diff --git a/core/math/big/build.bat b/core/math/big/build.bat index eb6f581aa..73dce208f 100644 --- a/core/math/big/build.bat +++ b/core/math/big/build.bat @@ -1,8 +1,9 @@ @echo off -:odin run . -vet +:odin run . -vet -o:speed -no-bounds-check : -o:size :odin build . -build-mode:shared -show-timings -o:minimal -no-bounds-check -define:MATH_BIG_EXE=false && python test.py -fast-tests :odin build . -build-mode:shared -show-timings -o:size -no-bounds-check -define:MATH_BIG_EXE=false && python test.py -fast-tests :odin build . -build-mode:shared -show-timings -o:size -define:MATH_BIG_EXE=false && python test.py -fast-tests -odin build . -build-mode:shared -show-timings -o:speed -no-bounds-check -define:MATH_BIG_EXE=false && python test.py -fast-tests +odin build . -build-mode:shared -show-timings -o:speed -no-bounds-check -define:MATH_BIG_EXE=false && python test.py +: -fast-tests :odin build . -build-mode:shared -show-timings -o:speed -define:MATH_BIG_EXE=false && python test.py -fast-tests \ No newline at end of file diff --git a/core/math/big/example.odin b/core/math/big/example.odin index 4fbf44664..994fbf55a 100644 --- a/core/math/big/example.odin +++ b/core/math/big/example.odin @@ -206,16 +206,18 @@ demo :: proc() { a, b, c, d, e, f := &Int{}, &Int{}, &Int{}, &Int{}, &Int{}, &Int{}; defer destroy(a, b, c, d, e, f); - atoi(a, "12980742146337069150589594264770969721", 10); + atoi(a, "615037959146039477924633848896619112832171971562900618409305032006863881436080", 10); print("a: ", a, 10, true, true, true); - atoi(b, "4611686018427387904", 10); + atoi(b, "378271691190525325893712245607881659587045836991909505715443874842659307597325888631898626653926188084180707310543535657996185416604973577488563643125766400", 10); print("b: ", b, 10, true, true, true); - if err := internal_divmod(c, d, a, b); err != nil { - fmt.printf("Error: %v\n", err); - } - print("c: ", c); - print("c: ", d); + factorial(c, 10_000); + + // 120CCAA2076ADF69F75A97695E6C1C2A4E6F377DF92226E43B + cs, _ := itoa(c, 16); + defer delete(cs); + + print("c: ", c, 10, true, true, true); } main :: proc() { diff --git a/core/math/big/helpers.odin b/core/math/big/helpers.odin index ab686b914..e50579ac0 100644 --- a/core/math/big/helpers.odin +++ b/core/math/big/helpers.odin @@ -432,18 +432,16 @@ int_init_multi :: proc(integers: ..^Int, allocator := context.allocator) -> (err init_multi :: proc { int_init_multi, }; -copy_digits :: proc(dest, src: ^Int, digits: int, allocator := context.allocator) -> (err: Error) { +copy_digits :: proc(dest, src: ^Int, digits: int, offset := int(0), allocator := context.allocator) -> (err: Error) { context.allocator = allocator; - digits := digits; /* Check that `src` is usable and `dest` isn't immutable. */ assert_if_nil(dest, src); #force_inline internal_clear_if_uninitialized(src) or_return; - digits = min(digits, len(src.digit), len(dest.digit)); - return #force_inline internal_copy_digits(dest, src, digits); + return #force_inline internal_copy_digits(dest, src, digits, offset); } /* diff --git a/core/math/big/internal.odin b/core/math/big/internal.odin index 2c988f91e..31831f492 100644 --- a/core/math/big/internal.odin +++ b/core/math/big/internal.odin @@ -36,8 +36,6 @@ import "core:mem" import "core:intrinsics" import rnd "core:math/rand" -import "core:fmt" - /* Low-level addition, unsigned. Handbook of Applied Cryptography, algorithm 14.7. @@ -651,7 +649,6 @@ internal_int_mul :: proc(dest, src, multiplier: ^Int, allocator := context.alloc Fast comba? */ err = #force_inline _private_int_sqr_comba(dest, src); - //err = #force_inline _private_int_sqr(dest, src); } else { err = #force_inline _private_int_sqr(dest, src); } @@ -678,9 +675,13 @@ internal_int_mul :: proc(dest, src, multiplier: ^Int, allocator := context.alloc max_used >= 2 * min_used { // err = s_mp_mul_balance(a,b,c); } else if false && min_used >= MUL_TOOM_CUTOFF { - // err = s_mp_mul_toom(a, b, c); - } else if false && min_used >= MUL_KARATSUBA_CUTOFF { - // err = s_mp_mul_karatsuba(a, b, c); + /* + Toom path commented out until it no longer fails Factorial 10k or 100k, + as reveaved in the long test. + */ + err = #force_inline _private_int_mul_toom(dest, src, multiplier); + } else if min_used >= MUL_KARATSUBA_CUTOFF { + err = #force_inline _private_int_mul_karatsuba(dest, src, multiplier); } else if digits < _WARRAY && min_used <= _MAX_COMBA { /* Can we use the fast multiplier? @@ -1628,16 +1629,13 @@ internal_int_set_from_integer :: proc(dest: ^Int, src: $T, minimize := false, al internal_set :: proc { internal_int_set_from_integer, internal_int_copy }; -internal_copy_digits :: #force_inline proc(dest, src: ^Int, digits: int) -> (err: Error) { +internal_copy_digits :: #force_inline proc(dest, src: ^Int, digits: int, offset := int(0)) -> (err: Error) { #force_inline internal_error_if_immutable(dest) or_return; /* If dest == src, do nothing */ - if (dest == src) { return nil; } - - #force_inline mem.copy_non_overlapping(&dest.digit[0], &src.digit[0], size_of(DIGIT) * digits); - return nil; + return #force_inline _private_copy_digits(dest, src, digits, offset); } /* diff --git a/core/math/big/private.odin b/core/math/big/private.odin index a99d6119f..a46fab230 100644 --- a/core/math/big/private.odin +++ b/core/math/big/private.odin @@ -89,6 +89,245 @@ _private_int_mul :: proc(dest, a, b: ^Int, digits: int, allocator := context.all return internal_clamp(dest); } + +/* + Multiplication using the Toom-Cook 3-way algorithm. + + Much more complicated than Karatsuba but has a lower asymptotic running time of O(N**1.464). + This algorithm is only particularly useful on VERY large inputs. + (We're talking 1000s of digits here...). + + This file contains code from J. Arndt's book "Matters Computational" + and the accompanying FXT-library with permission of the author. + + Setup from: + Chung, Jaewook, and M. Anwar Hasan. "Asymmetric squaring formulae." + 18th IEEE Symposium on Computer Arithmetic (ARITH'07). IEEE, 2007. + + The interpolation from above needed one temporary variable more than the interpolation here: + + Bodrato, Marco, and Alberto Zanoni. "What about Toom-Cook matrices optimality." + Centro Vito Volterra Universita di Roma Tor Vergata (2006) +*/ +_private_int_mul_toom :: proc(dest, a, b: ^Int, allocator := context.allocator) -> (err: Error) { + context.allocator = allocator; + + S1, S2, T1, a0, a1, a2, b0, b1, b2 := &Int{}, &Int{}, &Int{}, &Int{}, &Int{}, &Int{}, &Int{}, &Int{}, &Int{}; + defer destroy(S1, S2, T1, a0, a1, a2, b0, b1, b2); + + /* + Init temps. + */ + internal_init_multi(S1, S2, T1) or_return; + + /* + B + */ + B := min(a.used, b.used) / 3; + + /* + a = a2 * x^2 + a1 * x + a0; + */ + internal_grow(a0, B) or_return; + internal_grow(a1, B) or_return; + internal_grow(a2, a.used - 2 * B) or_return; + + a0.used, a1.used = B, B; + a2.used = a.used - 2 * B; + + internal_copy_digits(a0, a, a0.used) or_return; + internal_copy_digits(a1, a, a1.used, B) or_return; + internal_copy_digits(a2, a, a2.used, 2 * B) or_return; + + internal_clamp(a0); + internal_clamp(a1); + internal_clamp(a2); + + /* + b = b2 * x^2 + b1 * x + b0; + */ + internal_grow(b0, B) or_return; + internal_grow(b1, B) or_return; + internal_grow(b2, b.used - 2 * B) or_return; + + b0.used, b1.used = B, B; + b2.used = b.used - 2 * B; + + internal_copy_digits(b0, b, b0.used) or_return; + internal_copy_digits(b1, b, b1.used, B) or_return; + internal_copy_digits(b2, b, b2.used, 2 * B) or_return; + + internal_clamp(b0); + internal_clamp(b1); + internal_clamp(b2); + + /* + \\ S1 = (a2+a1+a0) * (b2+b1+b0); + */ + internal_add(T1, a2, a1) or_return; /* T1 = a2 + a1; */ + internal_add(S2, T1, a0) or_return; /* S2 = T1 + a0; */ + internal_add(dest, b2, b1) or_return; /* dest = b2 + b1; */ + internal_add(S1, dest, b0) or_return; /* S1 = c + b0; */ + internal_mul(S1, S1, S2) or_return; /* S1 = S1 * S2; */ + + /* + \\S2 = (4*a2+2*a1+a0) * (4*b2+2*b1+b0); + */ + internal_add(T1, T1, a2) or_return; /* T1 = T1 + a2; */ + internal_int_shl1(T1, T1) or_return; /* T1 = T1 << 1; */ + internal_add(T1, T1, a0) or_return; /* T1 = T1 + a0; */ + internal_add(dest, dest, b2) or_return; /* c = c + b2; */ + internal_int_shl1(dest, dest) or_return; /* c = c << 1; */ + internal_add(dest, dest, b0) or_return; /* c = c + b0; */ + internal_mul(S2, T1, dest) or_return; /* S2 = T1 * c; */ + + /* + \\S3 = (a2-a1+a0) * (b2-b1+b0); + */ + internal_sub(a1, a2, a1) or_return; /* a1 = a2 - a1; */ + internal_add(a1, a1, a0) or_return; /* a1 = a1 + a0; */ + internal_sub(b1, b2, b1) or_return; /* b1 = b2 - b1; */ + internal_add(b1, b1, b0) or_return; /* b1 = b1 + b0; */ + internal_mul(a1, a1, b1) or_return; /* a1 = a1 * b1; */ + internal_mul(b1, a2, b2) or_return; /* b1 = a2 * b2; */ + + /* + \\S2 = (S2 - S3) / 3; + */ + internal_sub(S2, S2, a1) or_return; /* S2 = S2 - a1; */ + _private_int_div_3(S2, S2) or_return; /* S2 = S2 / 3; \\ this is an exact division */ + internal_sub(a1, S1, a1) or_return; /* a1 = S1 - a1; */ + internal_int_shr1(a1, a1) or_return; /* a1 = a1 >> 1; */ + internal_mul(a0, a0, b0) or_return; /* a0 = a0 * b0; */ + internal_sub(S1, S1, a0) or_return; /* S1 = S1 - a0; */ + internal_sub(S2, S2, S1) or_return; /* S2 = S2 - S1; */ + internal_int_shr1(S2, S2) or_return; /* S2 = S2 >> 1; */ + internal_sub(S1, S1, a1) or_return; /* S1 = S1 - a1; */ + internal_sub(S1, S1, b1) or_return; /* S1 = S1 - b1; */ + internal_int_shl1(T1, b1) or_return; /* T1 = b1 << 1; */ + internal_sub(S2, S2, T1) or_return; /* S2 = S2 - T1; */ + internal_sub(a1, a1, S2) or_return; /* a1 = a1 - S2; */ + + /* + P = b1*x^4+ S2*x^3+ S1*x^2+ a1*x + a0; + */ + internal_shl_digit(b1, 4 * B) or_return; + internal_shl_digit(S2, 3 * B) or_return; + internal_add(b1, b1, S2) or_return; + internal_shl_digit(S1, 2 * B) or_return; + internal_add(b1, b1, S1) or_return; + internal_shl_digit(a1, 1 * B) or_return; + internal_add(b1, b1, a1) or_return; + internal_add(dest, b1, a0) or_return; + + /* + a * b - P + */ + return nil; +} + +/* + product = |a| * |b| using Karatsuba Multiplication using three half size multiplications. + + Let `B` represent the radix [e.g. 2**_DIGIT_BITS] and let `n` represent + half of the number of digits in the min(a,b) + + `a` = `a1` * `B`**`n` + `a0` + `b` = `b`1 * `B`**`n` + `b0` + + Then, a * b => 1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0 + + Note that a1b1 and a0b0 are used twice and only need to be computed once. + So in total three half size (half # of digit) multiplications are performed, + a0b0, a1b1 and (a1+b1)(a0+b0) + + Note that a multiplication of half the digits requires 1/4th the number of + single precision multiplications, so in total after one call 25% of the + single precision multiplications are saved. + + Note also that the call to `internal_mul` can end up back in this function + if the a0, a1, b0, or b1 are above the threshold. + + This is known as divide-and-conquer and leads to the famous O(N**lg(3)) or O(N**1.584) + work which is asymptopically lower than the standard O(N**2) that the + baseline/comba methods use. Generally though, the overhead of this method doesn't pay off + until a certain size is reached, of around 80 used DIGITs. +*/ +_private_int_mul_karatsuba :: proc(dest, a, b: ^Int, allocator := context.allocator) -> (err: Error) { + context.allocator = allocator; + + x0, x1, y0, y1, t1, x0y0, x1y1 := &Int{}, &Int{}, &Int{}, &Int{}, &Int{}, &Int{}, &Int{}; + defer destroy(x0, x1, y0, y1, t1, x0y0, x1y1); + + /* + min # of digits, divided by two. + */ + B := min(a.used, b.used) >> 1; + + /* + Init all the temps. + */ + internal_grow(x0, B) or_return; + internal_grow(x1, a.used - B) or_return; + internal_grow(y0, B) or_return; + internal_grow(y1, b.used - B) or_return; + internal_grow(t1, B * 2) or_return; + internal_grow(x0y0, B * 2) or_return; + internal_grow(x1y1, B * 2) or_return; + + /* + Now shift the digits. + */ + x0.used, y0.used = B, B; + x1.used = a.used - B; + y1.used = b.used - B; + + /* + We copy the digits directly instead of using higher level functions + since we also need to shift the digits. + */ + internal_copy_digits(x0, a, x0.used); + internal_copy_digits(y0, b, y0.used); + internal_copy_digits(x1, a, x1.used, B); + internal_copy_digits(y1, b, y1.used, B); + + /* + Only need to clamp the lower words since by definition the + upper words x1/y1 must have a known number of digits. + */ + clamp(x0); + clamp(y0); + + /* + Now calc the products x0y0 and x1y1, + after this x0 is no longer required, free temp [x0==t2]! + */ + internal_mul(x0y0, x0, y0) or_return; /* x0y0 = x0*y0 */ + internal_mul(x1y1, x1, y1) or_return; /* x1y1 = x1*y1 */ + internal_add(t1, x1, x0) or_return; /* now calc x1+x0 and */ + internal_add(x0, y1, y0) or_return; /* t2 = y1 + y0 */ + internal_mul(t1, t1, x0) or_return; /* t1 = (x1 + x0) * (y1 + y0) */ + + /* + Add x0y0. + */ + internal_add(x0, x0y0, x1y1) or_return; /* t2 = x0y0 + x1y1 */ + internal_sub(t1, t1, x0) or_return; /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */ + + /* + shift by B. + */ + internal_shl_digit(t1, B) or_return; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))< (log: int, err: Error Copies DIGITs from `src` to `dest`. Assumes `src` and `dest` to not be `nil` and have been initialized. */ -_private_copy_digits :: proc(dest, src: ^Int, digits: int) -> (err: Error) { +_private_copy_digits :: proc(dest, src: ^Int, digits: int, offset := int(0)) -> (err: Error) { digits := digits; /* If dest == src, do nothing @@ -1639,7 +1878,7 @@ _private_copy_digits :: proc(dest, src: ^Int, digits: int) -> (err: Error) { } digits = min(digits, len(src.digit), len(dest.digit)); - mem.copy_non_overlapping(&dest.digit[0], &src.digit[0], size_of(DIGIT) * digits); + mem.copy_non_overlapping(&dest.digit[0], &src.digit[offset], size_of(DIGIT) * digits); return nil; } diff --git a/core/math/big/test.py b/core/math/big/test.py index 5b5a86cff..4798d110f 100644 --- a/core/math/big/test.py +++ b/core/math/big/test.py @@ -446,6 +446,7 @@ TESTS = { test_mul: [ [ 1234, 5432], [ 0xd3b4e926aaba3040e1c12b5ea553b5, 0x1a821e41257ed9281bee5bc7789ea7 ], + [ 1 << 21_105, 1 << 21_501 ], ], test_sqr: [ [ 5432], @@ -531,7 +532,7 @@ TESTS = { if not args.fast_tests: TESTS[test_factorial].append( # This one on its own takes around 800ms, so we exclude it for FAST_TESTS - [ 100_000 ], + [ 10_000 ], ) total_passes = 0