUL_TOOM33_THRESHOLD_LIMIT);
limbs_mul_greater_to_out_toom_22(out, xs, ys, scratch);
} else if len < MUL_TOOM44_THRESHOLD {
let mut scratch = vec![0; limbs_mul_greater_to_out_toom_33_scratch_len(len)];
limbs_mul_greater_to_out_toom_33(out, xs, ys, &mut scratch);
} else if len < MUL_TOOM6H_THRESHOLD {
let mut scratch = vec![0; limbs_mul_greater_to_out_toom_44_scratch_len(len)];
limbs_mul_greater_to_out_toom_44(out, xs, ys, &mut scratch);
} else if len < MUL_TOOM8H_THRESHOLD {
let mut scratch = vec![0; limbs_mul_same_length_to_out_toom_6h_scratch_len(len)];
limbs_mul_greater_to_out_toom_6h(out, xs, ys, &mut scratch);
} else if len < MUL_FFT_THRESHOLD {
let mut scratch = vec![0; limbs_mul_same_length_to_out_toom_8h_scratch_len(len)];
limbs_mul_greater_to_out_toom_8h(out, xs, ys, &mut scratch);
} else {
limbs_mul_greater_to_out_fft(out, xs, ys);
}
}}
// This is equivalent to `TOOM44_OK` from `mpn/generic/mul.c`, GMP 6.2.1.
const fn toom44_ok(xs_len: usize, ys_len: usize) -> bool {
12 + 3 * xs_len < ys_len << 2
}
// Interpreting two slices of `Limb`s as the limbs (in ascending order) of two `Natural`s, writes
// the `xs.len() + ys.len()` least-significant limbs of the product of the `Natural`s to an output
// slice. The output must be at least as long as `xs.len() + ys.len()`, `xs` must be as least as
// long as `ys`, and `ys` cannot be empty. Returns the result limb at index
// `xs.len() + ys.len() - 1` (which may be zero).
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `xs.len()`.
//
// # Panics
// Panics if `out` is too short, `xs` is shorter than `ys`, or `ys` is empty.
//
// This is equivalent to `mpn_mul` from `mpn/generic/mul.c`, GMP 6.2.1.
pub_crate_test! {limbs_mul_greater_to_out(out: &mut [Limb], xs: &[Limb], ys: &[Limb]) -> Limb {
let xs_len = xs.len();
let ys_len = ys.len();
assert!(xs_len >= ys_len);
assert_ne!(ys_len, 0);
assert!(out.len() >= xs_len + ys_len);
if xs_len == ys_len {
limbs_mul_same_length_to_out(out, xs, ys);
} else if ys_len < MUL_TOOM22_THRESHOLD {
// Plain schoolbook multiplication. Unless xs_len is very large, or else if
// `limbs_mul_same_length_to_out` applies, perform basecase multiply directly.
limbs_mul_greater_to_out_basecase(out, xs, ys);
} else if ys_len < MUL_TOOM33_THRESHOLD {
let toom_x2_scratch_len = 9 * ys_len / 2 + (usize::wrapping_from(Limb::WIDTH) << 1);
let mut scratch = vec![0; toom_x2_scratch_len];
if xs_len >= 3 * ys_len {
limbs_mul_greater_to_out_toom_42(out, &xs[..ys_len << 1], ys, &mut scratch);
let two_ys_len = ys_len << 1;
let three_ys_len = two_ys_len + ys_len;
// The maximum `scratch2` usage is for the `limbs_mul_greater_to_out_toom_x2` result.
let mut scratch2 = vec![0; two_ys_len << 1];
let mut xs = &xs[two_ys_len..];
let mut out_offset = two_ys_len;
while xs.len() >= three_ys_len {
let out = &mut out[out_offset..];
let (xs_lo, xs_hi) = xs.split_at(two_ys_len);
limbs_mul_greater_to_out_toom_42(&mut scratch2, xs_lo, ys, &mut scratch);
let (scratch2_lo, scratch2_hi) = scratch2.split_at(ys_len);
out[ys_len..three_ys_len].copy_from_slice(&scratch2_hi[..two_ys_len]);
assert!(!limbs_slice_add_greater_in_place_left(out, scratch2_lo));
xs = xs_hi;
out_offset += two_ys_len;
}
let xs_len = xs.len();
let out = &mut out[out_offset..];
// ys_len <= xs_len < 3 * ys_len
let four_xs_len = xs_len << 2;
if four_xs_len < 5 * ys_len {
limbs_mul_greater_to_out_toom_22(&mut scratch2, xs, ys, &mut scratch);
} else if four_xs_len < 7 * ys_len {
limbs_mul_greater_to_out_toom_32(&mut scratch2, xs, ys, &mut scratch);
} else {
limbs_mul_greater_to_out_toom_42(&mut scratch2, xs, ys, &mut scratch);
}
let (scratch2_lo, scratch2_hi) = scratch2.split_at(ys_len);
out[ys_len..ys_len + xs_len].copy_from_slice(&scratch2_hi[..xs_len]);
assert!(!limbs_slice_add_greater_in_place_left(out, scratch2_lo));
} else if 4 * xs_len < 5 * ys_len {
limbs_mul_greater_to_out_toom_22(out, xs, ys, &mut scratch);
} else if 4 * xs_len < 7 * ys_len {
limbs_mul_greater_to_out_toom_32(out, xs, ys, &mut scratch);
} else {
limbs_mul_greater_to_out_toom_42(out, xs, ys, &mut scratch);
}
} else if (xs_len + ys_len) >> 1 < MUL_FFT_THRESHOLD || 3 * ys_len < MUL_FFT_THRESHOLD {
// Handle the largest operands that are not in the FFT range. The 2nd condition makes very
// unbalanced operands avoid the FFT code (except perhaps as coefficient products of the
// Toom code).
if ys_len < MUL_TOOM44_THRESHOLD || !toom44_ok(xs_len, ys_len) {
// Use ToomX3 variants
let toom_x3_scratch_len = (ys_len << 2) + usize::wrapping_from(Limb::WIDTH);
let mut scratch = vec![0; toom_x3_scratch_len];
if xs_len << 1 >= 5 * ys_len {
// The maximum scratch2 usage is for the `limbs_mul_to_out` result.
let mut scratch2 = vec![0; (7 * ys_len) >> 1];
let two_ys_len = ys_len << 1;
let (xs_lo, mut xs) = xs.split_at(two_ys_len);
if ys_len < MUL_TOOM42_TO_TOOM63_THRESHOLD {
limbs_mul_greater_to_out_toom_42(out, xs_lo, ys, &mut scratch);
} else {
limbs_mul_greater_to_out_toom_63(out, xs_lo, ys, &mut scratch);
}
let mut out_offset = two_ys_len;
// xs_len >= 2.5 * ys_len
while xs.len() << 1 >= 5 * ys_len {
let out = &mut out[out_offset..];
let (xs_lo, xs_hi) = xs.split_at(two_ys_len);
if ys_len < MUL_TOOM42_TO_TOOM63_THRESHOLD {
limbs_mul_greater_to_out_toom_42(&mut scratch2, xs_lo, ys, &mut scratch);
} else {
limbs_mul_greater_to_out_toom_63(&mut scratch2, xs_lo, ys, &mut scratch);
}
let (scratch2_lo, scratch2_hi) = scratch2.split_at(ys_len);
out[ys_len..ys_len + two_ys_len].copy_from_slice(&scratch2_hi[..two_ys_len]);
assert!(!limbs_slice_add_greater_in_place_left(out, scratch2_lo));
xs = xs_hi;
out_offset += two_ys_len;
}
let xs_len = xs.len();
let out = &mut out[out_offset..];
// ys_len / 2 <= xs_len < 2.5 * ys_len
limbs_mul_to_out(&mut scratch2, xs, ys);
let (scratch2_lo, scratch2_hi) = scratch2.split_at(ys_len);
out[ys_len..xs_len + ys_len].copy_from_slice(&scratch2_hi[..xs_len]);
assert!(!limbs_slice_add_greater_in_place_left(out, scratch2_lo));
} else if 6 * xs_len < 7 * ys_len {
limbs_mul_greater_to_out_toom_33(out, xs, ys, &mut scratch);
} else if xs_len << 1 < 3 * ys_len {
if ys_len < MUL_TOOM32_TO_TOOM43_THRESHOLD {
limbs_mul_greater_to_out_toom_32(out, xs, ys, &mut scratch);
} else {
limbs_mul_greater_to_out_toom_43(out, xs, ys, &mut scratch);
}
} else if 6 * xs_len < 11 * ys_len {
if xs_len << 2 < 7 * ys_len {
if ys_len < MUL_TOOM32_TO_TOOM53_THRESHOLD {
limbs_mul_greater_to_out_toom_32(out, xs, ys, &mut scratch);
} else {
limbs_mul_greater_to_out_toom_53(out, xs, ys, &mut scratch);
}
} else if ys_len < MUL_TOOM42_TO_TOOM53_THRESHOLD {
limbs_mul_greater_to_out_toom_42(out, xs, ys, &mut scratch);
} else {
limbs_mul_greater_to_out_toom_53(out, xs, ys, &mut scratch);
}
} else if ys_len < MUL_TOOM42_TO_TOOM63_THRESHOLD {
limbs_mul_greater_to_out_toom_42(out, xs, ys, &mut scratch);
} else {
limbs_mul_greater_to_out_toom_63(out, xs, ys, &mut scratch);
}
} else if ys_len < MUL_TOOM6H_THRESHOLD {
let mut scratch = vec![0; limbs_mul_greater_to_out_toom_44_scratch_len(xs_len)];
limbs_mul_greater_to_out_toom_44(out, xs, ys, &mut scratch);
} else if ys_len < MUL_TOOM8H_THRESHOLD {
let mut scratch = vec![0; limbs_mul_greater_to_out_toom_6h_scratch_len(xs_len, ys_len)];
limbs_mul_greater_to_out_toom_6h(out, xs, ys, &mut scratch);
} else {
let mut scratch = vec![0; limbs_mul_greater_to_out_toom_8h_scratch_len(xs_len, ys_len)];
limbs_mul_greater_to_out_toom_8h(out, xs, ys, &mut scratch);
}
} else {
limbs_mul_greater_to_out_fft(out, xs, ys);
}
out[xs_len + ys_len - 1]
}}
// Interpreting two slices of `Limb`s as the limbs (in ascending order) of two `Natural`s, writes
// the `xs.len() + ys.len()` least-significant limbs of the product of the `Natural`s to an output
// slice. The output must be at least as long as `xs.len() + ys.len()`, and neither slice can be
// empty. Returns the result limb at index `xs.len() + ys.len() - 1` (which may be zero).
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `max(xs.len(), ys.len())`.
//
// # Panics
// Panics if `out` is too short or either slice is empty.
//
// This is equivalent to `mpn_mul` from `mpn/generic/mul.c`, GMP 6.2.1, where `un` may be less than
// `vn`.
pub_crate_test! {limbs_mul_to_out(out: &mut [Limb], xs: &[Limb], ys: &[Limb]) -> Limb {
if xs.len() >= ys.len() {
limbs_mul_greater_to_out(out, xs, ys)
} else {
limbs_mul_greater_to_out(out, ys, xs)
}
}}
// Interpreting two slices of `Limb`s as the limbs (in ascending order) of two `Natural`s, writes
// the `xs.len() + ys.len()` least-significant limbs of the product of the `Natural`s to an output
// slice. The output must be at least as long as `xs.len() + ys.len()`, `xs` must be as least as
// long as `ys`, and `ys` cannot be empty. Returns the result limb at index
// `xs.len() + ys.len() - 1` (which may be zero).
//
// This uses the basecase, quadratic, schoolbook algorithm, and it is most critical code for
// multiplication. All multiplies rely on this, both small and huge. Small ones arrive here
// immediately, and huge ones arrive here as this is the base case for Karatsuba's recursive
// algorithm.
//
// # Worst-case complexity
// $T(n) = O(n^2)$
//
// $M(n) = O(1)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `max(xs.len(), ys.len())`.
//
// # Panics
// Panics if `out` is too short, `xs` is shorter than `ys`, or `ys` is empty.
//
// This is equivalent to `mpn_mul_basecase` from `mpn/generic/mul_basecase.c`, GMP 6.2.1.
pub_crate_test! {limbs_mul_greater_to_out_basecase(out: &mut [Limb], xs: &[Limb], ys: &[Limb]) {
let xs_len = xs.len();
let ys_len = ys.len();
assert_ne!(ys_len, 0);
assert!(xs_len >= ys_len);
assert!(out.len() >= xs_len + ys_len);
// We first multiply by the low order limb. This result can be stored, not added, to out.
out[xs_len] = limbs_mul_limb_to_out(out, xs, ys[0]);
// Now accumulate the product of xs and the next higher limb from ys.
let window_size = xs_len + 1;
for i in 1..ys_len {
let (out_last, out_init) = out[i..i + window_size].split_last_mut().unwrap();
*out_last = limbs_slice_add_mul_limb_same_length_in_place_left(out_init, xs, ys[i]);
}
}}
impl Mul<Natural> for Natural {
type Output = Natural;
/// Multiplies two [`Natural`]s, taking both by value.
///
/// $$
/// f(x, y) = xy.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is
/// `max(self.significant_bits(), other.significant_bits())`.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::basic::traits::{One, Zero};
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// assert_eq!(Natural::ONE * Natural::from(123u32), 123);
/// assert_eq!(Natural::from(123u32) * Natural::ZERO, 0);
/// assert_eq!(Natural::from(123u32) * Natural::from(456u32), 56088);
/// assert_eq!(
/// (Natural::from_str("123456789000").unwrap() * Natural::from_str("987654321000")
/// .unwrap()).to_string(),
/// "121932631112635269000000"
/// );
/// ```
#[inline]
fn mul(mut self, other: Natural) -> Natural {
self *= other;
self
}
}
impl<'a> Mul<&'a Natural> for Natural {
type Output = Natural;
/// Multiplies two [`Natural`]s, taking the first by value and the second by reference.
///
/// $$
/// f(x, y) = xy.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is
/// `max(self.significant_bits(), other.significant_bits())`.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::basic::traits::{One, Zero};
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// assert_eq!(Natural::ONE * &Natural::from(123u32), 123);
/// assert_eq!(Natural::from(123u32) * &Natural::ZERO, 0);
/// assert_eq!(Natural::from(123u32) * &Natural::from(456u32), 56088);
/// assert_eq!(
/// (Natural::from_str("123456789000").unwrap() * &Natural::from_str("987654321000")
/// .unwrap()).to_string(),
/// "121932631112635269000000"
/// );
/// ```
#[inline]
fn mul(mut self, other: &'a Natural) -> Natural {
self *= other;
self
}
}
impl<'a> Mul<Natural> for &'a Natural {
type Output = Natural;
/// Multiplies two [`Natural`]s, taking the first by reference and the second by value.
///
/// $$
/// f(x, y) = xy.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is
/// `max(self.significant_bits(), other.significant_bits())`.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::basic::traits::{One, Zero};
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// assert_eq!(&Natural::ONE * Natural::from(123u32), 123);
/// assert_eq!(&Natural::from(123u32) * Natural::ZERO, 0);
/// assert_eq!(&Natural::from(123u32) * Natural::from(456u32), 56088);
/// assert_eq!(
/// (&Natural::from_str("123456789000").unwrap() * Natural::from_str("987654321000")
/// .unwrap()).to_string(),
/// "121932631112635269000000"
/// );
/// ```
#[inline]
fn mul(self, mut other: Natural) -> Natural {
other *= self;
other
}
}
impl<'a, 'b> Mul<&'a Natural> for &'b Natural {
type Output = Natural;
/// Multiplies two [`Natural`]s, taking both by reference.
///
/// $$
/// f(x, y) = xy.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is
/// `max(self.significant_bits(), other.significant_bits())`.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::basic::traits::{One, Zero};
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// assert_eq!(&Natural::ONE * &Natural::from(123u32), 123);
/// assert_eq!(&Natural::from(123u32) * &Natural::ZERO, 0);
/// assert_eq!(&Natural::from(123u32) * &Natural::from(456u32), 56088);
/// assert_eq!(
/// (&Natural::from_str("123456789000").unwrap() * &Natural::from_str("987654321000")
/// .unwrap()).to_string(),
/// "121932631112635269000000"
/// );
/// ```
fn mul(self, other: &'a Natural) -> Natural {
match (self, other) {
(Natural(Small(x)), y) => y.mul_limb_ref(*x),
(x, Natural(Small(y))) => x.mul_limb_ref(*y),
(Natural(Large(ref xs)), Natural(Large(ref ys))) => {
Natural::from_owned_limbs_asc(limbs_mul(xs, ys))
}
}
}
}
impl MulAssign<Natural> for Natural {
/// Multiplies a [`Natural`] by a [`Natural`] in place, taking the [`Natural`] on the
/// right-hand side by value.
///
/// $$
/// x \gets = xy.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is
/// `max(self.significant_bits(), other.significant_bits())`.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::basic::traits::One;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// let mut x = Natural::ONE;
/// x *= Natural::from_str("1000").unwrap();
/// x *= Natural::from_str("2000").unwrap();
/// x *= Natural::from_str("3000").unwrap();
/// x *= Natural::from_str("4000").unwrap();
/// assert_eq!(x.to_string(), "24000000000000");
/// ```
fn mul_assign(&mut self, mut other: Natural) {
match (&mut *self, &mut other) {
(Natural(Small(x)), _) => {
other.mul_assign_limb(*x);
*self = other;
}
(_, Natural(Small(y))) => self.mul_assign_limb(*y),
(Natural(Large(ref mut xs)), Natural(Large(ref ys))) => {
*xs = limbs_mul(xs, ys);
self.trim();
}
}
}
}
impl<'a> MulAssign<&'a Natural> for Natural {
/// Multiplies a [`Natural`] by a [`Natural`] in place, taking the [`Natural`] on the
/// right-hand side by reference.
///
/// $$
/// x \gets = xy.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is
/// `max(self.significant_bits(), other.significant_bits())`.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::basic::traits::One;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// let mut x = Natural::ONE;
/// x *= &Natural::from_str("1000").unwrap();
/// x *= &Natural::from_str("2000").unwrap();
/// x *= &Natural::from_str("3000").unwrap();
/// x *= &Natural::from_str("4000").unwrap();
/// assert_eq!(x.to_string(), "24000000000000");
/// ```
fn mul_assign(&mut self, other: &'a Natural) {
match (&mut *self, other) {
(Natural(Small(x)), _) => *self = other.mul_limb_ref(*x),
(_, Natural(Small(y))) => self.mul_assign_limb(*y),
(Natural(Large(ref mut xs)), Natural(Large(ref ys))) => {
*xs = limbs_mul(xs, ys);
self.trim();
}
}
}
}
/// Code for the Schönhage-Strassen (FFT) multiplication algorithm.
pub mod fft;
/// Code for multiplying a many-limbed [`Natural`] by a single [limb](crate#limbs).
pub mod limb;
/// Code for computing only the lowest [limbs](crate#limbs) of the product of two [`Natural`]s.
pub mod mul_low;
/// Code for multiplying two [`Natural`]s modulo one less than a large power of 2; used by the
/// Schönhage-Strassen algorithm.
pub mod mul_mod;
/// Code for evaluating polynomials at various points; used in Toom-Cook multiplication.
pub mod poly_eval;
/// Code for reconstructing polynomials from their values at various points; used in Toom-Cook
/// multiplication.
pub mod poly_interpolate;
#[cfg(feature = "test_build")]
pub mod product_of_limbs;
#[cfg(not(feature = "test_build"))]
pub(crate) mod product_of_limbs;
/// Code for computing only the lowest [limbs](crate#limbs) of the square of a [`Natural`].
pub mod square_mod;
/// Code for Toom-Cook multiplication.
pub mod toom;