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# This file is a part of Julia. License is MIT: https://julialang.org/license
## number-theoretic functions ##
"""
gcd(x,y)
Greatest common (positive) divisor (or zero if `x` and `y` are both zero).
# Examples
```jldoctest
julia> gcd(6,9)
3
julia> gcd(6,-9)
3
```
"""
function gcd(a::T, b::T) where T<:Integer
while b != 0
t = b
b = rem(a, b)
a = t
end
checked_abs(a)
end
# binary GCD (aka Stein's) algorithm
# about 1.7x (2.1x) faster for random Int64s (Int128s)
function gcd(a::T, b::T) where T<:Union{Int64,UInt64,Int128,UInt128}
@noinline throw1(a, b) = throw(OverflowError("gcd($a, $b) overflows"))
a == 0 && return abs(b)
b == 0 && return abs(a)
za = trailing_zeros(a)
zb = trailing_zeros(b)
k = min(za, zb)
u = unsigned(abs(a >> za))
v = unsigned(abs(b >> zb))
while u != v
if u > v
u, v = v, u
end
v -= u
v >>= trailing_zeros(v)
end
r = u << k
# T(r) would throw InexactError; we want OverflowError instead
r > typemax(T) && throw1(a, b)
r % T
end
"""
lcm(x,y)
Least common (non-negative) multiple.
# Examples
```jldoctest
julia> lcm(2,3)
6
julia> lcm(-2,3)
6
```
"""
function lcm(a::T, b::T) where T<:Integer
# explicit a==0 test is to handle case of lcm(0,0) correctly
if a == 0
return a
else
return checked_abs(a * div(b, gcd(b,a)))
end
end
gcd(a::Integer) = a
lcm(a::Integer) = a
gcd(a::Integer, b::Integer) = gcd(promote(a,b)...)
lcm(a::Integer, b::Integer) = lcm(promote(a,b)...)
gcd(a::Integer, b::Integer...) = gcd(a, gcd(b...))
lcm(a::Integer, b::Integer...) = lcm(a, lcm(b...))
gcd(abc::AbstractArray{<:Integer}) = reduce(gcd,abc)
lcm(abc::AbstractArray{<:Integer}) = reduce(lcm,abc)
# return (gcd(a,b),x,y) such that ax+by == gcd(a,b)
"""
gcdx(x,y)
Computes the greatest common (positive) divisor of `x` and `y` and their Bézout
coefficients, i.e. the integer coefficients `u` and `v` that satisfy
``ux+vy = d = gcd(x,y)``. ``gcdx(x,y)`` returns ``(d,u,v)``.
# Examples
```jldoctest
julia> gcdx(12, 42)
(6, -3, 1)
julia> gcdx(240, 46)
(2, -9, 47)
```
!!! note
Bézout coefficients are *not* uniquely defined. `gcdx` returns the minimal
Bézout coefficients that are computed by the extended Euclidean algorithm.
(Ref: D. Knuth, TAoCP, 2/e, p. 325, Algorithm X.)
For signed integers, these coefficients `u` and `v` are minimal in
the sense that ``|u| < |y/d|`` and ``|v| < |x/d|``. Furthermore,
the signs of `u` and `v` are chosen so that `d` is positive.
For unsigned integers, the coefficients `u` and `v` might be near
their `typemax`, and the identity then holds only via the unsigned
integers' modulo arithmetic.
"""
function gcdx(a::T, b::T) where T<:Integer
# a0, b0 = a, b
s0, s1 = oneunit(T), zero(T)
t0, t1 = s1, s0
# The loop invariant is: s0*a0 + t0*b0 == a
while b != 0
q = div(a, b)
a, b = b, rem(a, b)
s0, s1 = s1, s0 - q*s1
t0, t1 = t1, t0 - q*t1
end
a < 0 ? (-a, -s0, -t0) : (a, s0, t0)
end
gcdx(a::Integer, b::Integer) = gcdx(promote(a,b)...)
# multiplicative inverse of n mod m, error if none
"""
invmod(x,m)
Take the inverse of `x` modulo `m`: `y` such that ``x y = 1 \\pmod m``,
with ``div(x,y) = 0``. This is undefined for ``m = 0``, or if
``gcd(x,m) \\neq 1``.
# Examples
```jldoctest
julia> invmod(2,5)
3
julia> invmod(2,3)
2
julia> invmod(5,6)
5
```
"""
function invmod(n::T, m::T) where T<:Integer
g, x, y = gcdx(n, m)
g != 1 && throw(DomainError((n, m), "Greatest common divisor is $g."))
m == 0 && throw(DomainError(m, "`m` must not be 0."))
# Note that m might be negative here.
# For unsigned T, x might be close to typemax; add m to force a wrap-around.
r = mod(x + m, m)
# The postcondition is: mod(r * n, m) == mod(T(1), m) && div(r, m) == 0
r
end
invmod(n::Integer, m::Integer) = invmod(promote(n,m)...)
# ^ for any x supporting *
to_power_type(x) = convert(promote_op(*, typeof(x), typeof(x)), x)
@noinline throw_domerr_powbysq(::Any, p) = throw(DomainError(p,
string("Cannot raise an integer x to a negative power ", p, '.',
"\nConvert input to float.")))
@noinline throw_domerr_powbysq(::Integer, p) = throw(DomainError(p,
string("Cannot raise an integer x to a negative power ", p, '.',
"\nMake x a float by adding a zero decimal (e.g., 2.0^$p instead ",
"of 2^$p), or write 1/x^$(-p), float(x)^$p, or (x//1)^$p")))
@noinline throw_domerr_powbysq(::AbstractMatrix, p) = throw(DomainError(p,
string("Cannot raise an integer matrix x to a negative power ", p, '.',
"\nMake x a float matrix by adding a zero decimal ",
"(e.g., [2.0 1.0;1.0 0.0]^$p instead ",
"of [2 1;1 0]^$p), or write float(x)^$p or Rational.(x)^$p")))
function power_by_squaring(x_, p::Integer)
x = to_power_type(x_)
if p == 1
return copy(x)
elseif p == 0
return one(x)
elseif p == 2
return x*x
elseif p < 0
isone(x) && return copy(x)
isone(-x) && return iseven(p) ? one(x) : copy(x)
throw_domerr_powbysq(x, p)
end
t = trailing_zeros(p) + 1
p >>= t
while (t -= 1) > 0
x *= x
end
y = x
while p > 0
t = trailing_zeros(p) + 1
p >>= t
while (t -= 1) >= 0
x *= x
end
y *= x
end
return y
end
power_by_squaring(x::Bool, p::Unsigned) = ((p==0) | x)
function power_by_squaring(x::Bool, p::Integer)
p < 0 && !x && throw_domerr_powbysq(x, p)
return (p==0) | x
end
^(x::T, p::T) where {T<:Integer} = power_by_squaring(x,p)
^(x::Number, p::Integer) = power_by_squaring(x,p)
^(x, p::Integer) = power_by_squaring(x,p)
# x^p for any literal integer p is lowered to Base.literal_pow(^, x, Val(p))
# to enable compile-time optimizations specialized to p.
# However, we still need a fallback that calls the function ^ which may either
# mean Base.^ or something else, depending on context.
# We mark these @inline since if the target is marked @inline,
# we want to make sure that gets propagated,
# even if it is over the inlining threshold.
@inline literal_pow(f, x, ::Val{p}) where {p} = f(x,p)
# Restrict inlining to hardware-supported arithmetic types, which
# are fast enough to benefit from inlining.
const HWReal = Union{Int8,Int16,Int32,Int64,UInt8,UInt16,UInt32,UInt64,Float32,Float64}
const HWNumber = Union{HWReal, Complex{<:HWReal}, Rational{<:HWReal}}
# inference.jl has complicated logic to inline x^2 and x^3 for
# numeric types. In terms of Val we can do it much more simply.
# (The first argument prevents unexpected behavior if a function ^
# is defined that is not equal to Base.^)
@inline literal_pow(::typeof(^), x::HWNumber, ::Val{0}) = one(x)
@inline literal_pow(::typeof(^), x::HWNumber, ::Val{1}) = x
@inline literal_pow(::typeof(^), x::HWNumber, ::Val{2}) = x*x
@inline literal_pow(::typeof(^), x::HWNumber, ::Val{3}) = x*x*x
# b^p mod m
"""
powermod(x::Integer, p::Integer, m)
Compute ``x^p \\pmod m``.
# Examples
```jldoctest
julia> powermod(2, 6, 5)
4
julia> mod(2^6, 5)
4
julia> powermod(5, 2, 20)
5
julia> powermod(5, 2, 19)
6
julia> powermod(5, 3, 19)
11
```
"""
function powermod(x::Integer, p::Integer, m::T) where T<:Integer
p < 0 && return powermod(invmod(x, m), -p, m)
p == 0 && return mod(one(m),m)
(m == 1 || m == -1) && return zero(m)
b = oftype(m,mod(x,m)) # this also checks for divide by zero
t = prevpow2(p)
local r::T
r = 1
while true
if p >= t
r = mod(widemul(r,b),m)
p -= t
end
t >>>= 1
t <= 0 && break
r = mod(widemul(r,r),m)
end
return r
end
# optimization: promote the modulus m to BigInt only once (cf. widemul in generic powermod above)
powermod(x::Integer, p::Integer, m::Union{Int128,UInt128}) = oftype(m, powermod(x, p, big(m)))
# smallest power of 2 >= x
"""
nextpow2(n::Integer)
The smallest power of two not less than `n`. Returns 0 for `n==0`, and returns
`-nextpow2(-n)` for negative arguments.
# Examples
```jldoctest
julia> nextpow2(16)
16
julia> nextpow2(17)
32
```
"""
nextpow2(x::Unsigned) = oneunit(x)<<((sizeof(x)<<3)-leading_zeros(x-oneunit(x)))
nextpow2(x::Integer) = reinterpret(typeof(x),x < 0 ? -nextpow2(unsigned(-x)) : nextpow2(unsigned(x)))
"""
prevpow2(n::Integer)
The largest power of two not greater than `n`. Returns 0 for `n==0`, and returns
`-prevpow2(-n)` for negative arguments.
# Examples
```jldoctest
julia> prevpow2(5)
4
julia> prevpow2(0)
0
```
"""
prevpow2(x::Unsigned) = one(x) << unsigned((sizeof(x)<<3)-leading_zeros(x)-1)
prevpow2(x::Integer) = reinterpret(typeof(x),x < 0 ? -prevpow2(unsigned(-x)) : prevpow2(unsigned(x)))
"""
ispow2(n::Integer) -> Bool
Test whether `n` is a power of two.
# Examples
```jldoctest
julia> ispow2(4)
true
julia> ispow2(5)
false
```
"""
ispow2(x::Integer) = x > 0 && count_ones(x) == 1
"""
nextpow(a, x)
The smallest `a^n` not less than `x`, where `n` is a non-negative integer. `a` must be
greater than 1, and `x` must be greater than 0.
# Examples
```jldoctest
julia> nextpow(2, 7)
8
julia> nextpow(2, 9)
16
julia> nextpow(5, 20)
25
julia> nextpow(4, 16)
16
```
See also [`prevpow`](@ref).
"""
function nextpow(a::Real, x::Real)
a <= 1 && throw(DomainError(a, "`a` must be greater than 1."))
x <= 0 && throw(DomainError(x, "`x` must be positive."))
x <= 1 && return one(a)
n = ceil(Integer,log(a, x))
p = a^(n-1)
# guard against roundoff error, e.g., with a=5 and x=125
p >= x ? p : a^n
end
"""
prevpow(a, x)
The largest `a^n` not greater than `x`, where `n` is a non-negative integer.
`a` must be greater than 1, and `x` must not be less than 1.
# Examples
```jldoctest
julia> prevpow(2, 7)
4
julia> prevpow(2, 9)
8
julia> prevpow(5, 20)
5
julia> prevpow(4, 16)
16
```
See also [`nextpow`](@ref).
"""
function prevpow(a::Real, x::Real)
a <= 1 && throw(DomainError(a, "`a` must be greater than 1."))
x < 1 && throw(DomainError(x, "`x` must be ≥ 1."))
n = floor(Integer,log(a, x))
p = a^(n+1)
p <= x ? p : a^n
end
## ndigits (number of digits) in base 10 ##
# decimal digits in an unsigned integer
const powers_of_ten = [
0x0000000000000001, 0x000000000000000a, 0x0000000000000064, 0x00000000000003e8,
0x0000000000002710, 0x00000000000186a0, 0x00000000000f4240, 0x0000000000989680,
0x0000000005f5e100, 0x000000003b9aca00, 0x00000002540be400, 0x000000174876e800,
0x000000e8d4a51000, 0x000009184e72a000, 0x00005af3107a4000, 0x00038d7ea4c68000,
0x002386f26fc10000, 0x016345785d8a0000, 0x0de0b6b3a7640000, 0x8ac7230489e80000,
]
function ndigits0z(x::Base.BitUnsigned64)
lz = (sizeof(x)<<3)-leading_zeros(x)
nd = (1233*lz)>>12+1
nd -= x < powers_of_ten[nd]
end
function ndigits0z(x::UInt128)
n = 0
while x > 0x8ac7230489e80000
x = div(x,0x8ac7230489e80000)
n += 19
end
return n + ndigits0z(UInt64(x))
end
ndigits0z(x::BitSigned) = ndigits0z(unsigned(abs(x)))
ndigits0z(x::Integer) = ndigits0zpb(x, 10)
# TODO (when keywords args are fast): rename to ndigits and make pad a keyword
ndigits10(x::Integer, pad::Int=1) = max(pad, ndigits0z(x))
ndigits(x::Integer) = iszero(x) ? 1 : ndigits0z(x)
## ndigits with specified base ##
# The suffix "nb" stands for "negative base"
function ndigits0znb(x::Integer, b::Integer)
# precondition: b < -1 && !(typeof(x) <: Unsigned)
d = 0
while x != 0
x = cld(x,b)
d += 1
end
return d
end
ndigits0znb(x::Unsigned, b::Integer) = ndigits0znb(signed(x), b)
ndigits0znb(x::Bool, b::Integer) = x % Int
# The suffix "pb" stands for "positive base"
# TODO: allow b::Integer
function ndigits0zpb(x::Base.BitUnsigned, b::Int)
# precondition: b > 1
x == 0 && return 0
b < 0 && return ndigits0znb(signed(x), b)
b == 2 && return sizeof(x)<<3 - leading_zeros(x)
b == 8 && return (sizeof(x)<<3 - leading_zeros(x) + 2) ÷ 3
b == 16 && return sizeof(x)<<1 - leading_zeros(x)>>2
b == 10 && return ndigits0z(x)
d = 0
while x > typemax(Int)
x = div(x,b)
d += 1
end
x = div(x,b)
d += 1
m = 1
while m <= x
m *= b
d += 1
end
return d
end
ndigits0zpb(x::Base.BitSigned, b::Integer) = ndigits0zpb(unsigned(abs(x)), Int(b))
ndigits0zpb(x::Base.BitUnsigned, b::Integer) = ndigits0zpb(x, Int(b))
ndigits0zpb(x::Bool, b::Integer) = x % Int
# The suffix "0z" means that the output is 0 on input zero (cf. #16841)
"""
ndigits0z(n::Integer, b::Integer=10)
Return 0 if `n == 0`, otherwise compute the number of digits in
integer `n` written in base `b` (i.e. equal to `ndigits(n, b)`
in this case).
The base `b` must not be in `[-1, 0, 1]`.
# Examples
```jldoctest
julia> Base.ndigits0z(0, 16)
0
julia> Base.ndigits(0, 16)
1
julia> Base.ndigits0z(0)
0
julia> Base.ndigits0z(10, 2)
4
julia> Base.ndigits0z(10)
2
```
See also [`ndigits`](@ref).
"""
function ndigits0z(x::Integer, b::Integer)
if b < -1
ndigits0znb(x, b)
elseif b > 1
ndigits0zpb(x, b)
else
throw(DomainError(b, "The base `b` must not be in `[-1, 0, 1]`."))
end
end
"""
ndigits(n::Integer, b::Integer=10)
Compute the number of digits in integer `n` written in base `b`.
The base `b` must not be in `[-1, 0, 1]`.
# Examples
```jldoctest
julia> ndigits(12345)
5
julia> ndigits(1022, 16)
3
julia> base(16, 1022)
"3fe"
```
"""
ndigits(x::Integer, b::Integer, pad::Int=1) = max(pad, ndigits0z(x, b))
## integer to string functions ##
string(x::Union{Int8,Int16,Int32,Int64,Int128}) = dec(x)
function bin(x::Unsigned, pad::Int, neg::Bool)
i = neg + max(pad,sizeof(x)<<3-leading_zeros(x))
a = StringVector(i)
while i > neg
a[i] = '0'+(x&0x1)
x >>= 1
i -= 1
end
if neg; a[1]='-'; end
String(a)
end
function oct(x::Unsigned, pad::Int, neg::Bool)
i = neg + max(pad,div((sizeof(x)<<3)-leading_zeros(x)+2,3))
a = StringVector(i)
while i > neg
a[i] = '0'+(x&0x7)
x >>= 3
i -= 1
end
if neg; a[1]='-'; end
String(a)
end
function dec(x::Unsigned, pad::Int, neg::Bool)
i = neg + ndigits10(x, pad)
a = StringVector(i)
while i > neg
a[i] = '0'+rem(x,10)
x = oftype(x,div(x,10))
i -= 1
end
if neg; a[1]='-'; end
String(a)
end
function hex(x::Unsigned, pad::Int, neg::Bool)
i = neg + max(pad,(sizeof(x)<<1)-(leading_zeros(x)>>2))
a = StringVector(i)
while i > neg
d = x & 0xf
a[i] = '0'+d+39*(d>9)
x >>= 4
i -= 1
end
if neg; a[1]='-'; end
String(a)
end
const base36digits = ['0':'9';'a':'z']
const base62digits = ['0':'9';'A':'Z';'a':'z']
function base(b::Int, x::Integer, pad::Int, neg::Bool)
(x >= 0) | (b < 0) || throw(DomainError(x, "For negative `x`, `b` must be negative."))
2 <= abs(b) <= 62 || throw(ArgumentError("base must satisfy 2 ≤ abs(base) ≤ 62, got $b"))
digits = abs(b) <= 36 ? base36digits : base62digits
i = neg + ndigits(x, b, pad)
a = StringVector(i)
@inbounds while i > neg
if b > 0
a[i] = digits[1+rem(x,b)]
x = div(x,b)
else
a[i] = digits[1+mod(x,-b)]
x = cld(x,b)
end
i -= 1
end
if neg; a[1]='-'; end
String(a)
end
"""
base(base::Integer, n::Integer, pad::Integer=1)
Convert an integer `n` to a string in the given `base`,
optionally specifying a number of digits to pad to.
```jldoctest
julia> base(13,5,4)
"0005"
julia> base(5,13,4)
"0023"
```
"""
base(b::Integer, n::Integer, pad::Integer=1) =
base(Int(b), b > 0 ? unsigned(abs(n)) : convert(Signed, n), Int(pad), (b>0) & (n<0))
for sym in (:bin, :oct, :dec, :hex)
@eval begin
($sym)(x::Unsigned, p::Int) = ($sym)(x,p,false)
($sym)(x::Unsigned) = ($sym)(x,1,false)
($sym)(x::Char, p::Int) = ($sym)(unsigned(x),p,false)
($sym)(x::Char) = ($sym)(unsigned(x),1,false)
($sym)(x::Integer, p::Int) = ($sym)(unsigned(abs(x)),p,x<0)
($sym)(x::Integer) = ($sym)(unsigned(abs(x)),1,x<0)
end
end
"""
bin(n, pad::Int=1)
Convert an integer to a binary string, optionally specifying a number of digits to pad to.
```jldoctest
julia> bin(10,2)
"1010"
julia> bin(10,8)
"00001010"
```
"""
bin
"""
hex(n, pad::Int=1)
Convert an integer to a hexadecimal string, optionally specifying a number of
digits to pad to.
```jldoctest
julia> hex(20)
"14"
julia> hex(20, 3)
"014"
```
"""
hex
"""
oct(n, pad::Int=1)
Convert an integer to an octal string, optionally specifying a number of digits
to pad to.
```jldoctest
julia> oct(20)
"24"
julia> oct(20, 3)
"024"
```
"""
oct
"""
dec(n, pad::Int=1)
Convert an integer to a decimal string, optionally specifying a number of digits
to pad to.
# Examples
```jldoctest
julia> dec(20)
"20"
julia> dec(20, 3)
"020"
```
"""
dec
"""
bits(n)
A string giving the literal bit representation of a number.
# Examples
```jldoctest
julia> bits(4)
"0000000000000000000000000000000000000000000000000000000000000100"
julia> bits(2.2)
"0100000000000001100110011001100110011001100110011001100110011010"
```
"""
function bits end
bits(x::Union{Bool,Int8,UInt8}) = bin(reinterpret(UInt8,x),8)
bits(x::Union{Int16,UInt16,Float16}) = bin(reinterpret(UInt16,x),16)
bits(x::Union{Char,Int32,UInt32,Float32}) = bin(reinterpret(UInt32,x),32)
bits(x::Union{Int64,UInt64,Float64}) = bin(reinterpret(UInt64,x),64)
bits(x::Union{Int128,UInt128}) = bin(reinterpret(UInt128,x),128)
"""
digits([T<:Integer], n::Integer, base::T=10, pad::Integer=1)
Returns an array with element type `T` (default `Int`) of the digits of `n` in the given
base, optionally padded with zeros to a specified size. More significant digits are at
higher indexes, such that `n == sum([digits[k]*base^(k-1) for k=1:length(digits)])`.
# Examples
```jldoctest
julia> digits(10, 10)
2-element Array{Int64,1}:
0
1
julia> digits(10, 2)
4-element Array{Int64,1}:
0
1
0
1
julia> digits(10, 2, 6)
6-element Array{Int64,1}:
0
1
0
1
0
0
```
"""
digits(n::Integer, base::T=10, pad::Integer=1) where {T<:Integer} = digits(T, n, base, pad)
function digits(T::Type{<:Integer}, n::Integer, base::Integer=10, pad::Integer=1)
digits!(zeros(T, ndigits(n, base, pad)), n, base)
end
"""
hastypemax(T::Type) -> Bool
Return `true` if and only if `typemax(T)` is defined.
"""
hastypemax(::Base.BitIntegerType) = true
hastypemax(::Type{T}) where {T} = applicable(typemax, T)
"""
digits!(array, n::Integer, base::Integer=10)
Fills an array of the digits of `n` in the given base. More significant digits are at higher
indexes. If the array length is insufficient, the least significant digits are filled up to
the array length. If the array length is excessive, the excess portion is filled with zeros.
# Examples
```jldoctest
julia> digits!([2,2,2,2], 10, 2)
4-element Array{Int64,1}:
0
1
0
1
julia> digits!([2,2,2,2,2,2], 10, 2)
6-element Array{Int64,1}:
0
1
0
1
0
0
```
"""
function digits!(a::AbstractVector{T}, n::Integer, base::Integer=10) where T<:Integer
base < 0 && isa(n, Unsigned) && return digits!(a, convert(Signed, n), base)
2 <= abs(base) || throw(ArgumentError("base must be ≥ 2 or ≤ -2, got $base"))
hastypemax(T) && abs(base) - 1 > typemax(T) &&
throw(ArgumentError("type $T too small for base $base"))
for i in eachindex(a)
if base > 0
a[i] = rem(n, base)
n = div(n, base)
else
a[i] = mod(n, -base)
n = cld(n, base)
end
end
return a
end
"""
isqrt(n::Integer)
Integer square root: the largest integer `m` such that `m*m <= n`.
```jldoctest
julia> isqrt(5)
2
```
"""
isqrt(x::Integer) = oftype(x, trunc(sqrt(x)))
function isqrt(x::Union{Int64,UInt64,Int128,UInt128})
x==0 && return x
s = oftype(x, trunc(sqrt(x)))
# fix with a Newton iteration, since conversion to float discards
# too many bits.
s = (s + div(x,s)) >> 1
s*s > x ? s-1 : s
end
function factorial(n::Integer)
n < 0 && throw(DomainError(n, "`n` must be nonnegative."))
f::typeof(n*n) = 1
for i::typeof(n*n) = 2:n
f *= i
end
return f
end
"""
binomial(n, k)
Number of ways to choose `k` out of `n` items.
# Examples
```jldoctest
julia> binomial(5, 3)
10
julia> factorial(5) ÷ (factorial(5-3) * factorial(3))
10
```
"""
function binomial(n::T, k::T) where T<:Integer
n0, k0 = n, k
k < 0 && return zero(T)
sgn = one(T)
if n < 0
n = -n + k -1
if isodd(k)
sgn = -sgn
end
end
k > n && return zero(T)
(k == 0 || k == n) && return sgn
k == 1 && return sgn*n
if k > (n>>1)
k = (n - k)
end
x::T = nn = n - k + 1
nn += 1
rr = 2
while rr <= k
xt = div(widemul(x, nn), rr)
x = xt
x == xt || throw(OverflowError("binomial($n0, $k0) overflows"))
rr += 1
nn += 1
end
convert(T, copysign(x, sgn))
end