Namespace: go.std.math.big
v1.0Contents
Summary
Provides a lowlevel interface to the math/big package.
Package big implements arbitraryprecision arithmetic (big numbers).
The following numeric types are supported:
Int signed integers
Rat rational numbers
Float floatingpoint 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.
Index
 *Accuracy
 *ErrNaN
 *Float
 *Int
 *Rat
 *RoundingMode
 *Word
 Above
 Accuracy
 AwayFromZero
 Below
 ErrNaN
 Exact
 Float
 Int
 Jacobi
 MaxBase
 NewInt
 NewRat
 ParseFloat
 Rat
 RoundingMode
 ToNearestAway
 ToNearestEven
 ToNegativeInf
 ToPositiveInf
 ToZero
 Word
Legend

Constant
Variable
Function
Macro
Special form
GoType
GoVar
Receiver/Method
Constants
Constants are variables with :const true in their metadata. Joker currently does not recognize them as special; as such, it allows redefining them or their values.
Above
Int v1.0Constants describing the Accuracy of a Float.

AwayFromZero
Int v1.0no IEEE 7542008 equivalent

Below
Int v1.0Constants describing the Accuracy of a Float.

Exact
Int v1.0Constants describing the Accuracy of a Float.

MaxBase
Int v1.0MaxBase is the largest number base accepted for string conversions.

ToNearestAway
Int v1.0== IEEE 7542008 roundTiesToAway

ToNearestEven
Int v1.0== IEEE 7542008 roundTiesToEven

ToNegativeInf
Int v1.0== IEEE 7542008 roundTowardNegative

ToPositiveInf
Int v1.0== IEEE 7542008 roundTowardPositive

ToZero
Int v1.0== IEEE 7542008 roundTowardZero
Variables

(None.)
Functions, Macros, and Special Forms

Jacobi
Function v1.0(Jacobi x y)
Jacobi returns the Jacobi symbol (x/y), either +1, 1, or 0.
The y argument must be an odd integer.
Go input arguments: (x *Int, y *Int)
Go return type: int
Joker input arguments: [^(refto go.std.math.big/Int) x, ^(refto go.std.math.big/Int) y]
Joker return type: Int 
NewInt
Function v1.0(NewInt x)
NewInt allocates and returns a new Int set to x.
Go input arguments: (x int64)
Go return type: *Int
Joker input arguments: [^Number x]
Joker return type: (refto go.std.math.big/Int) 
NewRat
Function v1.0(NewRat a b)
NewRat creates a new Rat with numerator a and denominator b.
Go input arguments: (a int64, b int64)
Go return type: *Rat
Joker input arguments: [^Number a, ^Number b]
Joker return type: (refto go.std.math.big/Rat) 
ParseFloat
Function v1.0(ParseFloat s base prec mode)
ParseFloat is like f.Parse(s, base) with f set to the given precision
and rounding mode.
Go input arguments: (s string, base int, prec uint, mode RoundingMode)
Go return type: (f *Float, b int, err error)
Joker input arguments: [^String s, ^Int base, ^Number prec, ^go.std.math.big/RoundingMode mode]
Joker return type: [(refto go.std.math.big/Float) Int Error]
Types

*Accuracy
Concrete GoType v1.0 
*ErrNaN
Concrete GoType v1.0 
*Float
Concrete GoType v1.0 
Acc
Receiver for *Float v1.0([])
Acc returns the accuracy of x produced by the most recent operation.

GobEncode
Receiver for *Float v1.0([])
GobEncode implements the gob.GobEncoder interface.
The Float value and all its attributes (precision,
rounding mode, accuracy) are marshaled.

Int64
Receiver for *Float v1.0([])
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.

IsInf
Receiver for *Float v1.0([])
IsInf reports whether x is +Inf or Inf.

IsInt
Receiver for *Float v1.0([])
IsInt reports whether x is an integer.
±Inf values are not integers.

MarshalText
Receiver for *Float v1.0([])
MarshalText implements the encoding.TextMarshaler interface.
Only the Float value is marshaled (in full precision), other
attributes such as precision or accuracy are ignored.

MinPrec
Receiver for *Float v1.0([])
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.

Mode
Receiver for *Float v1.0([])
Mode returns the rounding mode of x.

Parse
Receiver for *Float v1.0([s base])
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 runtime 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.

Prec
Receiver for *Float v1.0([])
Prec returns the mantissa precision of x in bits.
The result may be 0 for x == 0 and x == Inf.

SetInf
Receiver for *Float v1.0([signbit])
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.

SetInt64
Receiver for *Float v1.0([x])
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).

SetMode
Receiver for *Float v1.0([mode])
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.

SetPrec
Receiver for *Float v1.0([prec])
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.

SetString
Receiver for *Float v1.0([s])
SetString sets z to the value of s and returns z and a boolean indicating
success. s must be a floatingpoint 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.

SetUint64
Receiver for *Float v1.0([x])
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).

Sign
Receiver for *Float v1.0([])
Sign returns:
1 if x < 0
0 if x is ±0
+1 if x > 0

Signbit
Receiver for *Float v1.0([])
Signbit reports whether x is negative or negative zero.

String
Receiver for *Float v1.0([])
String formats x like x.Text('g', 10).
(String must be called explicitly, Float.Format does not support %s verb.)

Text
Receiver for *Float v1.0([format prec])
Text converts the floatingpoint 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 (nonstandard)
'b' ddddddp±dd, decimal mantissa, decimal power of two exponent (nonstandard)
For the poweroftwo 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.

Uint64
Receiver for *Float v1.0([])
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.

*Int
Concrete GoType v1.0 
Binomial
Receiver for *Int v1.0([n k])
Binomial sets z to the binomial coefficient of (n, k) and returns z.

Bit
Receiver for *Int v1.0([i])
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.

BitLen
Receiver for *Int v1.0([])
BitLen returns the length of the absolute value of x in bits.
The bit length of 0 is 0.

Bits
Receiver for *Int v1.0([])
Bits provides raw (unchecked but fast) access to x by returning its
absolute value as a littleendian Word slice. The result and x share
the same underlying array.
Bits is intended to support implementation of missing lowlevel Int
functionality outside this package; it should be avoided otherwise.

Bytes
Receiver for *Int v1.0([])
Bytes returns the absolute value of x as a bigendian byte slice.

GobEncode
Receiver for *Int v1.0([])
GobEncode implements the gob.GobEncoder interface.

Int64
Receiver for *Int v1.0([])
Int64 returns the int64 representation of x.
If x cannot be represented in an int64, the result is undefined.

IsInt64
Receiver for *Int v1.0([])
IsInt64 reports whether x can be represented as an int64.

IsUint64
Receiver for *Int v1.0([])
IsUint64 reports whether x can be represented as a uint64.

MarshalJSON
Receiver for *Int v1.0([])
MarshalJSON implements the json.Marshaler interface.

MarshalText
Receiver for *Int v1.0([])
MarshalText implements the encoding.TextMarshaler interface.

MulRange
Receiver for *Int v1.0([a b])
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.

ProbablyPrime
Receiver for *Int v1.0([n])
ProbablyPrime reports whether x is probably prime,
applying the MillerRabin test with n pseudorandomly chosen bases
as well as a BailliePSW 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 nonprime is at most ¼ⁿ.
ProbablyPrime is 100% accurate for inputs less than 2⁶⁴.
See Menezes et al., Handbook of Applied Cryptography, 1997, pp. 145149,
and FIPS 1864 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 BailliePSW test.
Before Go 1.8, ProbablyPrime applied only the MillerRabin tests, and ProbablyPrime(0) panicked.

SetInt64
Receiver for *Int v1.0([x])
SetInt64 sets z to x and returns z.

SetString
Receiver for *Int v1.0([s base])
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.

SetUint64
Receiver for *Int v1.0([x])
SetUint64 sets z to x and returns z.

Sign
Receiver for *Int v1.0([])
Sign returns:
1 if x < 0
0 if x == 0
+1 if x > 0

String
Receiver for *Int v1.0([])
String returns the decimal representation of x as generated by
x.Text(10).

Text
Receiver for *Int v1.0([base])
Text returns the string representation of x in the given base.
Base must be between 2 and 62, inclusive. The result uses the
lowercase letters 'a' to 'z' for digit values 10 to 35, and
the uppercase 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>".

TrailingZeroBits
Receiver for *Int v1.0([])
TrailingZeroBits returns the number of consecutive least significant zero
bits of x.

Uint64
Receiver for *Int v1.0([])
Uint64 returns the uint64 representation of x.
If x cannot be represented in a uint64, the result is undefined.

*Rat
Concrete GoType v1.0 
Denom
Receiver for *Rat v1.0([])
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.

FloatString
Receiver for *Rat v1.0([prec])
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.

GobEncode
Receiver for *Rat v1.0([])
GobEncode implements the gob.GobEncoder interface.

IsInt
Receiver for *Rat v1.0([])
IsInt reports whether the denominator of x is 1.

MarshalText
Receiver for *Rat v1.0([])
MarshalText implements the encoding.TextMarshaler interface.

Num
Receiver for *Rat v1.0([])
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.

RatString
Receiver for *Rat v1.0([])
RatString returns a string representation of x in the form "a/b" if b != 1,
and in the form "a" if b == 1.

SetFrac64
Receiver for *Rat v1.0([a b])
SetFrac64 sets z to a/b and returns z.
If b == 0, SetFrac64 panics.

SetInt64
Receiver for *Rat v1.0([x])
SetInt64 sets z to x and returns z.

SetString
Receiver for *Rat v1.0([s])
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
floatingpoint 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 uppercase variants) to denote a binary, octal, or
hexadecimal integer, respectively. The divisor may not be signed.
If a floatingpoint number is provided, it may be in decimal form or
use any of the same prefixes as above but for ``0'' to denote a nondecimal
mantissa. A leading ``0'' is considered a decimal leading 0; it does not
indicate octal representation in this case.
An optional base10 ``e'' or base2 ``p'' (or their uppercase 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.

SetUint64
Receiver for *Rat v1.0([x])
SetUint64 sets z to x and returns z.

Sign
Receiver for *Rat v1.0([])
Sign returns:
1 if x < 0
0 if x == 0
+1 if x > 0

String
Receiver for *Rat v1.0([])
String returns a string representation of x in the form "a/b" (even if b == 1).

*RoundingMode
Concrete GoType v1.0 
*Word
Concrete GoType v1.0 
Accuracy
Concrete GoType v1.0Accuracy describes the rounding error produced by the most recent
operation that generated a Float value, relative to the exact value.

String
Receiver for Accuracy v1.0([])

ErrNaN
Concrete GoType v1.0An ErrNaN panic is raised by a Float operation that would lead to
a NaN under IEEE754 rules. An ErrNaN implements the error interface.

Error
Receiver for ErrNaN v1.0([])

Float
Concrete GoType v1.0A nonzero finite Float represents a multiprecision 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 IEEE754 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 IEEE754 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.

Int
Concrete GoType v1.0An Int represents a signed multiprecision 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.

Rat
Concrete GoType v1.0A 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.

RoundingMode
Concrete GoType v1.0RoundingMode 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.

String
Receiver for RoundingMode v1.0([])

Word
Concrete GoType v1.0A Word represents a single digit of a multiprecision unsigned integer.