/std/internal/math/biguintnoasm.d
http://github.com/jcd/phobos · D · 360 lines · 270 code · 31 blank · 59 comment · 74 complexity · 65728d2d08a4409d5ccca9ab6fe3adca MD5 · raw file
- /** Arbitrary precision arithmetic ('bignum') for processors with no asm support
- *
- * All functions operate on arrays of uints, stored LSB first.
- * If there is a destination array, it will be the first parameter.
- * Currently, all of these functions are subject to change, and are
- * intended for internal use only.
- * This module is intended only to assist development of high-speed routines
- * on currently unsupported processors.
- * The X86 asm version is about 30 times faster than the D version(DMD).
- */
- /* Copyright Don Clugston 2008 - 2010.
- * Distributed under the Boost Software License, Version 1.0.
- * (See accompanying file LICENSE_1_0.txt or copy at
- * http://www.boost.org/LICENSE_1_0.txt)
- */
- module std.internal.math.biguintnoasm;
- public:
- alias uint BigDigit; // A Bignum is an array of BigDigits.
- // Limits for when to switch between multiplication algorithms.
- enum : int { KARATSUBALIMIT = 10 }; // Minimum value for which Karatsuba is worthwhile.
- enum : int { KARATSUBASQUARELIMIT=12 }; // Minimum value for which square Karatsuba is worthwhile
- /** Multi-byte addition or subtraction
- * dest[] = src1[] + src2[] + carry (0 or 1).
- * or dest[] = src1[] - src2[] - carry (0 or 1).
- * Returns carry or borrow (0 or 1).
- * Set op == '+' for addition, '-' for subtraction.
- */
- uint multibyteAddSub(char op)(uint[] dest, const(uint) [] src1,
- const (uint) [] src2, uint carry) pure
- {
- ulong c = carry;
- for (size_t i = 0; i < src2.length; ++i)
- {
- static if (op=='+') c = c + src1[i] + src2[i];
- else c = cast(ulong)src1[i] - src2[i] - c;
- dest[i] = cast(uint)c;
- c = (c > 0xFFFF_FFFF);
- }
- return cast(uint)c;
- }
- unittest
- {
- uint [] a = new uint[40];
- uint [] b = new uint[40];
- uint [] c = new uint[40];
- for (size_t i = 0; i < a.length; ++i)
- {
- if (i&1) a[i]=cast(uint)(0x8000_0000 + i);
- else a[i]=cast(uint)i;
- b[i]= 0x8000_0003;
- }
- c[19]=0x3333_3333;
- uint carry = multibyteAddSub!('+')(c[0..18], b[0..18], a[0..18], 0);
- assert(c[0]==0x8000_0003);
- assert(c[1]==4);
- assert(c[19]==0x3333_3333); // check for overrun
- assert(carry==1);
- for (size_t i = 0; i < a.length; ++i)
- {
- a[i] = b[i] = c[i] = 0;
- }
- a[8]=0x048D159E;
- b[8]=0x048D159E;
- a[10]=0x1D950C84;
- b[10]=0x1D950C84;
- a[5] =0x44444444;
- carry = multibyteAddSub!('-')(a[0..12], a[0..12], b[0..12], 0);
- assert(a[11] == 0);
- for (size_t i = 0; i < 10; ++i)
- if (i != 5)
- assert(a[i] == 0);
- for (size_t q = 3; q < 36; ++q)
- {
- for (size_t i = 0; i< a.length; ++i)
- {
- a[i] = b[i] = c[i] = 0;
- }
- a[q-2]=0x040000;
- b[q-2]=0x040000;
- carry = multibyteAddSub!('-')(a[0..q], a[0..q], b[0..q], 0);
- assert(a[q-2]==0);
- }
- }
- /** dest[] += carry, or dest[] -= carry.
- * op must be '+' or '-'
- * Returns final carry or borrow (0 or 1)
- */
- uint multibyteIncrementAssign(char op)(uint[] dest, uint carry) pure
- {
- static if (op=='+')
- {
- ulong c = carry;
- c += dest[0];
- dest[0] = cast(uint)c;
- if (c<=0xFFFF_FFFF)
- return 0;
- for (size_t i = 1; i < dest.length; ++i)
- {
- ++dest[i];
- if (dest[i] != 0)
- return 0;
- }
- return 1;
- }
- else
- {
- ulong c = carry;
- c = dest[0] - c;
- dest[0] = cast(uint)c;
- if (c<=0xFFFF_FFFF)
- return 0;
- for (size_t i = 1; i < dest.length; ++i)
- {
- --dest[i];
- if (dest[i] != 0xFFFF_FFFF)
- return 0;
- }
- return 1;
- }
- }
- /** dest[] = src[] << numbits
- * numbits must be in the range 1..31
- */
- uint multibyteShl(uint [] dest, const(uint) [] src, uint numbits) pure
- {
- ulong c = 0;
- for (size_t i = 0; i < dest.length; ++i)
- {
- c += (cast(ulong)(src[i]) << numbits);
- dest[i] = cast(uint)c;
- c >>>= 32;
- }
- return cast(uint)c;
- }
- /** dest[] = src[] >> numbits
- * numbits must be in the range 1..31
- */
- void multibyteShr(uint [] dest, const(uint) [] src, uint numbits) pure
- {
- ulong c = 0;
- for(ptrdiff_t i = dest.length; i!=0; --i)
- {
- c += (src[i-1] >>numbits) + (cast(ulong)(src[i-1]) << (64 - numbits));
- dest[i-1] = cast(uint)c;
- c >>>= 32;
- }
- }
- unittest
- {
- uint [] aa = [0x1222_2223, 0x4555_5556, 0x8999_999A, 0xBCCC_CCCD, 0xEEEE_EEEE];
- multibyteShr(aa[0..$-2], aa, 4);
- assert(aa[0] == 0x6122_2222 && aa[1] == 0xA455_5555 && aa[2] == 0x0899_9999);
- assert(aa[3] == 0xBCCC_CCCD);
- aa = [0x1222_2223, 0x4555_5556, 0x8999_999A, 0xBCCC_CCCD, 0xEEEE_EEEE];
- multibyteShr(aa[0..$-1], aa, 4);
- assert(aa[0] == 0x6122_2222 && aa[1] == 0xA455_5555
- && aa[2] == 0xD899_9999 && aa[3] == 0x0BCC_CCCC);
- aa = [0xF0FF_FFFF, 0x1222_2223, 0x4555_5556, 0x8999_999A, 0xBCCC_CCCD,
- 0xEEEE_EEEE];
- multibyteShl(aa[1..4], aa[1..$], 4);
- assert(aa[0] == 0xF0FF_FFFF && aa[1] == 0x2222_2230
- && aa[2]==0x5555_5561 && aa[3]==0x9999_99A4 && aa[4]==0x0BCCC_CCCD);
- }
- /** dest[] = src[] * multiplier + carry.
- * Returns carry.
- */
- uint multibyteMul(uint[] dest, const(uint)[] src, uint multiplier, uint carry)
- pure
- {
- assert(dest.length == src.length);
- ulong c = carry;
- for(size_t i = 0; i < src.length; ++i)
- {
- c += cast(ulong)(src[i]) * multiplier;
- dest[i] = cast(uint)c;
- c>>=32;
- }
- return cast(uint)c;
- }
- unittest
- {
- uint [] aa = [0xF0FF_FFFF, 0x1222_2223, 0x4555_5556, 0x8999_999A,
- 0xBCCC_CCCD, 0xEEEE_EEEE];
- multibyteMul(aa[1..4], aa[1..4], 16, 0);
- assert(aa[0] == 0xF0FF_FFFF && aa[1] == 0x2222_2230 && aa[2]==0x5555_5561
- && aa[3]==0x9999_99A4 && aa[4]==0x0BCCC_CCCD);
- }
- /**
- * dest[] += src[] * multiplier + carry(0..FFFF_FFFF).
- * Returns carry out of MSB (0..FFFF_FFFF).
- */
- uint multibyteMulAdd(char op)(uint [] dest, const(uint)[] src,
- uint multiplier, uint carry) pure
- {
- assert(dest.length == src.length);
- ulong c = carry;
- for(size_t i = 0; i < src.length; ++i)
- {
- static if(op=='+')
- {
- c += cast(ulong)(multiplier) * src[i] + dest[i];
- dest[i] = cast(uint)c;
- c >>= 32;
- }
- else
- {
- c += cast(ulong)multiplier * src[i];
- ulong t = cast(ulong)dest[i] - cast(uint)c;
- dest[i] = cast(uint)t;
- c = cast(uint)((c>>32) - (t>>32));
- }
- }
- return cast(uint)c;
- }
- unittest
- {
- uint [] aa = [0xF0FF_FFFF, 0x1222_2223, 0x4555_5556, 0x8999_999A,
- 0xBCCC_CCCD, 0xEEEE_EEEE];
- uint [] bb = [0x1234_1234, 0xF0F0_F0F0, 0x00C0_C0C0, 0xF0F0_F0F0,
- 0xC0C0_C0C0];
- multibyteMulAdd!('+')(bb[1..$-1], aa[1..$-2], 16, 5);
- assert(bb[0] == 0x1234_1234 && bb[4] == 0xC0C0_C0C0);
- assert(bb[1] == 0x2222_2230 + 0xF0F0_F0F0 + 5
- && bb[2] == 0x5555_5561 + 0x00C0_C0C0 + 1
- && bb[3] == 0x9999_99A4 + 0xF0F0_F0F0 );
- }
- /**
- Sets result = result[0..left.length] + left * right
- It is defined in this way to allow cache-efficient multiplication.
- This function is equivalent to:
- ----
- for (size_t i = 0; i< right.length; ++i) {
- dest[left.length + i] = multibyteMulAdd(dest[i..left.length+i],
- left, right[i], 0);
- }
- ----
- */
- void multibyteMultiplyAccumulate(uint [] dest, const(uint)[] left, const(uint)
- [] right) pure
- {
- for (size_t i = 0; i < right.length; ++i)
- {
- dest[left.length + i] = multibyteMulAdd!('+')(dest[i..left.length+i],
- left, right[i], 0);
- }
- }
- /** dest[] /= divisor.
- * overflow is the initial remainder, and must be in the range 0..divisor-1.
- */
- uint multibyteDivAssign(uint [] dest, uint divisor, uint overflow) pure
- {
- ulong c = cast(ulong)overflow;
- for(ptrdiff_t i = dest.length-1; i>= 0; --i)
- {
- c = (c<<32) + cast(ulong)(dest[i]);
- uint q = cast(uint)(c/divisor);
- c -= divisor * q;
- dest[i] = q;
- }
- return cast(uint)c;
- }
- unittest
- {
- uint [] aa = new uint[101];
- for (uint i = 0; i < aa.length; ++i)
- aa[i] = 0x8765_4321 * (i+3);
- uint overflow = multibyteMul(aa, aa, 0x8EFD_FCFB, 0x33FF_7461);
- uint r = multibyteDivAssign(aa, 0x8EFD_FCFB, overflow);
- for (uint i=0; i<aa.length; ++i)
- {
- assert(aa[i] == 0x8765_4321 * (i+3));
- }
- assert(r == 0x33FF_7461);
- }
- // Set dest[2*i..2*i+1]+=src[i]*src[i]
- void multibyteAddDiagonalSquares(uint[] dest, const(uint)[] src) pure
- {
- ulong c = 0;
- for(size_t i = 0; i < src.length; ++i)
- {
- // At this point, c is 0 or 1, since FFFF*FFFF+FFFF_FFFF = 1_0000_0000.
- c += cast(ulong)(src[i]) * src[i] + dest[2*i];
- dest[2*i] = cast(uint)c;
- c = (c>>=32) + dest[2*i+1];
- dest[2*i+1] = cast(uint)c;
- c >>= 32;
- }
- }
- // Does half a square multiply. (square = diagonal + 2*triangle)
- void multibyteTriangleAccumulate(uint[] dest, const(uint)[] x) pure
- {
- // x[0]*x[1...$] + x[1]*x[2..$] + ... + x[$-2]x[$-1..$]
- dest[x.length] = multibyteMul(dest[1 .. x.length], x[1..$], x[0], 0);
- if (x.length < 4)
- {
- if (x.length == 3)
- {
- ulong c = cast(ulong)(x[$-1]) * x[$-2] + dest[2*x.length-3];
- dest[2*x.length - 3] = cast(uint)c;
- c >>= 32;
- dest[2*x.length - 2] = cast(uint)c;
- }
- return;
- }
- for (size_t i = 2; i < x.length - 2; ++i)
- {
- dest[i-1+ x.length] = multibyteMulAdd!('+')(
- dest[i+i-1 .. i+x.length-1], x[i..$], x[i-1], 0);
- }
- // Unroll the last two entries, to reduce loop overhead:
- ulong c = cast(ulong)(x[$-3]) * x[$-2] + dest[2*x.length-5];
- dest[2*x.length-5] = cast(uint)c;
- c >>= 32;
- c += cast(ulong)(x[$-3]) * x[$-1] + dest[2*x.length-4];
- dest[2*x.length-4] = cast(uint)c;
- c >>= 32;
- c += cast(ulong)(x[$-1]) * x[$-2];
- dest[2*x.length-3] = cast(uint)c;
- c >>= 32;
- dest[2*x.length-2] = cast(uint)c;
- }
- void multibyteSquare(BigDigit[] result, const(BigDigit) [] x) pure
- {
- multibyteTriangleAccumulate(result, x);
- result[$-1] = multibyteShl(result[1..$-1], result[1..$-1], 1); // mul by 2
- result[0] = 0;
- multibyteAddDiagonalSquares(result, x);
- }