...

# Package big

`import "math/big"`
Overview
Index
Examples

## Overview ▾

Package big implements multi-precision arithmetic (big numbers). The following numeric types are supported:

```Int    signed integers
Rat    rational numbers
Float  floating-point numbers
```

Methods are typically of the form:

```func (z *T) Unary(x *T) *T        // z = op x
func (z *T) Binary(x, y *T) *T    // z = x op y
func (x *T) M() T1                // v = x.M()
```

with T one of Int, Rat, or Float. For unary and binary operations, the result is the receiver (usually named z in that case); if it is one of the operands x or y it may be overwritten (and its memory reused). To enable chaining of operations, the result is also returned. Methods returning a result other than *Int, *Rat, or *Float take an operand as the receiver (usually named x in that case).

Example (Fibonacci)

This example demonstrates how to use big.Int to compute the smallest Fibonacci number with 100 decimal digits and to test whether it is prime.

```1344719667586153181419716641724567886890850696275767987106294472017884974410332069524504824747437757
false
```

Example (Sqrt2)

This example shows how to use big.Float to compute the square root of 2 with a precision of 200 bits, and how to print the result as a decimal number.

```sqrt(2) = 1.41421356237309504880168872420969807856967187537695
error = 0.000000e+00
```

## Index ▾

Constants
func Jacobi(x, y *Int) int
type Accuracy
func (i Accuracy) String() string
type ErrNaN
func (err ErrNaN) Error() string
type Float
func NewFloat(x float64) *Float
func ParseFloat(s string, base int, prec uint, mode RoundingMode) (f *Float, b int, err error)
func (z *Float) Abs(x *Float) *Float
func (x *Float) Acc() Accuracy
func (z *Float) Add(x, y *Float) *Float
func (x *Float) Append(buf []byte, fmt byte, prec int) []byte
func (x *Float) Cmp(y *Float) int
func (z *Float) Copy(x *Float) *Float
func (x *Float) Float32() (float32, Accuracy)
func (x *Float) Float64() (float64, Accuracy)
func (x *Float) Format(s fmt.State, format rune)
func (x *Float) Int(z *Int) (*Int, Accuracy)
func (x *Float) Int64() (int64, Accuracy)
func (x *Float) IsInf() bool
func (x *Float) IsInt() bool
func (x *Float) MantExp(mant *Float) (exp int)
func (x *Float) MinPrec() uint
func (x *Float) Mode() RoundingMode
func (z *Float) Mul(x, y *Float) *Float
func (z *Float) Neg(x *Float) *Float
func (z *Float) Parse(s string, base int) (f *Float, b int, err error)
func (x *Float) Prec() uint
func (z *Float) Quo(x, y *Float) *Float
func (x *Float) Rat(z *Rat) (*Rat, Accuracy)
func (z *Float) Set(x *Float) *Float
func (z *Float) SetFloat64(x float64) *Float
func (z *Float) SetInf(signbit bool) *Float
func (z *Float) SetInt(x *Int) *Float
func (z *Float) SetInt64(x int64) *Float
func (z *Float) SetMantExp(mant *Float, exp int) *Float
func (z *Float) SetMode(mode RoundingMode) *Float
func (z *Float) SetPrec(prec uint) *Float
func (z *Float) SetRat(x *Rat) *Float
func (z *Float) SetString(s string) (*Float, bool)
func (z *Float) SetUint64(x uint64) *Float
func (x *Float) Sign() int
func (x *Float) Signbit() bool
func (x *Float) String() string
func (z *Float) Sub(x, y *Float) *Float
func (x *Float) Text(format byte, prec int) string
func (x *Float) Uint64() (uint64, Accuracy)
type Int
func NewInt(x int64) *Int
func (z *Int) Abs(x *Int) *Int
func (z *Int) Add(x, y *Int) *Int
func (z *Int) And(x, y *Int) *Int
func (z *Int) AndNot(x, y *Int) *Int
func (z *Int) Binomial(n, k int64) *Int
func (x *Int) Bit(i int) uint
func (x *Int) BitLen() int
func (x *Int) Bits() []Word
func (x *Int) Bytes() []byte
func (x *Int) Cmp(y *Int) (r int)
func (z *Int) Div(x, y *Int) *Int
func (z *Int) DivMod(x, y, m *Int) (*Int, *Int)
func (z *Int) Exp(x, y, m *Int) *Int
func (x *Int) Format(s fmt.State, ch rune)
func (z *Int) GCD(x, y, a, b *Int) *Int
func (z *Int) GobDecode(buf []byte) error
func (x *Int) GobEncode() ([]byte, error)
func (x *Int) Int64() int64
func (z *Int) Lsh(x *Int, n uint) *Int
func (z *Int) MarshalJSON() ([]byte, error)
func (z *Int) MarshalText() (text []byte, err error)
func (z *Int) Mod(x, y *Int) *Int
func (z *Int) ModInverse(g, n *Int) *Int
func (z *Int) ModSqrt(x, p *Int) *Int
func (z *Int) Mul(x, y *Int) *Int
func (z *Int) MulRange(a, b int64) *Int
func (z *Int) Neg(x *Int) *Int
func (z *Int) Not(x *Int) *Int
func (z *Int) Or(x, y *Int) *Int
func (x *Int) ProbablyPrime(n int) bool
func (z *Int) Quo(x, y *Int) *Int
func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int)
func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int
func (z *Int) Rem(x, y *Int) *Int
func (z *Int) Rsh(x *Int, n uint) *Int
func (z *Int) Scan(s fmt.ScanState, ch rune) error
func (z *Int) Set(x *Int) *Int
func (z *Int) SetBit(x *Int, i int, b uint) *Int
func (z *Int) SetBits(abs []Word) *Int
func (z *Int) SetBytes(buf []byte) *Int
func (z *Int) SetInt64(x int64) *Int
func (z *Int) SetString(s string, base int) (*Int, bool)
func (z *Int) SetUint64(x uint64) *Int
func (x *Int) Sign() int
func (x *Int) String() string
func (z *Int) Sub(x, y *Int) *Int
func (x *Int) Uint64() uint64
func (z *Int) UnmarshalJSON(text []byte) error
func (z *Int) UnmarshalText(text []byte) error
func (z *Int) Xor(x, y *Int) *Int
type Rat
func NewRat(a, b int64) *Rat
func (z *Rat) Abs(x *Rat) *Rat
func (z *Rat) Add(x, y *Rat) *Rat
func (x *Rat) Cmp(y *Rat) int
func (x *Rat) Denom() *Int
func (x *Rat) Float32() (f float32, exact bool)
func (x *Rat) Float64() (f float64, exact bool)
func (x *Rat) FloatString(prec int) string
func (z *Rat) GobDecode(buf []byte) error
func (x *Rat) GobEncode() ([]byte, error)
func (z *Rat) Inv(x *Rat) *Rat
func (x *Rat) IsInt() bool
func (r *Rat) MarshalText() (text []byte, err error)
func (z *Rat) Mul(x, y *Rat) *Rat
func (z *Rat) Neg(x *Rat) *Rat
func (x *Rat) Num() *Int
func (z *Rat) Quo(x, y *Rat) *Rat
func (x *Rat) RatString() string
func (z *Rat) Scan(s fmt.ScanState, ch rune) error
func (z *Rat) Set(x *Rat) *Rat
func (z *Rat) SetFloat64(f float64) *Rat
func (z *Rat) SetFrac(a, b *Int) *Rat
func (z *Rat) SetFrac64(a, b int64) *Rat
func (z *Rat) SetInt(x *Int) *Rat
func (z *Rat) SetInt64(x int64) *Rat
func (z *Rat) SetString(s string) (*Rat, bool)
func (x *Rat) Sign() int
func (x *Rat) String() string
func (z *Rat) Sub(x, y *Rat) *Rat
func (r *Rat) UnmarshalText(text []byte) error
type RoundingMode
func (i RoundingMode) String() string
type Word
Bugs

## Constants

```const (
MaxExp  = math.MaxInt32  // largest supported exponent
MinExp  = math.MinInt32  // smallest supported exponent
MaxPrec = math.MaxUint32 // largest (theoretically) supported precision; likely memory-limited
)```

Exponent and precision limits.

`const MaxBase = 'z' - 'a' + 10 + 1`

MaxBase is the largest number base accepted for string conversions.

## func Jacobi

`func Jacobi(x, y *Int) int`

Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0. The y argument must be an odd integer.

## type Accuracy

`type Accuracy int8`

Accuracy describes the rounding error produced by the most recent operation that generated a Float value, relative to the exact value.

```const (
Below Accuracy = -1
Exact Accuracy = 0
Above Accuracy = +1
)```

Constants describing the Accuracy of a Float.

### func (Accuracy) String

`func (i Accuracy) String() string`

## type ErrNaN

```type ErrNaN struct {
// contains filtered or unexported fields
}```

An ErrNaN panic is raised by a Float operation that would lead to a NaN under IEEE-754 rules. An ErrNaN implements the error interface.

### func (ErrNaN) Error

`func (err ErrNaN) Error() string`

## type Float

```type Float struct {
// contains filtered or unexported fields
}```

A nonzero finite Float represents a multi-precision floating point number

```sign × mantissa × 2**exponent
```

with 0.5 <= mantissa < 1.0, and MinExp <= exponent <= MaxExp. A Float may also be zero (+0, -0) or infinite (+Inf, -Inf). All Floats are ordered, and the ordering of two Floats x and y is defined by x.Cmp(y).

Each Float value also has a precision, rounding mode, and accuracy. The precision is the maximum number of mantissa bits available to represent the value. The rounding mode specifies how a result should be rounded to fit into the mantissa bits, and accuracy describes the rounding error with respect to the exact result.

Unless specified otherwise, all operations (including setters) that specify a *Float variable for the result (usually via the receiver with the exception of MantExp), round the numeric result according to the precision and rounding mode of the result variable.

If the provided result precision is 0 (see below), it is set to the precision of the argument with the largest precision value before any rounding takes place, and the rounding mode remains unchanged. Thus, uninitialized Floats provided as result arguments will have their precision set to a reasonable value determined by the operands and their mode is the zero value for RoundingMode (ToNearestEven).

By setting the desired precision to 24 or 53 and using matching rounding mode (typically ToNearestEven), Float operations produce the same results as the corresponding float32 or float64 IEEE-754 arithmetic for operands that correspond to normal (i.e., not denormal) float32 or float64 numbers. Exponent underflow and overflow lead to a 0 or an Infinity for different values than IEEE-754 because Float exponents have a much larger range.

The zero (uninitialized) value for a Float is ready to use and represents the number +0.0 exactly, with precision 0 and rounding mode ToNearestEven.

### func NewFloat

`func NewFloat(x float64) *Float`

NewFloat allocates and returns a new Float set to x, with precision 53 and rounding mode ToNearestEven. NewFloat panics with ErrNaN if x is a NaN.

### func ParseFloat

`func ParseFloat(s string, base int, prec uint, mode RoundingMode) (f *Float, b int, err error)`

ParseFloat is like f.Parse(s, base) with f set to the given precision and rounding mode.

### func (*Float) Abs

`func (z *Float) Abs(x *Float) *Float`

Abs sets z to the (possibly rounded) value |x| (the absolute value of x) and returns z.

### func (*Float) Acc

`func (x *Float) Acc() Accuracy`

Acc returns the accuracy of x produced by the most recent operation.

### func (*Float) Add

`func (z *Float) Add(x, y *Float) *Float`

Add sets z to the rounded sum x+y and returns z. If z's precision is 0, it is changed to the larger of x's or y's precision before the operation. Rounding is performed according to z's precision and rounding mode; and z's accuracy reports the result error relative to the exact (not rounded) result. Add panics with ErrNaN if x and y are infinities with opposite signs. The value of z is undefined in that case.

BUG(gri) When rounding ToNegativeInf, the sign of Float values rounded to 0 is incorrect.

Example

```x = 1000 (0x.fap+10, prec = 64, acc = Exact)
y = 2.718281828 (0x.adf85458248cd8p+2, prec = 53, acc = Exact)
z = 1002.718282 (0x.faadf854p+10, prec = 32, acc = Below)
```

### func (*Float) Append

`func (x *Float) Append(buf []byte, fmt byte, prec int) []byte`

Append appends to buf the string form of the floating-point number x, as generated by x.Text, and returns the extended buffer.

### func (*Float) Cmp

`func (x *Float) Cmp(y *Float) int`

Cmp compares x and y and returns:

```-1 if x <  y
0 if x == y (incl. -0 == 0, -Inf == -Inf, and +Inf == +Inf)
+1 if x >  y
```

Example

```   x     y  cmp
---------------
-Inf  -Inf    0
-Inf  -1.2   -1
-Inf    -0   -1
-Inf     0   -1
-Inf   1.2   -1
-Inf  +Inf   -1

-1.2  -Inf    1
-1.2  -1.2    0
-1.2    -0   -1
-1.2     0   -1
-1.2   1.2   -1
-1.2  +Inf   -1

-0  -Inf    1
-0  -1.2    1
-0    -0    0
-0     0    0
-0   1.2   -1
-0  +Inf   -1

0  -Inf    1
0  -1.2    1
0    -0    0
0     0    0
0   1.2   -1
0  +Inf   -1

1.2  -Inf    1
1.2  -1.2    1
1.2    -0    1
1.2     0    1
1.2   1.2    0
1.2  +Inf   -1

+Inf  -Inf    1
+Inf  -1.2    1
+Inf    -0    1
+Inf     0    1
+Inf   1.2    1
+Inf  +Inf    0
```

### func (*Float) Copy

`func (z *Float) Copy(x *Float) *Float`

Copy sets z to x, with the same precision, rounding mode, and accuracy as x, and returns z. x is not changed even if z and x are the same.

### func (*Float) Float32

`func (x *Float) Float32() (float32, Accuracy)`

Float32 returns the float32 value nearest to x. If x is too small to be represented by a float32 (|x| < math.SmallestNonzeroFloat32), the result is (0, Below) or (-0, Above), respectively, depending on the sign of x. If x is too large to be represented by a float32 (|x| > math.MaxFloat32), the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.

### func (*Float) Float64

`func (x *Float) Float64() (float64, Accuracy)`

Float64 returns the float64 value nearest to x. If x is too small to be represented by a float64 (|x| < math.SmallestNonzeroFloat64), the result is (0, Below) or (-0, Above), respectively, depending on the sign of x. If x is too large to be represented by a float64 (|x| > math.MaxFloat64), the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.

### func (*Float) Format

`func (x *Float) Format(s fmt.State, format rune)`

Format implements fmt.Formatter. It accepts all the regular formats for floating-point numbers ('e', 'E', 'f', 'F', 'g', 'G') as well as 'b', 'p', and 'v'. See (*Float).Text for the interpretation of 'b' and 'p'. The 'v' format is handled like 'g'. Format also supports specification of the minimum precision in digits, the output field width, as well as the format verbs '+' and ' ' for sign control, '0' for space or zero padding, and '-' for left or right justification. See the fmt package for details.

BUG(gri) A missing precision for the 'g' format, or a negative

```(via '*') precision is not yet supported. Instead the
default precision (6) is used in that case (issue #10991).
```

### func (*Float) Int

`func (x *Float) Int(z *Int) (*Int, Accuracy)`

Int returns the result of truncating x towards zero; or nil if x is an infinity. The result is Exact if x.IsInt(); otherwise it is Below for x > 0, and Above for x < 0. If a non-nil *Int argument z is provided, Int stores the result in z instead of allocating a new Int.

### func (*Float) Int64

`func (x *Float) Int64() (int64, Accuracy)`

Int64 returns the integer resulting from truncating x towards zero. If math.MinInt64 <= x <= math.MaxInt64, the result is Exact if x is an integer, and Above (x < 0) or Below (x > 0) otherwise. The result is (math.MinInt64, Above) for x < math.MinInt64, and (math.MaxInt64, Below) for x > math.MaxInt64.

### func (*Float) IsInf

`func (x *Float) IsInf() bool`

IsInf reports whether x is +Inf or -Inf.

### func (*Float) IsInt

`func (x *Float) IsInt() bool`

IsInt reports whether x is an integer. ±Inf values are not integers.

### func (*Float) MantExp

`func (x *Float) MantExp(mant *Float) (exp int)`

MantExp breaks x into its mantissa and exponent components and returns the exponent. If a non-nil mant argument is provided its value is set to the mantissa of x, with the same precision and rounding mode as x. The components satisfy x == mant × 2**exp, with 0.5 <= |mant| < 1.0. Calling MantExp with a nil argument is an efficient way to get the exponent of the receiver.

Special cases are:

```(  ±0).MantExp(mant) = 0, with mant set to   ±0
(±Inf).MantExp(mant) = 0, with mant set to ±Inf
```

x and mant may be the same in which case x is set to its mantissa value.

### func (*Float) MinPrec

`func (x *Float) MinPrec() uint`

MinPrec returns the minimum precision required to represent x exactly (i.e., the smallest prec before x.SetPrec(prec) would start rounding x). The result is 0 for |x| == 0 and |x| == Inf.

### func (*Float) Mode

`func (x *Float) Mode() RoundingMode`

Mode returns the rounding mode of x.

### func (*Float) Mul

`func (z *Float) Mul(x, y *Float) *Float`

Mul sets z to the rounded product x*y and returns z. Precision, rounding, and accuracy reporting are as for Add. Mul panics with ErrNaN if one operand is zero and the other operand an infinity. The value of z is undefined in that case.

### func (*Float) Neg

`func (z *Float) Neg(x *Float) *Float`

Neg sets z to the (possibly rounded) value of x with its sign negated, and returns z.

### func (*Float) Parse

`func (z *Float) Parse(s string, base int) (f *Float, b int, err error)`

Parse parses s which must contain a text representation of a floating- point number with a mantissa in the given conversion base (the exponent is always a decimal number), or a string representing an infinite value.

It sets z to the (possibly rounded) value of the corresponding floating- point value, and returns z, the actual base b, and an error err, if any. If z's precision is 0, it is changed to 64 before rounding takes effect. The number must be of the form:

```	number   = [ sign ] [ prefix ] mantissa [ exponent ] | infinity .
sign     = "+" | "-" .
prefix   = "0" ( "x" | "X" | "b" | "B" ) .
mantissa = digits | digits "." [ digits ] | "." digits .
exponent = ( "E" | "e" | "p" ) [ sign ] digits .
digits   = digit { digit } .
digit    = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
infinity = [ sign ] ( "inf" | "Inf" ) .
```

The base argument must be 0, 2, 10, or 16. Providing an invalid base argument will lead to a run-time panic.

For base 0, the number prefix determines the actual base: A prefix of "0x" or "0X" selects base 16, and a "0b" or "0B" prefix selects base 2; otherwise, the actual base is 10 and no prefix is accepted. The octal prefix "0" is not supported (a leading "0" is simply considered a "0").

A "p" exponent indicates a binary (rather then decimal) exponent; for instance "0x1.fffffffffffffp1023" (using base 0) represents the maximum float64 value. For hexadecimal mantissae, the exponent must be binary, if present (an "e" or "E" exponent indicator cannot be distinguished from a mantissa digit).

The returned *Float f is nil and the value of z is valid but not defined if an error is reported.

### func (*Float) Prec

`func (x *Float) Prec() uint`

Prec returns the mantissa precision of x in bits. The result may be 0 for |x| == 0 and |x| == Inf.

### func (*Float) Quo

`func (z *Float) Quo(x, y *Float) *Float`

Quo sets z to the rounded quotient x/y and returns z. Precision, rounding, and accuracy reporting are as for Add. Quo panics with ErrNaN if both operands are zero or infinities. The value of z is undefined in that case.

### func (*Float) Rat

`func (x *Float) Rat(z *Rat) (*Rat, Accuracy)`

Rat returns the rational number corresponding to x; or nil if x is an infinity. The result is Exact is x is not an Inf. If a non-nil *Rat argument z is provided, Rat stores the result in z instead of allocating a new Rat.

### func (*Float) Set

`func (z *Float) Set(x *Float) *Float`

Set sets z to the (possibly rounded) value of x and returns z. If z's precision is 0, it is changed to the precision of x before setting z (and rounding will have no effect). Rounding is performed according to z's precision and rounding mode; and z's accuracy reports the result error relative to the exact (not rounded) result.

### func (*Float) SetFloat64

`func (z *Float) SetFloat64(x float64) *Float`

SetFloat64 sets z to the (possibly rounded) value of x and returns z. If z's precision is 0, it is changed to 53 (and rounding will have no effect). SetFloat64 panics with ErrNaN if x is a NaN.

### func (*Float) SetInf

`func (z *Float) SetInf(signbit bool) *Float`

SetInf sets z to the infinite Float -Inf if signbit is set, or +Inf if signbit is not set, and returns z. The precision of z is unchanged and the result is always Exact.

### func (*Float) SetInt

`func (z *Float) SetInt(x *Int) *Float`

SetInt sets z to the (possibly rounded) value of x and returns z. If z's precision is 0, it is changed to the larger of x.BitLen() or 64 (and rounding will have no effect).

### func (*Float) SetInt64

`func (z *Float) SetInt64(x int64) *Float`

SetInt64 sets z to the (possibly rounded) value of x and returns z. If z's precision is 0, it is changed to 64 (and rounding will have no effect).

### func (*Float) SetMantExp

`func (z *Float) SetMantExp(mant *Float, exp int) *Float`

SetMantExp sets z to mant × 2**exp and and returns z. The result z has the same precision and rounding mode as mant. SetMantExp is an inverse of MantExp but does not require 0.5 <= |mant| < 1.0. Specifically:

```mant := new(Float)
new(Float).SetMantExp(mant, x.SetMantExp(mant)).Cmp(x).Eql() is true
```

Special cases are:

```z.SetMantExp(  ±0, exp) =   ±0
z.SetMantExp(±Inf, exp) = ±Inf
```

z and mant may be the same in which case z's exponent is set to exp.

### func (*Float) SetMode

`func (z *Float) SetMode(mode RoundingMode) *Float`

SetMode sets z's rounding mode to mode and returns an exact z. z remains unchanged otherwise. z.SetMode(z.Mode()) is a cheap way to set z's accuracy to Exact.

### func (*Float) SetPrec

`func (z *Float) SetPrec(prec uint) *Float`

SetPrec sets z's precision to prec and returns the (possibly) rounded value of z. Rounding occurs according to z's rounding mode if the mantissa cannot be represented in prec bits without loss of precision. SetPrec(0) maps all finite values to ±0; infinite values remain unchanged. If prec > MaxPrec, it is set to MaxPrec.

### func (*Float) SetRat

`func (z *Float) SetRat(x *Rat) *Float`

SetRat sets z to the (possibly rounded) value of x and returns z. If z's precision is 0, it is changed to the largest of a.BitLen(), b.BitLen(), or 64; with x = a/b.

### func (*Float) SetString

`func (z *Float) SetString(s string) (*Float, bool)`

SetString sets z to the value of s and returns z and a boolean indicating success. s must be a floating-point number of the same format as accepted by Parse, with base argument 0.

### func (*Float) SetUint64

`func (z *Float) SetUint64(x uint64) *Float`

SetUint64 sets z to the (possibly rounded) value of x and returns z. If z's precision is 0, it is changed to 64 (and rounding will have no effect).

### func (*Float) Sign

`func (x *Float) Sign() int`

Sign returns:

```-1 if x <   0
0 if x is ±0
+1 if x >   0
```

### func (*Float) Signbit

`func (x *Float) Signbit() bool`

Signbit returns true if x is negative or negative zero.

### func (*Float) String

`func (x *Float) String() string`

String formats x like x.Text('g', 10).

### func (*Float) Sub

`func (z *Float) Sub(x, y *Float) *Float`

Sub sets z to the rounded difference x-y and returns z. Precision, rounding, and accuracy reporting are as for Add. Sub panics with ErrNaN if x and y are infinities with equal signs. The value of z is undefined in that case.

### func (*Float) Text

`func (x *Float) Text(format byte, prec int) string`

Text converts the floating-point number x to a string according to the given format and precision prec. The format is one of:

```'e'	-d.dddde±dd, decimal exponent, at least two (possibly 0) exponent digits
'E'	-d.ddddE±dd, decimal exponent, at least two (possibly 0) exponent digits
'f'	-ddddd.dddd, no exponent
'g'	like 'e' for large exponents, like 'f' otherwise
'G'	like 'E' for large exponents, like 'f' otherwise
'b'	-ddddddp±dd, binary exponent
'p'	-0x.dddp±dd, binary exponent, hexadecimal mantissa
```

For the binary exponent formats, the mantissa is printed in normalized form:

```'b'	decimal integer mantissa using x.Prec() bits, or -0
'p'	hexadecimal fraction with 0.5 <= 0.mantissa < 1.0, or -0
```

If format is a different character, Text returns a "%" followed by the unrecognized format character.

The precision prec controls the number of digits (excluding the exponent) printed by the 'e', 'E', 'f', 'g', and 'G' formats. For 'e', 'E', and 'f' it is the number of digits after the decimal point. For 'g' and 'G' it is the total number of digits. A negative precision selects the smallest number of digits necessary to identify the value x uniquely. The prec value is ignored for the 'b' or 'p' format.

BUG(gri) Float.Text does not accept negative precisions (issue #10991).

### func (*Float) Uint64

`func (x *Float) Uint64() (uint64, Accuracy)`

Uint64 returns the unsigned integer resulting from truncating x towards zero. If 0 <= x <= math.MaxUint64, the result is Exact if x is an integer and Below otherwise. The result is (0, Above) for x < 0, and (math.MaxUint64, Below) for x > math.MaxUint64.

## type Int

```type Int struct {
// contains filtered or unexported fields
}```

An Int represents a signed multi-precision integer. The zero value for an Int represents the value 0.

### func NewInt

`func NewInt(x int64) *Int`

NewInt allocates and returns a new Int set to x.

### func (*Int) Abs

`func (z *Int) Abs(x *Int) *Int`

Abs sets z to |x| (the absolute value of x) and returns z.

### func (*Int) Add

`func (z *Int) Add(x, y *Int) *Int`

Add sets z to the sum x+y and returns z.

### func (*Int) And

`func (z *Int) And(x, y *Int) *Int`

And sets z = x & y and returns z.

### func (*Int) AndNot

`func (z *Int) AndNot(x, y *Int) *Int`

AndNot sets z = x &^ y and returns z.

### func (*Int) Binomial

`func (z *Int) Binomial(n, k int64) *Int`

Binomial sets z to the binomial coefficient of (n, k) and returns z.

### func (*Int) Bit

`func (x *Int) Bit(i int) uint`

Bit returns the value of the i'th bit of x. That is, it returns (x>>i)&1. The bit index i must be >= 0.

### func (*Int) BitLen

`func (x *Int) BitLen() int`

BitLen returns the length of the absolute value of x in bits. The bit length of 0 is 0.

### func (*Int) Bits

`func (x *Int) Bits() []Word`

Bits provides raw (unchecked but fast) access to x by returning its absolute value as a little-endian Word slice. The result and x share the same underlying array. Bits is intended to support implementation of missing low-level Int functionality outside this package; it should be avoided otherwise.

### func (*Int) Bytes

`func (x *Int) Bytes() []byte`

Bytes returns the absolute value of x as a big-endian byte slice.

### func (*Int) Cmp

`func (x *Int) Cmp(y *Int) (r int)`

Cmp compares x and y and returns:

```-1 if x <  y
0 if x == y
+1 if x >  y
```

### func (*Int) Div

`func (z *Int) Div(x, y *Int) *Int`

Div sets z to the quotient x/y for y != 0 and returns z. If y == 0, a division-by-zero run-time panic occurs. Div implements Euclidean division (unlike Go); see DivMod for more details.

### func (*Int) DivMod

`func (z *Int) DivMod(x, y, m *Int) (*Int, *Int)`

DivMod sets z to the quotient x div y and m to the modulus x mod y and returns the pair (z, m) for y != 0. If y == 0, a division-by-zero run-time panic occurs.

DivMod implements Euclidean division and modulus (unlike Go):

```q = x div y  such that
m = x - y*q  with 0 <= m < |q|
```

(See Raymond T. Boute, “The Euclidean definition of the functions div and mod”. ACM Transactions on Programming Languages and Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992. ACM press.) See QuoRem for T-division and modulus (like Go).

### func (*Int) Exp

`func (z *Int) Exp(x, y, m *Int) *Int`

Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z. If y <= 0, the result is 1 mod |m|; if m == nil or m == 0, z = x**y. See Knuth, volume 2, section 4.6.3.

### func (*Int) Format

`func (x *Int) Format(s fmt.State, ch rune)`

Format is a support routine for fmt.Formatter. It accepts the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal). Also supported are the full suite of package fmt's format verbs for integral types, including '+', '-', and ' ' for sign control, '#' for leading zero in octal and for hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X" respectively, specification of minimum digits precision, output field width, space or zero padding, and left or right justification.

### func (*Int) GCD

`func (z *Int) GCD(x, y, a, b *Int) *Int`

GCD sets z to the greatest common divisor of a and b, which both must be > 0, and returns z. If x and y are not nil, GCD sets x and y such that z = a*x + b*y. If either a or b is <= 0, GCD sets z = x = y = 0.

### func (*Int) GobDecode

`func (z *Int) GobDecode(buf []byte) error`

GobDecode implements the gob.GobDecoder interface.

### func (*Int) GobEncode

`func (x *Int) GobEncode() ([]byte, error)`

GobEncode implements the gob.GobEncoder interface.

### func (*Int) Int64

`func (x *Int) Int64() int64`

Int64 returns the int64 representation of x. If x cannot be represented in an int64, the result is undefined.

### func (*Int) Lsh

`func (z *Int) Lsh(x *Int, n uint) *Int`

Lsh sets z = x << n and returns z.

### func (*Int) MarshalJSON

`func (z *Int) MarshalJSON() ([]byte, error)`

MarshalJSON implements the json.Marshaler interface.

### func (*Int) MarshalText

`func (z *Int) MarshalText() (text []byte, err error)`

MarshalText implements the encoding.TextMarshaler interface.

### func (*Int) Mod

`func (z *Int) Mod(x, y *Int) *Int`

Mod sets z to the modulus x%y for y != 0 and returns z. If y == 0, a division-by-zero run-time panic occurs. Mod implements Euclidean modulus (unlike Go); see DivMod for more details.

### func (*Int) ModInverse

`func (z *Int) ModInverse(g, n *Int) *Int`

ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ and returns z. If g and n are not relatively prime, the result is undefined.

### func (*Int) ModSqrt

`func (z *Int) ModSqrt(x, p *Int) *Int`

ModSqrt sets z to a square root of x mod p if such a square root exists, and returns z. The modulus p must be an odd prime. If x is not a square mod p, ModSqrt leaves z unchanged and returns nil. This function panics if p is not an odd integer.

### func (*Int) Mul

`func (z *Int) Mul(x, y *Int) *Int`

Mul sets z to the product x*y and returns z.

### func (*Int) MulRange

`func (z *Int) MulRange(a, b int64) *Int`

MulRange sets z to the product of all integers in the range [a, b] inclusively and returns z. If a > b (empty range), the result is 1.

### func (*Int) Neg

`func (z *Int) Neg(x *Int) *Int`

Neg sets z to -x and returns z.

### func (*Int) Not

`func (z *Int) Not(x *Int) *Int`

Not sets z = ^x and returns z.

### func (*Int) Or

`func (z *Int) Or(x, y *Int) *Int`

Or sets z = x | y and returns z.

### func (*Int) ProbablyPrime

`func (x *Int) ProbablyPrime(n int) bool`

ProbablyPrime performs n Miller-Rabin tests to check whether x is prime. If it returns true, x is prime with probability 1 - 1/4^n. If it returns false, x is not prime. n must be > 0.

### func (*Int) Quo

`func (z *Int) Quo(x, y *Int) *Int`

Quo sets z to the quotient x/y for y != 0 and returns z. If y == 0, a division-by-zero run-time panic occurs. Quo implements truncated division (like Go); see QuoRem for more details.

### func (*Int) QuoRem

`func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int)`

QuoRem sets z to the quotient x/y and r to the remainder x%y and returns the pair (z, r) for y != 0. If y == 0, a division-by-zero run-time panic occurs.

QuoRem implements T-division and modulus (like Go):

```q = x/y      with the result truncated to zero
r = x - y*q
```

(See Daan Leijen, “Division and Modulus for Computer Scientists”.) See DivMod for Euclidean division and modulus (unlike Go).

### func (*Int) Rand

`func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int`

Rand sets z to a pseudo-random number in [0, n) and returns z.

### func (*Int) Rem

`func (z *Int) Rem(x, y *Int) *Int`

Rem sets z to the remainder x%y for y != 0 and returns z. If y == 0, a division-by-zero run-time panic occurs. Rem implements truncated modulus (like Go); see QuoRem for more details.

### func (*Int) Rsh

`func (z *Int) Rsh(x *Int, n uint) *Int`

Rsh sets z = x >> n and returns z.

### func (*Int) Scan

`func (z *Int) Scan(s fmt.ScanState, ch rune) error`

Scan is a support routine for fmt.Scanner; it sets z to the value of the scanned number. It accepts the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).

Example

```18446744073709551617
```

### func (*Int) Set

`func (z *Int) Set(x *Int) *Int`

Set sets z to x and returns z.

### func (*Int) SetBit

`func (z *Int) SetBit(x *Int, i int, b uint) *Int`

SetBit sets z to x, with x's i'th bit set to b (0 or 1). That is, if b is 1 SetBit sets z = x | (1 << i); if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1, SetBit will panic.

### func (*Int) SetBits

`func (z *Int) SetBits(abs []Word) *Int`

SetBits provides raw (unchecked but fast) access to z by setting its value to abs, interpreted as a little-endian Word slice, and returning z. The result and abs share the same underlying array. SetBits is intended to support implementation of missing low-level Int functionality outside this package; it should be avoided otherwise.

### func (*Int) SetBytes

`func (z *Int) SetBytes(buf []byte) *Int`

SetBytes interprets buf as the bytes of a big-endian unsigned integer, sets z to that value, and returns z.

### func (*Int) SetInt64

`func (z *Int) SetInt64(x int64) *Int`

SetInt64 sets z to x and returns z.

### func (*Int) SetString

`func (z *Int) SetString(s string, base int) (*Int, bool)`

SetString sets z to the value of s, interpreted in the given base, and returns z and a boolean indicating success. If SetString fails, the value of z is undefined but the returned value is nil.

The base argument must be 0 or a value between 2 and MaxBase. If the base is 0, the string prefix determines the actual conversion base. A prefix of “0x” or “0X” selects base 16; the “0” prefix selects base 8, and a “0b” or “0B” prefix selects base 2. Otherwise the selected base is 10.

Example

```420
```

### func (*Int) SetUint64

`func (z *Int) SetUint64(x uint64) *Int`

SetUint64 sets z to x and returns z.

### func (*Int) Sign

`func (x *Int) Sign() int`

Sign returns:

```-1 if x <  0
0 if x == 0
+1 if x >  0
```

### func (*Int) String

`func (x *Int) String() string`

### func (*Int) Sub

`func (z *Int) Sub(x, y *Int) *Int`

Sub sets z to the difference x-y and returns z.

### func (*Int) Uint64

`func (x *Int) Uint64() uint64`

Uint64 returns the uint64 representation of x. If x cannot be represented in a uint64, the result is undefined.

### func (*Int) UnmarshalJSON

`func (z *Int) UnmarshalJSON(text []byte) error`

UnmarshalJSON implements the json.Unmarshaler interface.

### func (*Int) UnmarshalText

`func (z *Int) UnmarshalText(text []byte) error`

UnmarshalText implements the encoding.TextUnmarshaler interface.

### func (*Int) Xor

`func (z *Int) Xor(x, y *Int) *Int`

Xor sets z = x ^ y and returns z.

## type Rat

```type Rat struct {
// contains filtered or unexported fields
}```

A Rat represents a quotient a/b of arbitrary precision. The zero value for a Rat represents the value 0.

### func NewRat

`func NewRat(a, b int64) *Rat`

NewRat creates a new Rat with numerator a and denominator b.

### func (*Rat) Abs

`func (z *Rat) Abs(x *Rat) *Rat`

Abs sets z to |x| (the absolute value of x) and returns z.

### func (*Rat) Add

`func (z *Rat) Add(x, y *Rat) *Rat`

Add sets z to the sum x+y and returns z.

### func (*Rat) Cmp

`func (x *Rat) Cmp(y *Rat) int`

Cmp compares x and y and returns:

```-1 if x <  y
0 if x == y
+1 if x >  y
```

### func (*Rat) Denom

`func (x *Rat) Denom() *Int`

Denom returns the denominator of x; it is always > 0. The result is a reference to x's denominator; it may change if a new value is assigned to x, and vice versa.

### func (*Rat) Float32

`func (x *Rat) Float32() (f float32, exact bool)`

Float32 returns the nearest float32 value for x and a bool indicating whether f represents x exactly. If the magnitude of x is too large to be represented by a float32, f is an infinity and exact is false. The sign of f always matches the sign of x, even if f == 0.

### func (*Rat) Float64

`func (x *Rat) Float64() (f float64, exact bool)`

Float64 returns the nearest float64 value for x and a bool indicating whether f represents x exactly. If the magnitude of x is too large to be represented by a float64, f is an infinity and exact is false. The sign of f always matches the sign of x, even if f == 0.

### func (*Rat) FloatString

`func (x *Rat) FloatString(prec int) string`

FloatString returns a string representation of x in decimal form with prec digits of precision after the decimal point. The last digit is rounded to nearest, with halves rounded away from zero.

### func (*Rat) GobDecode

`func (z *Rat) GobDecode(buf []byte) error`

GobDecode implements the gob.GobDecoder interface.

### func (*Rat) GobEncode

`func (x *Rat) GobEncode() ([]byte, error)`

GobEncode implements the gob.GobEncoder interface.

### func (*Rat) Inv

`func (z *Rat) Inv(x *Rat) *Rat`

Inv sets z to 1/x and returns z.

### func (*Rat) IsInt

`func (x *Rat) IsInt() bool`

IsInt reports whether the denominator of x is 1.

### func (*Rat) MarshalText

`func (r *Rat) MarshalText() (text []byte, err error)`

MarshalText implements the encoding.TextMarshaler interface.

### func (*Rat) Mul

`func (z *Rat) Mul(x, y *Rat) *Rat`

Mul sets z to the product x*y and returns z.

### func (*Rat) Neg

`func (z *Rat) Neg(x *Rat) *Rat`

Neg sets z to -x and returns z.

### func (*Rat) Num

`func (x *Rat) Num() *Int`

Num returns the numerator of x; it may be <= 0. The result is a reference to x's numerator; it may change if a new value is assigned to x, and vice versa. The sign of the numerator corresponds to the sign of x.

### func (*Rat) Quo

`func (z *Rat) Quo(x, y *Rat) *Rat`

Quo sets z to the quotient x/y and returns z. If y == 0, a division-by-zero run-time panic occurs.

### func (*Rat) RatString

`func (x *Rat) RatString() string`

RatString returns a string representation of x in the form "a/b" if b != 1, and in the form "a" if b == 1.

### func (*Rat) Scan

`func (z *Rat) Scan(s fmt.ScanState, ch rune) error`

Scan is a support routine for fmt.Scanner. It accepts the formats 'e', 'E', 'f', 'F', 'g', 'G', and 'v'. All formats are equivalent.

Example

```3/2
```

### func (*Rat) Set

`func (z *Rat) Set(x *Rat) *Rat`

Set sets z to x (by making a copy of x) and returns z.

### func (*Rat) SetFloat64

`func (z *Rat) SetFloat64(f float64) *Rat`

SetFloat64 sets z to exactly f and returns z. If f is not finite, SetFloat returns nil.

### func (*Rat) SetFrac

`func (z *Rat) SetFrac(a, b *Int) *Rat`

SetFrac sets z to a/b and returns z.

### func (*Rat) SetFrac64

`func (z *Rat) SetFrac64(a, b int64) *Rat`

SetFrac64 sets z to a/b and returns z.

### func (*Rat) SetInt

`func (z *Rat) SetInt(x *Int) *Rat`

SetInt sets z to x (by making a copy of x) and returns z.

### func (*Rat) SetInt64

`func (z *Rat) SetInt64(x int64) *Rat`

SetInt64 sets z to x and returns z.

### func (*Rat) SetString

`func (z *Rat) SetString(s string) (*Rat, bool)`

SetString sets z to the value of s and returns z and a boolean indicating success. s can be given as a fraction "a/b" or as a floating-point number optionally followed by an exponent. If the operation failed, the value of z is undefined but the returned value is nil.

Example

```3.142
```

### func (*Rat) Sign

`func (x *Rat) Sign() int`

Sign returns:

```-1 if x <  0
0 if x == 0
+1 if x >  0
```

### func (*Rat) String

`func (x *Rat) String() string`

String returns a string representation of x in the form "a/b" (even if b == 1).

### func (*Rat) Sub

`func (z *Rat) Sub(x, y *Rat) *Rat`

Sub sets z to the difference x-y and returns z.

### func (*Rat) UnmarshalText

`func (r *Rat) UnmarshalText(text []byte) error`

UnmarshalText implements the encoding.TextUnmarshaler interface.

## type RoundingMode

`type RoundingMode byte`

RoundingMode determines how a Float value is rounded to the desired precision. Rounding may change the Float value; the rounding error is described by the Float's Accuracy.

```const (
ToNearestEven RoundingMode = iota // == IEEE 754-2008 roundTiesToEven
ToNearestAway                     // == IEEE 754-2008 roundTiesToAway
ToZero                            // == IEEE 754-2008 roundTowardZero
AwayFromZero                      // no IEEE 754-2008 equivalent
ToNegativeInf                     // == IEEE 754-2008 roundTowardNegative
ToPositiveInf                     // == IEEE 754-2008 roundTowardPositive
)```

The following rounding modes are supported.

### func (RoundingMode) String

`func (i RoundingMode) String() string`

## type Word

`type Word uintptr`

A Word represents a single digit of a multi-precision unsigned integer.

## Bugs

• When rounding ToNegativeInf, the sign of Float values rounded to 0 is incorrect.
• Float.Text does not accept negative precisions (issue #10991).
• A missing precision for the 'g' format, or a negative (via '*') precision is not yet supported. Instead the default precision (6) is used in that case (issue #10991).