import "math/big"
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(1.25) // z2 := 5/4 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.
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).
Code:
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
}
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
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.
Code:
// 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
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.
Code:
// 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
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 = 10 + ('z' - 'a' + 1) + ('Z' - 'A' + 1)
MaxBase is the largest number base accepted for string conversions.
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 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 (i Accuracy) String() string
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 (err ErrNaN) Error() string
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.
Operations always take pointer arguments (*Float) rather than Float values, and each unique Float value requires its own unique *Float pointer. To "copy" a Float value, an existing (or newly allocated) Float must be set to a new value using the Float.Set method; shallow copies of Floats are not supported and may lead to errors.
Code:
// 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(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(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 (z *Float) Abs(x *Float) *Float
Abs sets z to the (possibly rounded) value |x| (the absolute value of x) and returns z.
func (x *Float) Acc() Accuracy
Acc returns the accuracy of x produced by the most recent operation.
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.
Code:
// 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 (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 (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
Code:
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 (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 (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 (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 (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', 'x') 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 (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 (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 (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 (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 (x *Float) IsInf() bool
IsInf reports whether x is +Inf or -Inf.
func (x *Float) IsInt() bool
IsInt reports whether x is an integer. ±Inf values are not integers.
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 (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 (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 (x *Float) Mode() RoundingMode
Mode returns the rounding mode of x.
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 (z *Float) Neg(x *Float) *Float
Neg sets z to the (possibly rounded) value of x with its sign negated, and returns z.
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.
For base 0, an underscore character ``_'' may appear between a base prefix and an adjacent digit, and between successive digits; such underscores do not change the value of the number, or the returned digit count. Incorrect placement of underscores is reported as an error if there are no other errors. If base != 0, underscores are not recognized and thus terminate scanning like any other character that is not a valid radix point or digit.
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 ] ( float | "inf" | "Inf" ) .
sign = "+" | "-" .
float = ( mantissa | prefix pmantissa ) [ exponent ] .
prefix = "0" [ "b" | "B" | "o" | "O" | "x" | "X" ] .
mantissa = digits "." [ digits ] | digits | "." digits .
pmantissa = [ "_" ] digits "." [ digits ] | [ "_" ] digits | "." digits .
exponent = ( "e" | "E" | "p" | "P" ) [ sign ] digits .
digits = digit { [ "_" ] digit } .
digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
The base argument must be 0, 2, 8, 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 ``0b'' or ``0B'' selects base 2, ``0o'' or ``0O'' selects base 8, and ``0x'' or ``0X'' selects base 16. 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" or "P" exponent indicates a base 2 (rather then base 10) exponent; for instance, "0x1.fffffffffffffp1023" (using base 0) represents the maximum float64 value. For hexadecimal mantissae, the exponent character must be one of 'p' or 'P', 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 (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 (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 (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 (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.
Code:
// 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) }
Output:
1.19282e+99
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 (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 (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 (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 (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 (z *Float) SetMantExp(mant *Float, exp int) *Float
SetMantExp sets z to mant × 2**exp 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 (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 (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 (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 (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 (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 (x *Float) Sign() int
Sign returns:
-1 if x < 0 0 if x is ±0 +1 if x > 0
func (x *Float) Signbit() bool
Signbit reports whether x is negative or negative zero.
func (z *Float) Sqrt(x *Float) *Float
Sqrt sets z to the rounded square root of x, and returns it.
If z's precision is 0, it is changed to x's precision before the operation. Rounding is performed according to z's precision and rounding mode.
The function panics if z < 0. The value of z is undefined in that case.
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 (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 (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 'x' -0xd.dddddp±dd, hexadecimal mantissa, decimal power of two exponent 'p' -0x.dddp±dd, hexadecimal mantissa, decimal power of two exponent (non-standard) 'b' -ddddddp±dd, decimal mantissa, decimal power of two exponent (non-standard)
For the power-of-two exponent formats, the mantissa is printed in normalized form:
'x' hexadecimal mantissa in [1, 2), or 0 'p' hexadecimal mantissa in [½, 1), or 0 'b' decimal integer mantissa using x.Prec() bits, or 0
Note that the 'x' form is the one used by most other languages and libraries.
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', 'G', and 'x' formats. For 'e', 'E', 'f', and 'x', 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' and 'p' formats.
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 (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 struct {
// contains filtered or unexported fields
}
An Int represents a signed multi-precision integer. The zero value for an Int represents the value 0.
Operations always take pointer arguments (*Int) rather than Int values, and each unique Int value requires its own unique *Int pointer. To "copy" an Int value, an existing (or newly allocated) Int must be set to a new value using the Int.Set method; shallow copies of Ints are not supported and may lead to errors.
func NewInt(x int64) *Int
NewInt allocates and returns a new Int set to x.
func (z *Int) Abs(x *Int) *Int
Abs sets z to |x| (the absolute value of x) and returns z.
func (z *Int) Add(x, y *Int) *Int
Add sets z to the sum x+y and returns z.
func (z *Int) And(x, y *Int) *Int
And sets z = x & y and returns z.
func (z *Int) AndNot(x, y *Int) *Int
AndNot sets z = x &^ y and returns z.
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 (z *Int) Binomial(n, k int64) *Int
Binomial sets z to the binomial coefficient of (n, k) and returns z.
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 (x *Int) BitLen() int
BitLen returns the length of the absolute value of x in bits. The bit length of 0 is 0.
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 (x *Int) Bytes() []byte
Bytes returns the absolute value of x as a big-endian byte slice.
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 (x *Int) CmpAbs(y *Int) int
CmpAbs compares the absolute values of x and y and returns:
-1 if |x| < |y| 0 if |x| == |y| +1 if |x| > |y|
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 (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 (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 m == nil or m == 0, z = x**y unless y <= 0 then z = 1. If m > 0, y < 0, and x and n are not relatively prime, z is unchanged and nil is returned.
Modular exponentation of inputs of a particular size is not a cryptographically constant-time operation.
func (x *Int) Format(s fmt.State, ch rune)
Format implements fmt.Formatter. It accepts the formats 'b' (binary), 'o' (octal with 0 prefix), 'O' (octal with 0o prefix), '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 (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 or y are not nil, GCD sets their value such that z = a*x + b*y. If either a or b is <= 0, GCD sets z = x = y = 0.
func (z *Int) GobDecode(buf []byte) error
GobDecode implements the gob.GobDecoder interface.
func (x *Int) GobEncode() ([]byte, error)
GobEncode implements the gob.GobEncoder interface.
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 (x *Int) IsInt64() bool
IsInt64 reports whether x can be represented as an int64.
func (x *Int) IsUint64() bool
IsUint64 reports whether x can be represented as a uint64.
func (z *Int) Lsh(x *Int, n uint) *Int
Lsh sets z = x << n and returns z.
func (x *Int) MarshalJSON() ([]byte, error)
MarshalJSON implements the json.Marshaler interface.
func (x *Int) MarshalText() (text []byte, err error)
MarshalText implements the encoding.TextMarshaler interface.
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 (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, g has no multiplicative inverse in the ring ℤ/nℤ. In this case, z is unchanged and the return value is nil.
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 (z *Int) Mul(x, y *Int) *Int
Mul sets z to the product x*y and returns z.
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 (z *Int) Neg(x *Int) *Int
Neg sets z to -x and returns z.
func (z *Int) Not(x *Int) *Int
Not sets z = ^x and returns z.
func (z *Int) Or(x, y *Int) *Int
Or sets z = x | y and returns z.
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 (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 (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 (z *Int) Rand(rnd *rand.Rand, n *Int) *Int
Rand sets z to a pseudo-random number in [0, n) and returns z.
As this uses the math/rand package, it must not be used for security-sensitive work. Use crypto/rand.Int instead.
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 (z *Int) Rsh(x *Int, n uint) *Int
Rsh sets z = x >> n and returns z.
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).
Code:
// 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 (z *Int) Set(x *Int) *Int
Set sets z to x and returns z.
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 (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 (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 (z *Int) SetInt64(x int64) *Int
SetInt64 sets z to x and returns z.
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. For base 0, the number prefix determines the actual base: A prefix of ``0b'' or ``0B'' selects base 2, ``0'', ``0o'' or ``0O'' selects base 8, and ``0x'' or ``0X'' selects base 16. Otherwise, the selected base is 10 and no prefix is accepted.
For bases <= 36, lower and upper case letters are considered the same: The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35. For bases > 36, the upper case letters 'A' to 'Z' represent the digit values 36 to 61.
For base 0, an underscore character ``_'' may appear between a base prefix and an adjacent digit, and between successive digits; such underscores do not change the value of the number. Incorrect placement of underscores is reported as an error if there are no other errors. If base != 0, underscores are not recognized and act like any other character that is not a valid digit.
Code:
i := new(big.Int)
i.SetString("644", 8) // octal
fmt.Println(i)
Output:
420
func (z *Int) SetUint64(x uint64) *Int
SetUint64 sets z to x and returns z.
func (x *Int) Sign() int
Sign returns:
-1 if x < 0 0 if x == 0 +1 if x > 0
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 (x *Int) String() string
String returns the decimal representation of x as generated by x.Text(10).
func (z *Int) Sub(x, y *Int) *Int
Sub sets z to the difference x-y and returns z.
func (x *Int) Text(base int) string
Text returns the string representation of x in the given base. Base must be between 2 and 62, inclusive. The result uses the lower-case letters 'a' to 'z' for digit values 10 to 35, and the upper-case letters 'A' to 'Z' for digit values 36 to 61. No prefix (such as "0x") is added to the string. If x is a nil pointer it returns "<nil>".
func (x *Int) TrailingZeroBits() uint
TrailingZeroBits returns the number of consecutive least significant zero bits of |x|.
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 (z *Int) UnmarshalJSON(text []byte) error
UnmarshalJSON implements the json.Unmarshaler interface.
func (z *Int) UnmarshalText(text []byte) error
UnmarshalText implements the encoding.TextUnmarshaler interface.
func (z *Int) Xor(x, y *Int) *Int
Xor sets z = x ^ y and returns z.
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.
Operations always take pointer arguments (*Rat) rather than Rat values, and each unique Rat value requires its own unique *Rat pointer. To "copy" a Rat value, an existing (or newly allocated) Rat must be set to a new value using the Rat.Set method; shallow copies of Rats are not supported and may lead to errors.
func NewRat(a, b int64) *Rat
NewRat creates a new Rat with numerator a and denominator b.
func (z *Rat) Abs(x *Rat) *Rat
Abs sets z to |x| (the absolute value of x) and returns z.
func (z *Rat) Add(x, y *Rat) *Rat
Add sets z to the sum x+y and returns z.
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 (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 (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 (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 (x *Rat) FloatString(prec int) string
FloatString returns a string representation of x in decimal form with prec digits of precision after the radix point. The last digit is rounded to nearest, with halves rounded away from zero.
func (z *Rat) GobDecode(buf []byte) error
GobDecode implements the gob.GobDecoder interface.
func (x *Rat) GobEncode() ([]byte, error)
GobEncode implements the gob.GobEncoder interface.
func (z *Rat) Inv(x *Rat) *Rat
Inv sets z to 1/x and returns z.
func (x *Rat) IsInt() bool
IsInt reports whether the denominator of x is 1.
func (x *Rat) MarshalText() (text []byte, err error)
MarshalText implements the encoding.TextMarshaler interface.
func (z *Rat) Mul(x, y *Rat) *Rat
Mul sets z to the product x*y and returns z.
func (z *Rat) Neg(x *Rat) *Rat
Neg sets z to -x and returns z.
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 (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 (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 (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.
Code:
// 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 (z *Rat) Set(x *Rat) *Rat
Set sets z to x (by making a copy of x) and returns z.
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 (z *Rat) SetFrac(a, b *Int) *Rat
SetFrac sets z to a/b and returns z.
func (z *Rat) SetFrac64(a, b int64) *Rat
SetFrac64 sets z to a/b and returns z.
func (z *Rat) SetInt(x *Int) *Rat
SetInt sets z to x (by making a copy of x) and returns z.
func (z *Rat) SetInt64(x int64) *Rat
SetInt64 sets z to x and returns z.
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 (possibly signed) fraction "a/b", or as a floating-point number optionally followed by an exponent. If a fraction is provided, both the dividend and the divisor may be a decimal integer or independently use a prefix of ``0b'', ``0'' or ``0o'', or ``0x'' (or their upper-case variants) to denote a binary, octal, or hexadecimal integer, respectively. The divisor may not be signed. If a floating-point number is provided, it may be in decimal form or use any of the same prefixes as above but for ``0'' to denote a non-decimal mantissa. A leading ``0'' is considered a decimal leading 0; it does not indicate octal representation in this case. An optional base-10 ``e'' or base-2 ``p'' (or their upper-case variants) exponent may be provided as well, except for hexadecimal floats which only accept an (optional) ``p'' exponent (because an ``e'' or ``E'' cannot be distinguished from a mantissa digit). 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.
Code:
r := new(big.Rat)
r.SetString("355/113")
fmt.Println(r.FloatString(3))
Output:
3.142
func (z *Rat) SetUint64(x uint64) *Rat
SetUint64 sets z to x and returns z.
func (x *Rat) Sign() int
Sign returns:
-1 if x < 0 0 if x == 0 +1 if x > 0
func (x *Rat) String() string
String returns a string representation of x in the form "a/b" (even if b == 1).
func (z *Rat) Sub(x, y *Rat) *Rat
Sub sets z to the difference x-y and returns z.
func (z *Rat) UnmarshalText(text []byte) error
UnmarshalText implements the encoding.TextUnmarshaler interface.
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.
Code:
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 (i RoundingMode) String() string
type Word uint
A Word represents a single digit of a multi-precision unsigned integer.