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/** Arbitrary-precision ('bignum') arithmetic
*
* Performance is optimized for numbers below ~1000 decimal digits.
* For X86 machines, highly optimised assembly routines are used.
*
* The following algorithms are currently implemented:
* $(UL
* $(LI Karatsuba multiplication)
* $(LI Squaring is optimized independently of multiplication)
* $(LI Divide-and-conquer division)
* $(LI Binary exponentiation)
* )
*
* For very large numbers, consider using the $(WEB gmplib.org, GMP library) instead.
*
* License: <a href="http://www.boost.org/LICENSE_1_0.txt">Boost License 1.0</a>.
* Authors: Don Clugston
* Source: $(PHOBOSSRC std/_bigint.d)
*/
/* 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.bigint;
private import std.internal.math.biguintcore;
private import std.format : FormatSpec, FormatException;
private import std.traits;
/** A struct representing an arbitrary precision integer
*
* All arithmetic operations are supported, except
* unsigned shift right (>>>). Logical operations are not currently supported.
*
* BigInt implements value semantics using copy-on-write. This means that
* assignment is cheap, but operations such as x++ will cause heap
* allocation. (But note that for most bigint operations, heap allocation is
* inevitable anyway).
Example:
----------------------------------------------------
BigInt a = "9588669891916142";
BigInt b = "7452469135154800";
auto c = a * b;
assert(c == BigInt("71459266416693160362545788781600"));
auto d = b * a;
assert(d == BigInt("71459266416693160362545788781600"));
assert(d == c);
d = c * BigInt("794628672112");
assert(d == BigInt("56783581982794522489042432639320434378739200"));
auto e = c + d;
assert(e == BigInt("56783581982865981755459125799682980167520800"));
auto f = d + c;
assert(f == e);
auto g = f - c;
assert(g == d);
g = f - d;
assert(g == c);
e = 12345678;
g = c + e;
auto h = g / b;
auto i = g % b;
assert(h == a);
assert(i == e);
BigInt j = "-0x9A56_57f4_7B83_AB78";
j ^^= 11;
----------------------------------------------------
*
*/
struct BigInt
{
private:
BigUint data; // BigInt adds signed arithmetic to BigUint.
bool sign = false;
public:
/// Construct a BigInt from a decimal or hexadecimal string.
/// The number must be in the form of a D decimal or hex literal:
/// It may have a leading + or - sign; followed by "0x" if hexadecimal.
/// Underscores are permitted.
/// BUG: Should throw a IllegalArgumentException/ConvError if invalid character found
this(T : const(char)[] )(T s)
{
bool neg = false;
if (s[0] == '-') {
neg = true;
s = s[1..$];
} else if (s[0] == '+') {
s = s[1..$];
}
data = 0UL;
auto q = 0X3;
bool ok;
assert(isZero());
if (s.length > 2 && (s[0..2] == "0x" || s[0..2] == "0X"))
{
ok = data.fromHexString(s[2..$]);
} else {
ok = data.fromDecimalString(s);
}
assert(ok);
if (isZero())
neg = false;
sign = neg;
}
///
this(T)(T x) if (isIntegral!T)
{
data = data.init; // @@@: Workaround for compiler bug
opAssign(x);
}
///
BigInt opAssign(T)(T x) if (isIntegral!T)
{
data = cast(ulong)((x < 0) ? -x : x);
sign = (x < 0);
return this;
}
///
BigInt opAssign(T:BigInt)(T x)
{
data = x.data;
sign = x.sign;
return this;
}
// BigInt op= integer
BigInt opOpAssign(string op, T)(T y)
if ((op=="+" || op=="-" || op=="*" || op=="/" || op=="%"
|| op==">>" || op=="<<" || op=="^^") && isIntegral!T)
{
ulong u = cast(ulong)(y < 0 ? -y : y);
static if (op=="+")
{
data = BigUint.addOrSubInt(data, u, sign != (y<0), sign);
}
else static if (op=="-")
{
data = BigUint.addOrSubInt(data, u, sign == (y<0), sign);
}
else static if (op=="*")
{
if (y == 0) {
sign = false;
data = 0UL;
} else {
sign = ( sign != (y<0) );
data = BigUint.mulInt(data, u);
}
}
else static if (op=="/")
{
assert(y!=0, "Division by zero");
static assert(!is(T == long) && !is(T == ulong));
data = BigUint.divInt(data, cast(uint)u);
sign = data.isZero() ? false : sign ^ (y < 0);
}
else static if (op=="%")
{
assert(y!=0, "Division by zero");
static if (is(immutable(T) == immutable(long)) || is( immutable(T) == immutable(ulong) ))
{
this %= BigInt(y);
}
else
{
data = cast(ulong)BigUint.modInt(data, cast(uint)u);
}
// x%y always has the same sign as x.
// This is not the same as mathematical mod.
}
else static if (op==">>" || op=="<<")
{
// Do a left shift if y>0 and <<, or
// if y<0 and >>; else do a right shift.
if (y == 0)
return this;
else if ((y > 0) == (op=="<<"))
{
// Sign never changes during left shift
data = data.opShl(u);
} else
{
data = data.opShr(u);
if (data.isZero())
sign = false;
}
}
else static if (op=="^^")
{
sign = (y & 1) ? sign : false;
data = BigUint.pow(data, u);
}
else static assert(0, "BigInt " ~ op[0..$-1] ~ "= " ~ T.stringof ~ " is not supported");
return this;
}
// BigInt op= BigInt
BigInt opOpAssign(string op, T)(T y)
if ((op=="+" || op== "-" || op=="*" || op=="/" || op=="%")
&& is (T: BigInt))
{
static if (op == "+")
{
data = BigUint.addOrSub(data, y.data, sign != y.sign, &sign);
}
else static if (op == "-")
{
data = BigUint.addOrSub(data, y.data, sign == y.sign, &sign);
}
else static if (op == "*")
{
data = BigUint.mul(data, y.data);
sign = isZero() ? false : sign ^ y.sign;
}
else static if (op == "/")
{
y.checkDivByZero();
if (!isZero())
{
data = BigUint.div(data, y.data);
sign = isZero() ? false : sign ^ y.sign;
}
}
else static if (op == "%")
{
y.checkDivByZero();
if (!isZero())
{
data = BigUint.mod(data, y.data);
// x%y always has the same sign as x.
if (isZero())
sign = false;
}
}
else static assert(0, "BigInt " ~ op[0..$-1] ~ "= " ~ T.stringof ~ " is not supported");
return this;
}
// BigInt op BigInt
BigInt opBinary(string op, T)(T y)
if ((op=="+" || op == "*" || op=="-" || op=="/" || op=="%") && is (T: BigInt))
{
BigInt r = this;
return r.opOpAssign!(op)(y);
}
// BigInt op integer
BigInt opBinary(string op, T)(T y)
if ((op=="+" || op == "*" || op=="-" || op=="/"
|| op==">>" || op=="<<" || op=="^^") && isIntegral!T)
{
BigInt r = this;
return r.opOpAssign!(op)(y);
}
//
int opBinary(string op, T : int)(T y)
if (op == "%" && isIntegral!T)
{
assert(y!=0);
uint u = y < 0 ? -y : y;
int rem = BigUint.modInt(data, u);
// x%y always has the same sign as x.
// This is not the same as mathematical mod.
return sign ? -rem : rem;
}
// Commutative operators
BigInt opBinaryRight(string op, T)(T y)
if ((op=="+" || op=="*") && isIntegral!T)
{
return opBinary!(op)(y);
}
// BigInt = integer op BigInt
BigInt opBinaryRight(string op, T)(T y)
if (op == "-" && isIntegral!T)
{
ulong u = cast(ulong)(y < 0 ? -y : y);
BigInt r;
static if (op == "-")
{
r.sign = sign;
r.data = BigUint.addOrSubInt(data, u, sign == (y<0), r.sign);
r.negate();
}
return r;
}
// integer = integer op BigInt
T opBinaryRight(string op, T)(T x)
if ((op=="%" || op=="/") && isIntegral!T)
{
static if (op == "%")
{
checkDivByZero();
// x%y always has the same sign as x.
if (data.ulongLength() > 1)
return x;
ulong u = x < 0 ? -x : x;
ulong rem = u % data.peekUlong(0);
// x%y always has the same sign as x.
return cast(T)((x<0) ? -rem : rem);
}
else static if (op == "/")
{
checkDivByZero();
if (data.ulongLength() > 1)
return 0;
return cast(T)(x / data.peekUlong(0));
}
}
// const unary operations
BigInt opUnary(string op)() /*const*/ if (op=="+" || op=="-")
{
static if (op=="-")
{
BigInt r = this;
r.negate();
return r;
}
else static if (op=="+")
return this;
}
// non-const unary operations
BigInt opUnary(string op)() if (op=="++" || op=="--")
{
static if (op=="++")
{
data = BigUint.addOrSubInt(data, 1UL, sign, sign);
return this;
}
else static if (op=="--")
{
data = BigUint.addOrSubInt(data, 1UL, !sign, sign);
return this;
}
}
///
bool opEquals()(auto ref const BigInt y) const
{
return sign == y.sign && y.data == data;
}
///
bool opEquals(T)(T y) const if (isIntegral!T)
{
if (sign != (y<0))
return 0;
return data.opEquals(cast(ulong)( y>=0 ? y : -y));
}
///
int opCmp(T)(T y) if (isIntegral!T)
{
if (sign != (y<0) )
return sign ? -1 : 1;
int cmp = data.opCmp(cast(ulong)(y >= 0 ? y : -y));
return sign? -cmp: cmp;
}
///
int opCmp(T:BigInt)(T y)
{
if (sign!=y.sign)
return sign ? -1 : 1;
int cmp = data.opCmp(y.data);
return sign? -cmp: cmp;
}
/// Returns the value of this BigInt as a long,
/// or +- long.max if outside the representable range.
long toLong() pure const
{
return (sign ? -1 : 1) *
(data.ulongLength() == 1 && (data.peekUlong(0) <= cast(ulong)(long.max))
? cast(long)(data.peekUlong(0))
: long.max);
}
/// Returns the value of this BigInt as an int,
/// or +- int.max if outside the representable range.
int toInt() pure const
{
return (sign ? -1 : 1) *
(data.uintLength() == 1 && (data.peekUint(0) <= cast(uint)(int.max))
? cast(int)(data.peekUint(0))
: int.max);
}
/// Number of significant uints which are used in storing this number.
/// The absolute value of this BigInt is always < 2^^(32*uintLength)
@property size_t uintLength() pure const
{
return data.uintLength();
}
/// Number of significant ulongs which are used in storing this number.
/// The absolute value of this BigInt is always < 2^^(64*ulongLength)
@property size_t ulongLength() pure const
{
return data.ulongLength();
}
/** Convert the BigInt to string, passing it to 'sink'.
*
* $(TABLE The output format is controlled via formatString:
* $(TR $(TD "d") $(TD Decimal))
* $(TR $(TD "x") $(TD Hexadecimal, lower case))
* $(TR $(TD "X") $(TD Hexadecimal, upper case))
* $(TR $(TD "s") $(TD Default formatting (same as "d") ))
* $(TR $(TD null) $(TD Default formatting (same as "d") ))
* )
*/
void toString(scope void delegate(const (char)[]) sink, string formatString) const
{
auto f = FormatSpec!char(formatString);
f.writeUpToNextSpec(sink);
toString(sink, f);
}
void toString(scope void delegate(const(char)[]) sink, ref FormatSpec!char f) const
{
auto hex = (f.spec == 'x' || f.spec == 'X');
if (!(f.spec == 's' || f.spec == 'd' || hex))
throw new FormatException("Format specifier not understood: %" ~ f.spec);
char[] buff =
hex ? data.toHexString(0, '_', 0, f.flZero ? '0' : ' ')
: data.toDecimalString(0);
assert(buff.length > 0);
char signChar = isNegative() ? '-' : 0;
auto minw = buff.length + (signChar ? 1 : 0);
if (!hex && !signChar && (f.width == 0 || minw < f.width))
{
if (f.flPlus)
signChar = '+', ++minw;
else if (f.flSpace)
signChar = ' ', ++minw;
}
auto maxw = minw < f.width ? f.width : minw;
auto difw = maxw - minw;
if (!f.flDash && !f.flZero)
foreach (i; 0 .. difw)
sink(" ");
if (signChar)
sink((&signChar)[0..1]);
if (!f.flDash && f.flZero)
foreach (i; 0 .. difw)
sink("0");
sink(buff);
if (f.flDash)
foreach (i; 0 .. difw)
sink(" ");
}
/+
private:
/// Convert to a hexadecimal string, with an underscore every
/// 8 characters.
string toHex()
{
string buff = data.toHexString(1, '_');
if (isNegative())
buff[0] = '-';
else
buff = buff[1..$];
return buff;
}
+/
private:
void negate()
{
if (!data.isZero())
sign = !sign;
}
bool isZero() pure const
{
return data.isZero();
}
bool isNegative() pure const
{
return sign;
}
// Generate a runtime error if division by zero occurs
void checkDivByZero() pure const
{
if (isZero())
throw new Error("BigInt division by zero");
}
}
string toDecimalString(BigInt x)
{
string outbuff="";
void sink(const(char)[] s) { outbuff ~= s; }
x.toString(&sink, "%d");
return outbuff;
}
string toHex(BigInt x)
{
string outbuff="";
void sink(const(char)[] s) { outbuff ~= s; }
x.toString(&sink, "%x");
return outbuff;
}
unittest {
// Radix conversion
assert( toDecimalString(BigInt("-1_234_567_890_123_456_789"))
== "-1234567890123456789");
assert( toHex(BigInt("0x1234567890123456789")) == "123_45678901_23456789");
assert( toHex(BigInt("0x00000000000000000000000000000000000A234567890123456789"))
== "A23_45678901_23456789");
assert( toHex(BigInt("0x000_00_000000_000_000_000000000000_000000_")) == "0");
assert(BigInt(-0x12345678).toInt() == -0x12345678);
assert(BigInt(-0x12345678).toLong() == -0x12345678);
assert(BigInt(0x1234_5678_9ABC_5A5AL).ulongLength == 1);
assert(BigInt(0x1234_5678_9ABC_5A5AL).toLong() == 0x1234_5678_9ABC_5A5AL);
assert(BigInt(-0x1234_5678_9ABC_5A5AL).toLong() == -0x1234_5678_9ABC_5A5AL);
assert(BigInt(0xF234_5678_9ABC_5A5AL).toLong() == long.max);
assert(BigInt(-0x123456789ABCL).toInt() == -int.max);
char[] s1 = "123".dup; // bug 8164
assert(BigInt(s1) == 123);
char[] s2 = "0xABC".dup;
assert(BigInt(s2) == 2748);
assert((BigInt(-2) + BigInt(1)) == BigInt(-1));
BigInt a = ulong.max - 5;
auto b = -long.max % a;
assert( b == -long.max % (ulong.max - 5));
b = long.max / a;
assert( b == long.max /(ulong.max - 5));
assert(BigInt(1) - 1 == 0);
assert((-4) % BigInt(5) == -4); // bug 5928
assert(BigInt(-4) % BigInt(5) == -4);
assert(BigInt(2)/BigInt(-3) == BigInt(0)); // bug 8022
}
unittest // Recursive division, bug 5568
{
enum Z = 4843;
BigInt m = (BigInt(1) << (Z*8) ) - 1;
m -= (BigInt(1) << (Z*6)) - 1;
BigInt oldm = m;
BigInt a = (BigInt(1) << (Z*4) )-1;
BigInt b = m % a;
m /= a;
m *= a;
assert( m + b == oldm);
m = (BigInt(1) << (4846 + 4843) ) - 1;
a = (BigInt(1) << 4846 ) - 1;
b = (BigInt(1) << (4846*2 + 4843)) - 1;
BigInt c = (BigInt(1) << (4846*2 + 4843*2)) - 1;
BigInt w = c - b + a;
assert(w % m == 0);
// Bug 6819. ^^
BigInt z1 = BigInt(10)^^64;
BigInt w1 = BigInt(10)^^128;
assert(z1^^2 == w1);
BigInt z2 = BigInt(1)<<64;
BigInt w2 = BigInt(1)<<128;
assert(z2^^2 == w2);
// Bug 7993
BigInt n7793 = 10;
assert( n7793 / 1 == 10);
// Bug 7973
auto a7973 = 10_000_000_000_000_000;
const c7973 = 10_000_000_000_000_000;
immutable i7973 = 10_000_000_000_000_000;
BigInt v7973 = 2551700137;
v7973 %= a7973;
assert(v7973 == 2551700137);
v7973 %= c7973;
assert(v7973 == 2551700137);
v7973 %= i7973;
assert(v7973 == 2551700137);
// 8165
BigInt[2] a8165;
a8165[0] = a8165[1] = 1;
}
unittest
{
import std.array;
import std.format;
immutable string[][] table = [
/* fmt, +10 -10 */
["%d", "10", "-10"],
["%+d", "+10", "-10"],
["%-d", "10", "-10"],
["%+-d", "+10", "-10"],
["%4d", " 10", " -10"],
["%+4d", " +10", " -10"],
["%-4d", "10 ", "-10 "],
["%+-4d", "+10 ", "-10 "],
["%04d", "0010", "-010"],
["%+04d", "+010", "-010"],
["%-04d", "10 ", "-10 "],
["%+-04d", "+10 ", "-10 "],
["% 04d", " 010", "-010"],
["%+ 04d", "+010", "-010"],
["%- 04d", " 10 ", "-10 "],
["%+- 04d", "+10 ", "-10 "],
];
auto w1 = appender!(char[])();
auto w2 = appender!(char[])();
foreach (entry; table)
{
immutable fmt = entry[0];
formattedWrite(w1, fmt, BigInt(10));
formattedWrite(w2, fmt, 10);
assert(w1.data == w2.data);
assert(w1.data == entry[1]);
w1.clear();
w2.clear();
formattedWrite(w1, fmt, BigInt(-10));
formattedWrite(w2, fmt, -10);
assert(w1.data == w2.data);
assert(w1.data == entry[2]);
w1.clear();
w2.clear();
}
}
unittest
{
import std.array;
import std.format;
immutable string[][] table = [
/* fmt, +10 -10 */
["%X", "A", "-A"],
["%+X", "A", "-A"],
["%-X", "A", "-A"],
["%+-X", "A", "-A"],
["%4X", " A", " -A"],
["%+4X", " A", " -A"],
["%-4X", "A ", "-A "],
["%+-4X", "A ", "-A "],
["%04X", "000A", "-00A"],
["%+04X", "000A", "-00A"],
["%-04X", "A ", "-A "],
["%+-04X", "A ", "-A "],
["% 04X", "000A", "-00A"],
["%+ 04X", "000A", "-00A"],
["%- 04X", "A ", "-A "],
["%+- 04X", "A ", "-A "],
];
auto w1 = appender!(char[])();
auto w2 = appender!(char[])();
foreach (entry; table)
{
immutable fmt = entry[0];
formattedWrite(w1, fmt, BigInt(10));
formattedWrite(w2, fmt, 10);
assert(w1.data == w2.data); // Equal only positive BigInt
assert(w1.data == entry[1]);
w1.clear();
w2.clear();
formattedWrite(w1, fmt, BigInt(-10));
//formattedWrite(w2, fmt, -10);
//assert(w1.data == w2.data);
assert(w1.data == entry[2]);
w1.clear();
//w2.clear();
}
}
// 6448
unittest
{
import std.array;
import std.format;
auto w1 = appender!string();
auto w2 = appender!string();
int x = 100;
formattedWrite(w1, "%010d", x);
BigInt bx = x;
formattedWrite(w2, "%010d", bx);
assert(w1.data == w2.data);
//8011
BigInt y = -3;
++y;
assert(y.toLong() == -2);
y = 1;
--y;
assert(y.toLong() == 0);
--y;
assert(y.toLong() == -1);
--y;
assert(y.toLong() == -2);
}
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