big

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 Go to latest Published: Jan 29, 2017 License: BSD-3-Clause Imports: 10 Imported by: 0

Documentation

Overview

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

Int    signed integers
Rat    rational numbers
Float  floating-point numbers

The zero value for an Int, Rat, or Float correspond to 0. Thus, new values can be declared in the usual ways and denote 0 without further initialization: 

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

Alternatively, new values can be allocated and initialized with factory functions of the form: 

func NewT(v V) *T

For instance, NewInt(x) returns an *Int set to the value of the int64 argument x, NewRat(a, b) returns a *Rat set to the fraction a/b where a and b are int64 values, and NewFloat(f) returns a *Float initialized to the float64 argument f. More flexibility is provided with explicit setters, for instance: 

var z1 Int
z1.SetUint64(123)                 // z1 := 123
z2 := new(Rat).SetFloat64(1.2)    // z2 := 6/5
z3 := new(Float).SetInt(z1)       // z3 := 123.0

Setters, numeric operations and predicates are represented as methods of the form: 

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)

with T one of Int, Rat, or Float. For unary and binary operations, the result is the receiver (usually named z in that case; see below); if it is one of the operands x or y it may be safely overwritten (and its memory reused).

Arithmetic expressions are typically written as a sequence of individual method calls, with each call corresponding to an operation. The receiver denotes the result and the method arguments are the operation's operands. For instance, given three *Int values a, b and c, the invocation 

c.Add(a, b)

computes the sum a + b and stores the result in c, overwriting whatever value was held in c before. Unless specified otherwise, operations permit aliasing of parameters, so it is perfectly ok to write 

sum.Add(sum, x)

to accumulate values x in a sum.

(By always passing in a result value via the receiver, memory use can be much better controlled. Instead of having to allocate new memory for each result, an operation can reuse the space allocated for the result value, and overwrite that value with the new result in the process.)

Notational convention: Incoming method parameters (including the receiver) are named consistently in the API to clarify their use. Incoming operands are usually named x, y, a, b, and so on, but never z. A parameter specifying the result is named z (typically the receiver).

For instance, the arguments for (*Int).Add are named x and y, and because the receiver specifies the result destination, it is called z: 

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

Methods of this form typically return the incoming receiver as well, to enable simple call chaining.

Methods which don't require a result value to be passed in (for instance, Int.Sign), simply return the result. In this case, the receiver is typically the first operand, named x: 

func (x *Int) Sign() int

Various methods support conversions between strings and corresponding numeric values, and vice versa: *Int, *Rat, and *Float values implement the Stringer interface for a (default) string representation of the value, but also provide SetString methods to initialize a value from a string in a variety of supported formats (see the respective SetString documentation).

Finally, *Int, *Rat, and *Float satisfy the fmt package's Scanner interface for scanning and (except for *Rat) the Formatter interface for formatted printing.

Example (EConvergents)

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). 

package main

import (
	"cmd/compile/internal/big"
	"fmt"
)

// 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 main() {
	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
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. 

package main

import (
	"cmd/compile/internal/big"
	"fmt"
)

func main() {
	// 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))

}

Output:

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. 

package main

import (
	"cmd/compile/internal/big"
	"fmt"
	"math"
)

func main() {
	// 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))

}

Output:

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

Index

Examples

Constants

View Source 
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.

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

MaxBase is the largest number base accepted for string conversions.

Variables

This section is empty.

Functions

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.

Types

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.

Example (Shift) 
package main

import (
	"cmd/compile/internal/big"
	"fmt"
)

func main() {
	// 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)
	}
}

Output:

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.

Example 
package main

import (
	"cmd/compile/internal/big"
	"fmt"
)

func main() {
	// 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())
}

Output:

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 
package main

import (
	"cmd/compile/internal/big"
	"fmt"
	"math"
)

func main() {
	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()
	}

}

Output:

   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 marshalled.

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. 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) 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.

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 

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) 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. See Knuth, volume 2, section 4.6.3.

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 performs n Miller-Rabin tests to check whether x is prime. If x is prime, it returns true. If x is not prime, it returns false with probability at least 1 - ¼ⁿ.

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

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 
package main

import (
	"cmd/compile/internal/big"
	"fmt"
	"log"
)

func main() {
	// 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)
	}
}

Output:

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 
package main

import (
	"cmd/compile/internal/big"
	"fmt"
)

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

Output:

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) 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 

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 (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.

Example 
package main

import (
	"cmd/compile/internal/big"
	"fmt"
	"log"
)

func main() {
	// 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)
	}
}

Output:

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 
package main

import (
	"cmd/compile/internal/big"
	"fmt"
)

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

Output:

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 

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.

Example 
package main

import (
	"cmd/compile/internal/big"
	"fmt"
)

func main() {
	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()
	}

}

Output:

   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

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
)

These constants define supported rounding modes.

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.

Notes

Bugs

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

Source Files

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