``````package big

Import Path
math/big (on go.dev)

Dependency Relation
imports 12 packages, and imported by 14 packages

Involved Source Files

accuracy_string.go
arith.go
arith_amd64.go
arith_decl.go
decimal.go
d doc.go
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

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

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 [fmt.Scanner] for scanning
and (except for *[Rat]) the Formatter interface for formatted printing.

float.go
floatconv.go
floatmarsh.go
ftoa.go
int.go
intconv.go
intmarsh.go
nat.go
natconv.go
natdiv.go
prime.go
rat.go
ratconv.go
ratmarsh.go
roundingmode_string.go
sqrt.go
arith_amd64.s
Code Examples

Float_Scan
package main

import (
"fmt"
"log"
"math/big"
)

func main() {
// 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)
}
}

Float_SetString
package main

import (
"fmt"
"math/big"
)

func main() {
f := new(big.Float)
f.SetString("3.14159")
fmt.Println(f)
}

Int_Scan
package main

import (
"fmt"
"log"
"math/big"
)

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)
}
}

Int_SetString
package main

import (
"fmt"
"math/big"
)

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

Rat_Scan
package main

import (
"fmt"
"log"
"math/big"
)

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)
}
}

Rat_SetString
package main

import (
"fmt"
"math/big"
)

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

_eConvergents
package main

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))

}

// 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))
}

}

_fibonacci
package main

import (
"fmt"
"math/big"
)

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.
// 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))

}

_sqrt2
package main

import (
"fmt"
"math"
"math/big"
)

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))

}

Package-Level Type Names (total 7)

/* sort by: alphabet | popularity */	 type Accuracy int8 (basic type)
Accuracy describes the rounding error produced by the most recent
operation that generated a [Float] value, relative to the exact value.

Methods (only one)
( Accuracy) String() string
Implements (at least 2)
Accuracy : expvar.Var
Accuracy : fmt.Stringer
As Outputs Of (at least 8)
func (*Float).Acc() Accuracy
func (*Float).Float32() (float32, Accuracy)
func (*Float).Float64() (float64, Accuracy)
func (*Float).Int(z *Int) (*Int, Accuracy)
func (*Float).Int64() (int64, Accuracy)
func (*Float).Rat(z *Rat) (*Rat, Accuracy)
func (*Float).Uint64() (uint64, Accuracy)
func (*Int).Float64() (float64, Accuracy)
As Types Of (total 3)
const Above
const Below
const Exact

type ErrNaN (struct)
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.

Methods (only one)
( ErrNaN) Error() string
Implements (at least one exported)
ErrNaN : error

type Float (struct)
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 [Float.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.

Methods (total 45)
(*Float) Abs(x *Float) *Float
Abs sets z to the (possibly rounded) value |x| (the absolute value of x)
and returns z.

(*Float) Acc() Accuracy
Acc returns the accuracy of x produced by the most recent
operation, unless explicitly documented otherwise by that
operation.

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.

(*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.

(*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

(*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.

(*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.

(*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.

(*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.

(*Float) GobDecode(buf []byte) error
GobDecode implements the [encoding/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.

(*Float) GobEncode() ([]byte, error)
GobEncode implements the [encoding/gob.GobEncoder] interface.
The [Float] value and all its attributes (precision,
rounding mode, accuracy) are marshaled.

(*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].

(*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].

(*Float) IsInf() bool
IsInf reports whether x is +Inf or -Inf.

(*Float) IsInt() bool
IsInt reports whether x is an integer.
±Inf values are not integers.

(*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.

(*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.

(*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.

(*Float) Mode() RoundingMode
Mode returns the rounding mode of x.

(*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 [Float.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.

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

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

(*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 [Float.Add].
Quo panics with [ErrNaN] if both operands are zero or infinities.
The value of z is undefined in that case.

(*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].

(*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.

(*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.

(*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.

(*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].

(*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).

(*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).

(*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 [Float.MantExp] but does
not require 0.5 <= |mant| < 1.0. Specifically, for a
given x of type *[Float], SetMantExp relates to [Float.MantExp]
as follows:

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.

(*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].

(*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].

(*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.

(*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 [Float.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.

(*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).

(*Float) Sign() int
Sign returns:

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

(*Float) Signbit() bool
Signbit reports whether x is negative or negative zero.

(*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, but z's accuracy is not computed. Specifically, the
result of z.Acc() is undefined.

The function panics if z < 0. The value of z is undefined in that
case.

(*Float) String() string
String formats x like x.Text('g', 10).
(String must be called explicitly, [Float.Format] does not support %s verb.)

(*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 [Float.Add].
Sub panics with [ErrNaN] if x and y are infinities with equal
signs. The value of z is undefined in that case.

(*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.

(*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].

(*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.

Implements (at least 8)
*Float : encoding.TextMarshaler
*Float : encoding.TextUnmarshaler
*Float : encoding/gob.GobDecoder
*Float : encoding/gob.GobEncoder
*Float : expvar.Var
*Float : fmt.Formatter
*Float : fmt.Scanner
*Float : fmt.Stringer
As Outputs Of (at least 22)
func NewFloat(x float64) *Float
func ParseFloat(s string, base int, prec uint, mode RoundingMode) (f *Float, b int, err error)
func (*Float).Abs(x *Float) *Float
func (*Float).Copy(x *Float) *Float
func (*Float).Mul(x, y *Float) *Float
func (*Float).Neg(x *Float) *Float
func (*Float).Parse(s string, base int) (f *Float, b int, err error)
func (*Float).Quo(x, y *Float) *Float
func (*Float).Set(x *Float) *Float
func (*Float).SetFloat64(x float64) *Float
func (*Float).SetInf(signbit bool) *Float
func (*Float).SetInt(x *Int) *Float
func (*Float).SetInt64(x int64) *Float
func (*Float).SetMantExp(mant *Float, exp int) *Float
func (*Float).SetMode(mode RoundingMode) *Float
func (*Float).SetPrec(prec uint) *Float
func (*Float).SetRat(x *Rat) *Float
func (*Float).SetString(s string) (*Float, bool)
func (*Float).SetUint64(x uint64) *Float
func (*Float).Sqrt(x *Float) *Float
func (*Float).Sub(x, y *Float) *Float
As Inputs Of (at least 12)
func (*Float).Abs(x *Float) *Float
func (*Float).Cmp(y *Float) int
func (*Float).Copy(x *Float) *Float
func (*Float).MantExp(mant *Float) (exp int)
func (*Float).Mul(x, y *Float) *Float
func (*Float).Neg(x *Float) *Float
func (*Float).Quo(x, y *Float) *Float
func (*Float).Set(x *Float) *Float
func (*Float).SetMantExp(mant *Float, exp int) *Float
func (*Float).Sqrt(x *Float) *Float
func (*Float).Sub(x, y *Float) *Float

type Int (struct)
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.

Note that methods may leak the Int's value through timing side-channels.
Because of this and because of the scope and complexity of the
implementation, Int is not well-suited to implement cryptographic operations.
The standard library avoids exposing non-trivial Int methods to
attacker-controlled inputs and the determination of whether a bug in math/big
is considered a security vulnerability might depend on the impact on the
standard library.

Methods (total 59)
(*Int) Abs(x *Int) *Int
Abs sets z to |x| (the absolute value of x) and returns z.

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

(*Int) And(x, y *Int) *Int
And sets z = x & y and returns z.

(*Int) AndNot(x, y *Int) *Int
AndNot sets z = x &^ y and returns z.

(*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.

(*Int) Binomial(n, k int64) *Int
Binomial sets z to the binomial coefficient C(n, k) and returns z.

(*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.

(*Int) BitLen() int
BitLen returns the length of the absolute value of x in bits.
The bit length of 0 is 0.

(*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.

(*Int) Bytes() []byte
Bytes returns the absolute value of x as a big-endian byte slice.

To use a fixed length slice, or a preallocated one, use [Int.FillBytes].

(*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

(*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|

(*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 [Int.DivMod] for more details.

(*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 [Int.QuoRem] for T-division and modulus (like Go).

(*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 m are not relatively prime, z is unchanged and nil is returned.

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

(*Int) FillBytes(buf []byte) []byte
FillBytes sets buf to the absolute value of x, storing it as a zero-extended
big-endian byte slice, and returns buf.

If the absolute value of x doesn't fit in buf, FillBytes will panic.

(*Int) Float64() (float64, Accuracy)
Float64 returns the float64 value nearest x,
and an indication of any rounding that occurred.

(*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
Also supported are the full suite of package fmt's format
flags for integral types, including '+' and ' ' for sign
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.

(*Int) GCD(x, y, a, b *Int) *Int
GCD sets z to the greatest common divisor of a and b and returns z.
If x or y are not nil, GCD sets their value such that z = a*x + b*y.

a and b may be positive, zero or negative. (Before Go 1.14 both had
to be > 0.) Regardless of the signs of a and b, z is always >= 0.

If a == b == 0, GCD sets z = x = y = 0.

If a == 0 and b != 0, GCD sets z = |b|, x = 0, y = sign(b) * 1.

If a != 0 and b == 0, GCD sets z = |a|, x = sign(a) * 1, y = 0.

(*Int) GobDecode(buf []byte) error
GobDecode implements the [encoding/gob.GobDecoder] interface.

(*Int) GobEncode() ([]byte, error)
GobEncode implements the [encoding/gob.GobEncoder] interface.

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

(*Int) IsInt64() bool
IsInt64 reports whether x can be represented as an int64.

(*Int) IsUint64() bool
IsUint64 reports whether x can be represented as a uint64.

(*Int) Lsh(x *Int, n uint) *Int
Lsh sets z = x << n and returns z.

(*Int) MarshalJSON() ([]byte, error)
MarshalJSON implements the [encoding/json.Marshaler] interface.

(*Int) MarshalText() (text []byte, err error)
MarshalText implements the [encoding.TextMarshaler] interface.

(*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 [Int.DivMod] for more details.

(*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. If n == 0, a division-by-zero run-time panic occurs.

(*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, its behavior is undefined if p is odd but not prime.

(*Int) Mul(x, y *Int) *Int
Mul sets z to the product x*y and returns 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.

(*Int) Neg(x *Int) *Int
Neg sets z to -x and returns z.

(*Int) Not(x *Int) *Int
Not sets z = ^x and returns z.

(*Int) Or(x, y *Int) *Int
Or sets z = x | y and returns z.

(*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.

(*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 [Int.QuoRem] for more details.

(*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).

(*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

(*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 [Int.QuoRem] for more details.

(*Int) Rsh(x *Int, n uint) *Int
Rsh sets z = x >> n and returns 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),

(*Int) Set(x *Int) *Int
Set sets z to x and returns 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.

(*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.

(*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.

(*Int) SetInt64(x int64) *Int
SetInt64 sets z to x and returns 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.

(*Int) SetUint64(x uint64) *Int
SetUint64 sets z to x and returns z.

(*Int) Sign() int
Sign returns:

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

(*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.

(*Int) String() string
String returns the decimal representation of x as generated by
x.Text(10).

(*Int) Sub(x, y *Int) *Int
Sub sets z to the difference x-y and returns z.

(*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>".

(*Int) TrailingZeroBits() uint
TrailingZeroBits returns the number of consecutive least significant zero
bits of |x|.

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

(*Int) UnmarshalJSON(text []byte) error
UnmarshalJSON implements the [encoding/json.Unmarshaler] interface.

(*Int) UnmarshalText(text []byte) error
UnmarshalText implements the [encoding.TextUnmarshaler] interface.

(*Int) Xor(x, y *Int) *Int
Xor sets z = x ^ y and returns z.

Implements (at least 11)
*Int : math/rand/v2.Source
*Int : encoding.TextMarshaler
*Int : encoding.TextUnmarshaler
*Int : encoding/gob.GobDecoder
*Int : encoding/gob.GobEncoder
*Int : encoding/json.Marshaler
*Int : encoding/json.Unmarshaler
*Int : expvar.Var
*Int : fmt.Formatter
*Int : fmt.Scanner
*Int : fmt.Stringer
As Outputs Of (at least 59)
func NewInt(x int64) *Int
func (*Float).Int(z *Int) (*Int, Accuracy)
func (*Int).Abs(x *Int) *Int
func (*Int).And(x, y *Int) *Int
func (*Int).AndNot(x, y *Int) *Int
func (*Int).Binomial(n, k int64) *Int
func (*Int).Div(x, y *Int) *Int
func (*Int).DivMod(x, y, m *Int) (*Int, *Int)
func (*Int).DivMod(x, y, m *Int) (*Int, *Int)
func (*Int).Exp(x, y, m *Int) *Int
func (*Int).GCD(x, y, a, b *Int) *Int
func (*Int).Lsh(x *Int, n uint) *Int
func (*Int).Mod(x, y *Int) *Int
func (*Int).ModInverse(g, n *Int) *Int
func (*Int).ModSqrt(x, p *Int) *Int
func (*Int).Mul(x, y *Int) *Int
func (*Int).MulRange(a, b int64) *Int
func (*Int).Neg(x *Int) *Int
func (*Int).Not(x *Int) *Int
func (*Int).Or(x, y *Int) *Int
func (*Int).Quo(x, y *Int) *Int
func (*Int).QuoRem(x, y, r *Int) (*Int, *Int)
func (*Int).QuoRem(x, y, r *Int) (*Int, *Int)
func (*Int).Rand(rnd *rand.Rand, n *Int) *Int
func (*Int).Rem(x, y *Int) *Int
func (*Int).Rsh(x *Int, n uint) *Int
func (*Int).Set(x *Int) *Int
func (*Int).SetBit(x *Int, i int, b uint) *Int
func (*Int).SetBits(abs []Word) *Int
func (*Int).SetBytes(buf []byte) *Int
func (*Int).SetInt64(x int64) *Int
func (*Int).SetString(s string, base int) (*Int, bool)
func (*Int).SetUint64(x uint64) *Int
func (*Int).Sqrt(x *Int) *Int
func (*Int).Sub(x, y *Int) *Int
func (*Int).Xor(x, y *Int) *Int
func (*Rat).Denom() *Int
func (*Rat).Num() *Int
func crypto/dsa.Sign(rand io.Reader, priv *dsa.PrivateKey, hash []byte) (r, s *Int, err error)
func crypto/ecdsa.Sign(rand io.Reader, priv *ecdsa.PrivateKey, hash []byte) (r, s *Int, err error)
func crypto/elliptic.GenerateKey(curve elliptic.Curve, rand io.Reader) (priv []byte, x, y *Int, err error)
func crypto/elliptic.Unmarshal(curve elliptic.Curve, data []byte) (x, y *Int)
func crypto/elliptic.UnmarshalCompressed(curve elliptic.Curve, data []byte) (x, y *Int)
func crypto/elliptic.Curve.Add(x1, y1, x2, y2 *Int) (x, y *Int)
func crypto/elliptic.Curve.Double(x1, y1 *Int) (x, y *Int)
func crypto/elliptic.Curve.ScalarBaseMult(k []byte) (x, y *Int)
func crypto/elliptic.Curve.ScalarMult(x1, y1 *Int, k []byte) (x, y *Int)
func crypto/elliptic.(*CurveParams).Add(x1, y1, x2, y2 *Int) (*Int, *Int)
func crypto/elliptic.(*CurveParams).Add(x1, y1, x2, y2 *Int) (*Int, *Int)
func crypto/elliptic.(*CurveParams).Double(x1, y1 *Int) (*Int, *Int)
func crypto/elliptic.(*CurveParams).Double(x1, y1 *Int) (*Int, *Int)
func crypto/elliptic.(*CurveParams).ScalarBaseMult(k []byte) (*Int, *Int)
func crypto/elliptic.(*CurveParams).ScalarBaseMult(k []byte) (*Int, *Int)
func crypto/elliptic.(*CurveParams).ScalarMult(Bx, By *Int, k []byte) (*Int, *Int)
func crypto/elliptic.(*CurveParams).ScalarMult(Bx, By *Int, k []byte) (*Int, *Int)
func crypto/internal/boring/bbig.Dec(b boring.BigInt) *Int
func crypto/rand.Int(rand io.Reader, max *Int) (n *Int, err error)
func crypto/rand.Prime(rand io.Reader, bits int) (*Int, error)
As Inputs Of (at least 49)
func Jacobi(x, y *Int) int
func (*Float).Int(z *Int) (*Int, Accuracy)
func (*Float).SetInt(x *Int) *Float
func (*Int).Abs(x *Int) *Int
func (*Int).And(x, y *Int) *Int
func (*Int).AndNot(x, y *Int) *Int
func (*Int).Cmp(y *Int) (r int)
func (*Int).CmpAbs(y *Int) int
func (*Int).Div(x, y *Int) *Int
func (*Int).DivMod(x, y, m *Int) (*Int, *Int)
func (*Int).Exp(x, y, m *Int) *Int
func (*Int).GCD(x, y, a, b *Int) *Int
func (*Int).Lsh(x *Int, n uint) *Int
func (*Int).Mod(x, y *Int) *Int
func (*Int).ModInverse(g, n *Int) *Int
func (*Int).ModSqrt(x, p *Int) *Int
func (*Int).Mul(x, y *Int) *Int
func (*Int).Neg(x *Int) *Int
func (*Int).Not(x *Int) *Int
func (*Int).Or(x, y *Int) *Int
func (*Int).Quo(x, y *Int) *Int
func (*Int).QuoRem(x, y, r *Int) (*Int, *Int)
func (*Int).Rand(rnd *rand.Rand, n *Int) *Int
func (*Int).Rem(x, y *Int) *Int
func (*Int).Rsh(x *Int, n uint) *Int
func (*Int).Set(x *Int) *Int
func (*Int).SetBit(x *Int, i int, b uint) *Int
func (*Int).Sqrt(x *Int) *Int
func (*Int).Sub(x, y *Int) *Int
func (*Int).Xor(x, y *Int) *Int
func (*Rat).SetFrac(a, b *Int) *Rat
func (*Rat).SetInt(x *Int) *Rat
func crypto/dsa.Verify(pub *dsa.PublicKey, hash []byte, r, s *Int) bool
func crypto/ecdsa.Verify(pub *ecdsa.PublicKey, hash []byte, r, s *Int) bool
func crypto/elliptic.Marshal(curve elliptic.Curve, x, y *Int) []byte
func crypto/elliptic.MarshalCompressed(curve elliptic.Curve, x, y *Int) []byte
func crypto/elliptic.Curve.Add(x1, y1, x2, y2 *Int) (x, y *Int)
func crypto/elliptic.Curve.Double(x1, y1 *Int) (x, y *Int)
func crypto/elliptic.Curve.IsOnCurve(x, y *Int) bool
func crypto/elliptic.Curve.ScalarMult(x1, y1 *Int, k []byte) (x, y *Int)
func crypto/elliptic.(*CurveParams).Add(x1, y1, x2, y2 *Int) (*Int, *Int)
func crypto/elliptic.(*CurveParams).Double(x1, y1 *Int) (*Int, *Int)
func crypto/elliptic.(*CurveParams).IsOnCurve(x, y *Int) bool
func crypto/elliptic.(*CurveParams).ScalarMult(Bx, By *Int, k []byte) (*Int, *Int)
func crypto/internal/bigmod.NewModulusFromBig(n *Int) (*bigmod.Modulus, error)
func crypto/internal/boring/bbig.Enc(b *Int) boring.BigInt
func crypto/rand.Int(rand io.Reader, max *Int) (n *Int, err error)

type Rat (struct)
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.

Methods (total 31)
(*Rat) Abs(x *Rat) *Rat
Abs sets z to |x| (the absolute value of x) and returns z.

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

(*Rat) Cmp(y *Rat) int
Cmp compares x and y and returns:

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

(*Rat) Denom() *Int
Denom returns the denominator of x; it is always > 0.
The result is a reference to x's denominator, unless
x is an uninitialized (zero value) [Rat], in which case
the result is a new [Int] of value 1. (To initialize x,
any operation that sets x will do, including x.Set(x).)
If the result is a reference to x's denominator it
may change if a new value is assigned to x, and vice versa.

(*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.

(*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.

(*Rat) FloatPrec() (n int, exact bool)
FloatPrec returns the number n of non-repeating digits immediately
following the decimal point of the decimal representation of x.
The boolean result indicates whether a decimal representation of x
with that many fractional digits is exact or rounded.

Examples:

x      n    exact    decimal representation n fractional digits
0      0    true     0
1      0    true     1
1/2    1    true     0.5
1/3    0    false    0       (0.333... rounded)
1/4    2    true     0.25
1/6    1    false    0.2     (0.166... rounded)

(*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.

(*Rat) GobDecode(buf []byte) error
GobDecode implements the [encoding/gob.GobDecoder] interface.

(*Rat) GobEncode() ([]byte, error)
GobEncode implements the [encoding/gob.GobEncoder] interface.

(*Rat) Inv(x *Rat) *Rat
Inv sets z to 1/x and returns z.
If x == 0, Inv panics.

(*Rat) IsInt() bool
IsInt reports whether the denominator of x is 1.

(*Rat) MarshalText() (text []byte, err error)
MarshalText implements the [encoding.TextMarshaler] interface.

(*Rat) Mul(x, y *Rat) *Rat
Mul sets z to the product x*y and returns z.

(*Rat) Neg(x *Rat) *Rat
Neg sets z to -x and returns z.

(*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.

(*Rat) Quo(x, y *Rat) *Rat
Quo sets z to the quotient x/y and returns z.
If y == 0, Quo panics.

(*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.

(*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.

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

(*Rat) SetFloat64(f float64) *Rat
SetFloat64 sets z to exactly f and returns z.
If f is not finite, SetFloat returns nil.

(*Rat) SetFrac(a, b *Int) *Rat
SetFrac sets z to a/b and returns z.
If b == 0, SetFrac panics.

(*Rat) SetFrac64(a, b int64) *Rat
SetFrac64 sets z to a/b and returns z.
If b == 0, SetFrac64 panics.

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

(*Rat) SetInt64(x int64) *Rat
SetInt64 sets z to x and returns 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). If the exponent's absolute value
is too large, the operation may fail.
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.

(*Rat) SetUint64(x uint64) *Rat
SetUint64 sets z to x and returns z.

(*Rat) Sign() int
Sign returns:

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

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

(*Rat) Sub(x, y *Rat) *Rat
Sub sets z to the difference x-y and returns z.

(*Rat) UnmarshalText(text []byte) error
UnmarshalText implements the [encoding.TextUnmarshaler] interface.

Implements (at least 7)
*Rat : encoding.TextMarshaler
*Rat : encoding.TextUnmarshaler
*Rat : encoding/gob.GobDecoder
*Rat : encoding/gob.GobEncoder
*Rat : expvar.Var
*Rat : fmt.Scanner
*Rat : fmt.Stringer
As Outputs Of (at least 17)
func NewRat(a, b int64) *Rat
func (*Float).Rat(z *Rat) (*Rat, Accuracy)
func (*Rat).Abs(x *Rat) *Rat
func (*Rat).Inv(x *Rat) *Rat
func (*Rat).Mul(x, y *Rat) *Rat
func (*Rat).Neg(x *Rat) *Rat
func (*Rat).Quo(x, y *Rat) *Rat
func (*Rat).Set(x *Rat) *Rat
func (*Rat).SetFloat64(f float64) *Rat
func (*Rat).SetFrac(a, b *Int) *Rat
func (*Rat).SetFrac64(a, b int64) *Rat
func (*Rat).SetInt(x *Int) *Rat
func (*Rat).SetInt64(x int64) *Rat
func (*Rat).SetString(s string) (*Rat, bool)
func (*Rat).SetUint64(x uint64) *Rat
func (*Rat).Sub(x, y *Rat) *Rat
As Inputs Of (at least 11)
func (*Float).Rat(z *Rat) (*Rat, Accuracy)
func (*Float).SetRat(x *Rat) *Float
func (*Rat).Abs(x *Rat) *Rat
func (*Rat).Cmp(y *Rat) int
func (*Rat).Inv(x *Rat) *Rat
func (*Rat).Mul(x, y *Rat) *Rat
func (*Rat).Neg(x *Rat) *Rat
func (*Rat).Quo(x, y *Rat) *Rat
func (*Rat).Set(x *Rat) *Rat
func (*Rat).Sub(x, y *Rat) *Rat

type RoundingMode byte (basic type)
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].

Methods (only one)
( RoundingMode) String() string
Implements (at least 2)
RoundingMode : expvar.Var
RoundingMode : fmt.Stringer
As Outputs Of (at least one exported)
func (*Float).Mode() RoundingMode
As Inputs Of (at least 2)
func ParseFloat(s string, base int, prec uint, mode RoundingMode) (f *Float, b int, err error)
func (*Float).SetMode(mode RoundingMode) *Float
As Types Of (total 6)
const AwayFromZero
const ToNearestAway
const ToNearestEven
const ToNegativeInf
const ToPositiveInf
const ToZero

type Word uint (basic type)
A Word represents a single digit of a multi-precision unsigned integer.

As Outputs Of (at least one exported)
func (*Int).Bits() []Word
As Inputs Of (at least one exported)
func (*Int).SetBits(abs []Word) *Int

Package-Level Functions (total 5)

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.

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 NewInt(x int64) *Int
NewInt allocates and returns a new [Int] set to x.

func NewRat(a, b int64) *Rat
NewRat creates a new [Rat] with numerator a and denominator b.

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.

Package-Level Constants (total 13)

const Above Accuracy = 1
Constants describing the [Accuracy] of a [Float].

const AwayFromZero RoundingMode = 3 // no IEEE 754-2008 equivalent
These constants define supported rounding modes.

const Below Accuracy = -1
Constants describing the [Accuracy] of a [Float].

const Exact Accuracy = 0
Constants describing the [Accuracy] of a [Float].

const MaxBase = 62
MaxBase is the largest number base accepted for string conversions.

const MaxExp = 2147483647 // largest supported exponent
Exponent and precision limits.

const MaxPrec = 4294967295 // largest (theoretically) supported precision; likely memory-limited
Exponent and precision limits.

const MinExp = -2147483648 // smallest supported exponent
Exponent and precision limits.

const ToNearestAway RoundingMode = 1 // == IEEE 754-2008 roundTiesToAway
These constants define supported rounding modes.

const ToNearestEven RoundingMode = 0 // == IEEE 754-2008 roundTiesToEven
These constants define supported rounding modes.

const ToNegativeInf RoundingMode = 4 // == IEEE 754-2008 roundTowardNegative
These constants define supported rounding modes.

const ToPositiveInf RoundingMode = 5 // == IEEE 754-2008 roundTowardPositive
These constants define supported rounding modes.

const ToZero RoundingMode = 2 // == IEEE 754-2008 roundTowardZero
These constants define supported rounding modes.

``````