/** 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;
nothrow:
@safe:
public:
alias BigDigit = uint; // 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 @nogc @safe
{
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;
}
@safe 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 @nogc @safe
{
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 @nogc @safe
{
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 @nogc @safe
{
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;
}
}
@safe 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..$], 4);
assert(aa[0] == 0xF0FF_FFFF && aa[1] == 0x2222_2230
&& aa[2]==0x5555_5561 && aa[3]==0x9999_99A4 && aa[4]==0x0BCCC_CCCD);
}
/** dest[] = src[] * multiplier + carry.
* Returns carry.
*/
uint multibyteMul(uint[] dest, const(uint)[] src, uint multiplier, uint carry)
pure @nogc @safe
{
assert(dest.length == src.length);
ulong c = carry;
for (size_t i = 0; i < src.length; ++i)
{
c += cast(ulong)(src[i]) * multiplier;
dest[i] = cast(uint) c;
c>>=32;
}
return cast(uint) c;
}
@safe unittest
{
uint [] aa = [0xF0FF_FFFF, 0x1222_2223, 0x4555_5556, 0x8999_999A,
0xBCCC_CCCD, 0xEEEE_EEEE];
multibyteMul(aa[1 .. 4], aa[1 .. 4], 16, 0);
assert(aa[0] == 0xF0FF_FFFF && aa[1] == 0x2222_2230 && aa[2]==0x5555_5561
&& aa[3]==0x9999_99A4 && aa[4]==0x0BCCC_CCCD);
}
/**
* dest[] += src[] * multiplier + carry(0 .. FFFF_FFFF).
* Returns carry out of MSB (0 .. FFFF_FFFF).
*/
uint multibyteMulAdd(char op)(uint [] dest, const(uint)[] src,
uint multiplier, uint carry) pure @nogc @safe
{
assert(dest.length == src.length);
ulong c = carry;
for (size_t i = 0; i < src.length; ++i)
{
static if (op=='+')
{
c += cast(ulong)(multiplier) * src[i] + dest[i];
dest[i] = cast(uint) c;
c >>= 32;
}
else
{
c += cast(ulong) multiplier * src[i];
ulong t = cast(ulong) dest[i] - cast(uint) c;
dest[i] = cast(uint) t;
c = cast(uint)((c >> 32) - (t >> 32));
}
}
return cast(uint) c;
}
@safe unittest
{
uint [] aa = [0xF0FF_FFFF, 0x1222_2223, 0x4555_5556, 0x8999_999A,
0xBCCC_CCCD, 0xEEEE_EEEE];
uint [] bb = [0x1234_1234, 0xF0F0_F0F0, 0x00C0_C0C0, 0xF0F0_F0F0,
0xC0C0_C0C0];
multibyteMulAdd!('+')(bb[1..$-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 @nogc @safe
{
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 @nogc @safe
{
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;
}
@safe 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 @nogc @safe
{
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 @nogc @safe
{
// 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 @nogc @safe
{
multibyteTriangleAccumulate(result, x);
result[$-1] = multibyteShl(result[1..$-1], result[1..$-1], 1); // mul by 2
result[0] = 0;
multibyteAddDiagonalSquares(result, x);
}