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# std.bigint

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:
• Karatsuba multiplication
• Squaring is optimized independently of multiplication
• Divide-and-conquer division
• Binary exponentiation
For very large numbers, consider using the GMP library instead.
Authors:
Don Clugston

Source: std/bigint.d

struct BigInt;
A struct representing an arbitrary precision integer.
All arithmetic operations are supported, except unsigned shift right (>>>). Bitwise operations (|, &, ^, ~) are supported, and behave as if BigInt was an infinite length 2's complement number.
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.)
Examples:
```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;
```
this(Range)(Range s)
if (isBidirectionalRange!Range && isSomeChar!(ElementType!Range) && !isInfinite!Range);
Construct a BigInt from a decimal or hexadecimal string. The number must be in the form of a decimal or hex literal. It may have a leading + or - sign, followed by 0x or 0X if hexadecimal. Underscores are permitted in any location after the 0x and/or the sign of the number.
Parameters:
 Range s a finite bidirectional range of any character type
Throws:
ConvException if the string doesn't represent a valid number
pure nothrow this(T)(T x)
if (isIntegral!T);
Construct a BigInt from a built-in integral type.
Examples:
```// @system due to failure in FreeBSD32
ulong data = 1_000_000_000_000;
auto bigData = BigInt(data);
assert(data == BigInt("1_000_000_000_000"));
```
pure nothrow this(T)(T x)
if (is(Unqual!T == BigInt));
Construct a BigInt from another BigInt.
Examples:
```const(BigInt) b1 = BigInt("1_234_567_890");
BigInt b2 = BigInt(b1);
assert(b2 == BigInt("1_234_567_890"));
```
pure nothrow BigInt opAssign(T)(T x)
if (isIntegral!T);
Assignment from built-in integer types.
Examples:
```auto b = BigInt("123");
b = 456;
assert(b == BigInt("456"));
```
pure @nogc BigInt opAssign(T : BigInt)(T x);
Assignment from another BigInt.
Examples:
```auto b1 = BigInt("123");
auto b2 = BigInt("456");
b2 = b1;
assert(b2 == BigInt("123"));
```
pure nothrow BigInt opOpAssign(string op, T)(T y)
if ((op == "+" || op == "-" || op == "*" || op == "/" || op == "%" || op == ">>" || op == "<<" || op == "^^" || op == "|" || op == "&" || op == "^") && isIntegral!T);
Implements assignment operators from built-in integers of the form BigInt op= integer.
Examples:
```//@system because opOpAssign is @system
auto b = BigInt("1_000_000_000");

b += 12345;
assert(b == BigInt("1_000_012_345"));

b /= 5;
assert(b == BigInt("200_002_469"));
```
pure nothrow BigInt opOpAssign(string op, T)(T y)
if ((op == "+" || op == "-" || op == "*" || op == "|" || op == "&" || op == "^" || op == "/" || op == "%") && is(T : BigInt));
Implements assignment operators of the form BigInt op= BigInt.
Examples:
```// @system because opOpAssign is @system
auto x = BigInt("123");
auto y = BigInt("321");
x += y;
assert(x == BigInt("444"));
```
const pure nothrow BigInt opBinary(string op, T)(T y)
if ((op == "+" || op == "*" || op == "-" || op == "|" || op == "&" || op == "^" || op == "/" || op == "%") && is(T : BigInt));
Implements binary operators between BigInts.
Examples:
```auto x = BigInt("123");
auto y = BigInt("456");
BigInt z = x * y;
assert(z == BigInt("56088"));
```
const pure nothrow BigInt opBinary(string op, T)(T y)
if ((op == "+" || op == "*" || op == "-" || op == "/" || op == "|" || op == "&" || op == "^" || op == ">>" || op == "<<" || op == "^^") && isIntegral!T);
Implements binary operators between BigInt's and built-in integers.
Examples:
```auto x = BigInt("123");
x *= 300;
assert(x == BigInt("36900"));
```
const pure nothrow auto opBinary(string op, T)(T y)
if (op == "%" && isIntegral!T);
Implements a narrowing remainder operation with built-in integer types.
This binary operator returns a narrower, built-in integer type where applicable, according to the following table.
 BigInt % long → long BigInt % ulong → BigInt BigInt % other type → int
Examples:
```auto  x  = BigInt("1_000_000_500");
long  l  = 1_000_000L;
ulong ul = 2_000_000UL;
int   i  = 500_000;
short s  = 30_000;

assert(is(typeof(x % l)  == long)   && x % l  == 500L);
assert(is(typeof(x % ul) == BigInt) && x % ul == BigInt(500));
assert(is(typeof(x % i)  == int)    && x % i  == 500);
assert(is(typeof(x % s)  == int)    && x % s  == 10500);
```
const pure nothrow BigInt opBinaryRight(string op, T)(T y)
if ((op == "+" || op == "*" || op == "|" || op == "&" || op == "^") && isIntegral!T);

const pure nothrow BigInt opBinaryRight(string op, T)(T y)
if (op == "-" && isIntegral!T);

const pure nothrow T opBinaryRight(string op, T)(T x)
if ((op == "%" || op == "/") && isIntegral!T);
Implements operators with built-in integers on the left-hand side and BigInt on the right-hand side.
Examples:
```auto x = BigInt("100");
BigInt y = 123 + x;
assert(y == BigInt("223"));

BigInt z = 123 - x;
assert(z == BigInt("23"));

// Dividing a built-in integer type by BigInt always results in
// something that fits in a built-in type, so the built-in type is
// returned, not BigInt.
assert(is(typeof(1000 / x) == int));
assert(1000 / x == 10);
```
const pure nothrow BigInt opUnary(string op)()
if (op == "+" || op == "-" || op == "~");

pure nothrow BigInt opUnary(string op)()
if (op == "++" || op == "--");
Implements BigInt unary operators.
Examples:
```auto x = BigInt("1234");
assert(-x == BigInt("-1234"));

++x;
assert(x == BigInt("1235"));
```
const pure @nogc bool opEquals()(auto ref const BigInt y);

const pure nothrow @nogc bool opEquals(T)(T y)
if (isIntegral!T);
Implements BigInt equality test with other BigInt's and built-in integer types.
Examples:
```auto x = BigInt("12345");
auto y = BigInt("12340");
int z = 12345;
int w = 54321;

assert(x == x);
assert(x != y);
assert(x == y + 5);
assert(x == z);
assert(x != w);
```
const pure nothrow @nogc T opCast(T : bool)();
Implements casting to bool.
Examples:
```// Non-zero values are regarded as true
auto x = BigInt("1");
auto y = BigInt("10");
assert(x);
assert(y);

// Zero value is regarded as false
auto z = BigInt("0");
assert(!z);
```
const T opCast(T : ulong)();
Implements casting to integer types.
Throws:
std.conv.ConvOverflowException if the number exceeds the target type's range.
Examples:
```import std.conv : to, ConvOverflowException;
import std.exception : assertThrown;

assert(BigInt("0").to!int == 0);

assert(BigInt("0").to!ubyte == 0);
assert(BigInt("255").to!ubyte == 255);
assertThrown!ConvOverflowException(BigInt("256").to!ubyte);
assertThrown!ConvOverflowException(BigInt("-1").to!ubyte);
```
const pure nothrow @nogc T opCast(T)()
if (is(Unqual!T == BigInt));
Implements casting to/from qualified BigInt's.

Warning: Casting to/from const or immutable may break type system guarantees. Use with care.

Examples:
```const(BigInt) x = BigInt("123");
BigInt y = cast() x;    // cast away const
assert(y == x);
```
const pure nothrow @nogc int opCmp(ref const BigInt y);

const pure nothrow @nogc int opCmp(T)(T y)
if (isIntegral!T);

const pure nothrow @nogc int opCmp(T : BigInt)(const T y);
Implements 3-way comparisons of BigInt with BigInt or BigInt with built-in integers.
Examples:
```auto x = BigInt("100");
auto y = BigInt("10");
int z = 50;
const int w = 200;

assert(y < x);
assert(x > z);
assert(z > y);
assert(x < w);
```
const pure nothrow @nogc @safe long toLong();
Returns:
The value of this BigInt as a long, or +/- long.max if outside the representable range.
Examples:
```auto b = BigInt("12345");
long l = b.toLong();
assert(l == 12345);
```
const pure nothrow @nogc @safe int toInt();
Returns:
The value of this BigInt as an int, or +/- int.max if outside the representable range.
Examples:
```auto big = BigInt("5_000_000");
auto i = big.toInt();
assert(i == 5_000_000);

// Numbers that are too big to fit into an int will be clamped to int.max.
auto tooBig = BigInt("5_000_000_000");
i = tooBig.toInt();
assert(i == int.max);
```
const pure nothrow @nogc @property @safe size_t uintLength();
Number of significant uints which are used in storing this number.
The absolute value of this BigInt is always < 232*uintLength
const pure nothrow @nogc @property @safe size_t ulongLength();
Number of significant ulongs which are used in storing this number.
The absolute value of this BigInt is always < 264*ulongLength
const void toString(scope void delegate(const(char)[]) sink, string formatString);

const void toString(scope void delegate(const(char)[]) sink, ref FormatSpec!char f);
Convert the BigInt to string, passing it to the given sink.
Parameters:
void delegate(const(char)[]) sink A delegate for accepting possibly piecewise segments of the formatted string.
string formatString A format string specifying the output format.
 "d" Decimal "o" Octal "x" Hexadecimal, lower case "X" Hexadecimal, upper case "s" Default formatting (same as "d") null Default formatting (same as "d")
Examples:
toString is rarely directly invoked; the usual way of using it is via std.format.format:
```import std.format : format;

auto x = BigInt("1_000_000");
x *= 12345;

assert(format("%d", x) == "12345000000");
assert(format("%x", x) == "2_dfd1c040");
assert(format("%X", x) == "2_DFD1C040");
assert(format("%o", x) == "133764340100");
```
const nothrow @safe size_t toHash();
Returns:
A unique hash of the BigInt's value suitable for use in a hash table.
Examples:
toHash is rarely directly invoked; it is implicitly used when BigInt is used as the key of an associative array.
```string[BigInt] aa;
aa[BigInt(123)] = "abc";
aa[BigInt(456)] = "def";

assert(aa[BigInt(123)] == "abc");
assert(aa[BigInt(456)] == "def");
```
string toDecimalString(const(BigInt) x);
Parameters:
 const(BigInt) x The BigInt to convert to a decimal string.
Returns:
A string that represents the BigInt as a decimal number.
Examples:
```auto x = BigInt("123");
x *= 1000;
x += 456;

auto xstr = x.toDecimalString();
assert(xstr == "123456");
```
string toHex(const(BigInt) x);
Parameters:
 const(BigInt) x The BigInt to convert to a hexadecimal string.
Returns:
A string that represents the BigInt as a hexadecimal (base 16) number in upper case.
Examples:
```auto x = BigInt("123");
x *= 1000;
x += 456;

auto xstr = x.toHex();
assert(xstr == "1E240");
```
Unsigned!T absUnsign(T)(T x)
if (isIntegral!T);
Returns the absolute value of x converted to the corresponding unsigned type.
Parameters:
 T x The integral value to return the absolute value of.
Returns:
The absolute value of x.