Package big

Int    signed integers
Rat    rational numbers
Float  floating-point numbers

var x Int        // &x is an *Int of value 0
var r = &Rat{}   // r is a *Rat of value 0
y := new(Float)  // y is a *Float of value 0

func NewT(v V) *T

var z1 Int
z1.SetUint64(123)                 // z1 := 123
z2 := new(Rat).SetFloat64(1.25)   // z2 := 5/4
z3 := new(Float).SetInt(z1)       // z3 := 123.0

func (z *T) SetV(v V) *T          // z = v
func (z *T) Unary(x *T) *T        // z = unary x
func (z *T) Binary(x, y *T) *T    // z = x binary y
func (x *T) Pred() P              // p = pred(x)

c.Add(a, b)

sum.Add(sum, x)

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

func (x *Int) Sign() int

package big_test

import (
    "fmt"
    "math/big"
)

// Use the classic continued fraction for e
//     e = [1; 0, 1, 1, 2, 1, 1, ... 2n, 1, 1, ...]
// i.e., for the nth term, use
//     1          if   n mod 3 != 1
//  (n-1)/3 * 2   if   n mod 3 == 1
func recur(n, lim int64) *big.Rat {
    term := new(big.Rat)
    if n%3 != 1 {
        term.SetInt64(1)
    } else {
        term.SetInt64((n - 1) / 3 * 2)
    }

    if n > lim {
        return term
    }

    // Directly initialize frac as the fractional
    // inverse of the result of recur.
    frac := new(big.Rat).Inv(recur(n+1, lim))

    return term.Add(term, frac)
}

// This example demonstrates how to use big.Rat to compute the
// first 15 terms in the sequence of rational convergents for
// the constant e (base of natural logarithm).
func Example_eConvergents() {
    for i := 1; i <= 15; i++ {
        r := recur(0, int64(i))

        // Print r both as a fraction and as a floating-point number.
        // Since big.Rat implements fmt.Formatter, we can use %-13s to
        // get a left-aligned string representation of the fraction.
        fmt.Printf("%-13s = %s\n", r, r.FloatString(8))
    }

    // Output:
    // 2/1           = 2.00000000
    // 3/1           = 3.00000000
    // 8/3           = 2.66666667
    // 11/4          = 2.75000000
    // 19/7          = 2.71428571
    // 87/32         = 2.71875000
    // 106/39        = 2.71794872
    // 193/71        = 2.71830986
    // 1264/465      = 2.71827957
    // 1457/536      = 2.71828358
    // 2721/1001     = 2.71828172
    // 23225/8544    = 2.71828184
    // 25946/9545    = 2.71828182
    // 49171/18089   = 2.71828183
    // 517656/190435 = 2.71828183
}

// Initialize two big ints with the first two numbers in the sequence.
a := big.NewInt(0)
b := big.NewInt(1)

// Initialize limit as 10^99, the smallest integer with 100 digits.
var limit big.Int
limit.Exp(big.NewInt(10), big.NewInt(99), nil)

// Loop while a is smaller than 1e100.
for a.Cmp(&limit) < 0 {
    // Compute the next Fibonacci number, storing it in a.
    a.Add(a, b)
    // Swap a and b so that b is the next number in the sequence.
    a, b = b, a
}
fmt.Println(a) // 100-digit Fibonacci number

// Test a for primality.
// (ProbablyPrimes' argument sets the number of Miller-Rabin
// rounds to be performed. 20 is a good value.)
fmt.Println(a.ProbablyPrime(20))

1344719667586153181419716641724567886890850696275767987106294472017884974410332069524504824747437757
false

// We'll do computations with 200 bits of precision in the mantissa.
const prec = 200

// Compute the square root of 2 using Newton's Method. We start with
// an initial estimate for sqrt(2), and then iterate:
//     x_{n+1} = 1/2 * ( x_n + (2.0 / x_n) )

// Since Newton's Method doubles the number of correct digits at each
// iteration, we need at least log_2(prec) steps.
steps := int(math.Log2(prec))

// Initialize values we need for the computation.
two := new(big.Float).SetPrec(prec).SetInt64(2)
half := new(big.Float).SetPrec(prec).SetFloat64(0.5)

// Use 1 as the initial estimate.
x := new(big.Float).SetPrec(prec).SetInt64(1)

// We use t as a temporary variable. There's no need to set its precision
// since big.Float values with unset (== 0) precision automatically assume
// the largest precision of the arguments when used as the result (receiver)
// of a big.Float operation.
t := new(big.Float)

// Iterate.
for i := 0; i <= steps; i++ {
    t.Quo(two, x)  // t = 2.0 / x_n
    t.Add(x, t)    // t = x_n + (2.0 / x_n)
    x.Mul(half, t) // x_{n+1} = 0.5 * t
}

// We can use the usual fmt.Printf verbs since big.Float implements fmt.Formatter
fmt.Printf("sqrt(2) = %.50f\n", x)

// Print the error between 2 and x*x.
t.Mul(x, x) // t = x*x
fmt.Printf("error = %e\n", t.Sub(two, t))

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

Constants

Exponent and precision limits.

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

MaxBase is the largest number base accepted for string conversions.

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

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 ¶

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

type Accuracy int8

Constants describing the Accuracy of a Float.

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

func (Accuracy) String ¶

func (i Accuracy) String() string

type ErrNaN ¶

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.

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

func (ErrNaN) Error ¶

func (err ErrNaN) Error() string

type Float ¶

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.

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

// Implement Float "shift" by modifying the (binary) exponents directly.
for s := -5; s <= 5; s++ {
    x := big.NewFloat(0.5)
    x.SetMantExp(x, x.MantExp(nil)+s) // shift x by s
    fmt.Println(x)
}

0.015625
0.03125
0.0625
0.125
0.25
0.5
1
2
4
8
16

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.

// Operate on numbers of different precision.
var x, y, z big.Float
x.SetInt64(1000)          // x is automatically set to 64bit precision
y.SetFloat64(2.718281828) // y is automatically set to 53bit precision
z.SetPrec(32)
z.Add(&x, &y)
fmt.Printf("x = %.10g (%s, prec = %d, acc = %s)\n", &x, x.Text('p', 0), x.Prec(), x.Acc())
fmt.Printf("y = %.10g (%s, prec = %d, acc = %s)\n", &y, y.Text('p', 0), y.Prec(), y.Acc())
fmt.Printf("z = %.10g (%s, prec = %d, acc = %s)\n", &z, z.Text('p', 0), z.Prec(), z.Acc())

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

inf := math.Inf(1)
zero := 0.0

operands := []float64{-inf, -1.2, -zero, 0, +1.2, +inf}

fmt.Println("   x     y  cmp")
fmt.Println("---------------")
for _, x64 := range operands {
    x := big.NewFloat(x64)
    for _, y64 := range operands {
        y := big.NewFloat(y64)
        fmt.Printf("%4g  %4g  %3d\n", x, y, x.Cmp(y))
    }
    fmt.Println()
}

   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 ('b', 'e', 'E', 'f', 'F', 'g', 'G') as well as 'p' and 'v'. See (*Float).Text for the interpretation of '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 flags '+' and ' ' for sign control, '0' for space or zero padding, and '-' for left or right justification. See the fmt package for details.

func (*Float) GobDecode ¶

func (z *Float) GobDecode(buf []byte) error

GobDecode implements the gob.GobDecoder interface. The result is rounded per the precision and rounding mode of z unless z's precision is 0, in which case z is set exactly to the decoded value.

func (*Float) GobEncode ¶

func (x *Float) GobEncode() ([]byte, error)

GobEncode implements the gob.GobEncoder interface. The Float value and all its attributes (precision, rounding mode, accuracy) are marshaled.

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) MarshalText ¶

func (x *Float) MarshalText() (text []byte, err error)

MarshalText implements the encoding.TextMarshaler interface. Only the Float value is marshaled (in full precision), other attributes such as precision or accuracy are ignored.

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. The entire string (not just a prefix) must be consumed for success. 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 if 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) Scan ¶

func (z *Float) 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 formats whose verbs are supported by fmt.Scan for floating point values, which are: 'b' (binary), 'e', 'E', 'f', 'F', 'g' and 'G'. Scan doesn't handle ±Inf.

// The Scan function is rarely used directly;
// the fmt package recognizes it as an implementation of fmt.Scanner.
f := new(big.Float)
_, err := fmt.Sscan("1.19282e99", f)
if err != nil {
    log.Println("error scanning value:", err)
} else {
    fmt.Println(f)
}

1.19282e+99

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.MantExp(mant)).Cmp(x) == 0

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. The entire string (not just a prefix) must be valid for success. If the operation failed, the value of z is undefined but the returned value is nil.

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). (String must be called explicitly, Float.Format does not support %s verb.)

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 decimal digits necessary to identify the value x uniquely using x.Prec() mantissa bits. The prec value is ignored for the 'b' or 'p' format.

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.

func (*Float) UnmarshalText ¶

func (z *Float) UnmarshalText(text []byte) error

UnmarshalText implements the encoding.TextUnmarshaler interface. The result is rounded per the precision and rounding mode of z. If z's precision is 0, it is changed to 64 before rounding takes effect.

type Int ¶

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

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

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) Append ¶

func (x *Int) Append(buf []byte, base int) []byte

Append appends the string representation of x, as generated by x.Text(base), to buf and returns the extended buffer.

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 < |y|

(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.

Modular exponentation of inputs of a particular size is not a cryptographically constant-time operation.

func (*Int) Format ¶

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

Format implements 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 flags 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 '-' for 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 (x *Int) MarshalJSON() ([]byte, error)

MarshalJSON implements the json.Marshaler interface.

func (*Int) MarshalText ¶

func (x *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 reports whether x is probably prime, applying the Miller-Rabin test with n pseudorandomly chosen bases as well as a Baillie-PSW test.

If x is prime, ProbablyPrime returns true. If x is chosen randomly and not prime, ProbablyPrime probably returns false. The probability of returning true for a randomly chosen non-prime is at most ¼ⁿ.

ProbablyPrime is 100% accurate for inputs less than 2⁶⁴. See Menezes et al., Handbook of Applied Cryptography, 1997, pp. 145-149, and FIPS 186-4 Appendix F for further discussion of the error probabilities.

ProbablyPrime is not suitable for judging primes that an adversary may have crafted to fool the test.

As of Go 1.8, ProbablyPrime(0) is allowed and applies only a Baillie-PSW test. Before Go 1.8, ProbablyPrime applied only the Miller-Rabin tests, and ProbablyPrime(0) panicked.

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).

// The Scan function is rarely used directly;
// the fmt package recognizes it as an implementation of fmt.Scanner.
i := new(big.Int)
_, err := fmt.Sscan("18446744073709551617", i)
if err != nil {
    log.Println("error scanning value:", err)
} else {
    fmt.Println(i)
}

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. The entire string (not just a prefix) must be valid for 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.

i := new(big.Int)
i.SetString("644", 8) // octal
fmt.Println(i)

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) Sqrt ¶

func (z *Int) Sqrt(x *Int) *Int

Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z. It panics if x is negative.

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) Text ¶

func (x *Int) Text(base int) string

Text returns the string representation of x in the given base. Base must be between 2 and 36, inclusive. The result uses the lower-case letters 'a' to 'z' for digit values >= 10. No base prefix (such as "0x") is added to the string.

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 ¶

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

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

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 (x *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.

// The Scan function is rarely used directly;
// the fmt package recognizes it as an implementation of fmt.Scanner.
r := new(big.Rat)
_, err := fmt.Sscan("1.5000", r)
if err != nil {
    log.Println("error scanning value:", err)
} else {
    fmt.Println(r)
}

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. The entire string (not just a prefix) must be valid for success. If the operation failed, the value of z is un- defined but the returned value is nil.

r := new(big.Rat)
r.SetString("355/113")
fmt.Println(r.FloatString(3))

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 (z *Rat) UnmarshalText(text []byte) error

UnmarshalText implements the encoding.TextUnmarshaler interface.

type RoundingMode ¶

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.

type RoundingMode byte

These constants define supported rounding modes.

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
)

operands := []float64{2.6, 2.5, 2.1, -2.1, -2.5, -2.6}

fmt.Print("   x")
for mode := big.ToNearestEven; mode <= big.ToPositiveInf; mode++ {
    fmt.Printf("  %s", mode)
}
fmt.Println()

for _, f64 := range operands {
    fmt.Printf("%4g", f64)
    for mode := big.ToNearestEven; mode <= big.ToPositiveInf; mode++ {
        // sample operands above require 2 bits to represent mantissa
        // set binary precision to 2 to round them to integer values
        f := new(big.Float).SetPrec(2).SetMode(mode).SetFloat64(f64)
        fmt.Printf("  %*g", len(mode.String()), f)
    }
    fmt.Println()
}

   x  ToNearestEven  ToNearestAway  ToZero  AwayFromZero  ToNegativeInf  ToPositiveInf
 2.6              3              3       2             3              2              3
 2.5              2              3       2             3              2              3
 2.1              2              2       2             3              2              3
-2.1             -2             -2      -2            -3             -3             -2
-2.5             -2             -3      -2            -3             -3             -2
-2.6             -3             -3      -2            -3             -3             -2

func (RoundingMode) String ¶

func (i RoundingMode) String() string

type Word ¶

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

type Word uintptr

Bugs

• ☞

  When rounding ToNegativeInf, the sign of Float values rounded to 0 is incorrect.