# This file is a part of Julia. License is MIT: https://julialang.org/license
module GMP
export BigInt
import .Base: *, +, -, /, <, <<, >>, >>>, <=, ==, >, >=, ^, (~), (&), (|), xor, nand, nor,
binomial, cmp, convert, div, divrem, factorial, cld, fld, gcd, gcdx, lcm, mod,
ndigits, promote_rule, rem, show, isqrt, string, powermod, sum, prod,
trailing_zeros, trailing_ones, count_ones, count_zeros, tryparse_internal,
bin, oct, dec, hex, isequal, invmod, _prevpow2, _nextpow2, ndigits0zpb,
widen, signed, unsafe_trunc, trunc, iszero, isone, big, flipsign, signbit,
sign, hastypemax, isodd, iseven, digits!, hash, hash_integer
if Clong == Int32
const ClongMax = Union{Int8, Int16, Int32}
const CulongMax = Union{UInt8, UInt16, UInt32}
else
const ClongMax = Union{Int8, Int16, Int32, Int64}
const CulongMax = Union{UInt8, UInt16, UInt32, UInt64}
end
const CdoubleMax = Union{Float16, Float32, Float64}
version() = VersionNumber(unsafe_string(unsafe_load(cglobal((:__gmp_version, :libgmp), Ptr{Cchar}))))
bits_per_limb() = Int(unsafe_load(cglobal((:__gmp_bits_per_limb, :libgmp), Cint)))
const VERSION = version()
const BITS_PER_LIMB = bits_per_limb()
# GMP's mp_limb_t is by default a typedef of `unsigned long`, but can also be configured to be either
# `unsigned int` or `unsigned long long int`. The correct unsigned type is here named Limb, and must
# be used whenever mp_limb_t is in the signature of ccall'ed GMP functions.
if BITS_PER_LIMB == 32
const Limb = UInt32
const SLimbMax = Union{Int8, Int16, Int32}
const ULimbMax = Union{UInt8, UInt16, UInt32}
elseif BITS_PER_LIMB == 64
const Limb = UInt64
const SLimbMax = Union{Int8, Int16, Int32, Int64}
const ULimbMax = Union{UInt8, UInt16, UInt32, UInt64}
else
error("GMP: cannot determine the type mp_limb_t (__gmp_bits_per_limb == $BITS_PER_LIMB)")
end
"""
BigInt <: Signed
Arbitrary precision integer type.
"""
mutable struct BigInt <: Signed
alloc::Cint
size::Cint
d::Ptr{Limb}
function BigInt(; nbits::Integer=0)
b = MPZ.init2!(new(), nbits)
finalizer(cglobal((:__gmpz_clear, :libgmp)), b)
return b
end
end
"""
BigInt(x)
Create an arbitrary precision integer. `x` may be an `Int` (or anything that can be
converted to an `Int`). The usual mathematical operators are defined for this type, and
results are promoted to a [`BigInt`](@ref).
Instances can be constructed from strings via [`parse`](@ref), or using the `big`
string literal.
# Examples
```jldoctest
julia> parse(BigInt, "42")
42
julia> big"313"
313
julia> BigInt(10)^19
10000000000000000000
```
"""
BigInt(x)
"""
ALLOC_OVERFLOW_FUNCTION
A reference that holds a boolean, if true, indicating julia is linked with a patched GMP that
does not abort on huge allocation and throws OutOfMemoryError instead.
"""
const ALLOC_OVERFLOW_FUNCTION = Ref(false)
function __init__()
try
if version().major != VERSION.major || bits_per_limb() != BITS_PER_LIMB
msg = """The dynamically loaded GMP library (v\"$(version())\" with __gmp_bits_per_limb == $(bits_per_limb()))
does not correspond to the compile time version (v\"$VERSION\" with __gmp_bits_per_limb == $BITS_PER_LIMB).
Please rebuild Julia."""
bits_per_limb() != BITS_PER_LIMB ? @error(msg) : @warn(msg)
end
ccall((:__gmp_set_memory_functions, :libgmp), Cvoid,
(Ptr{Cvoid},Ptr{Cvoid},Ptr{Cvoid}),
cglobal(:jl_gc_counted_malloc),
cglobal(:jl_gc_counted_realloc_with_old_size),
cglobal(:jl_gc_counted_free_with_size))
ZERO.alloc, ZERO.size, ZERO.d = 0, 0, C_NULL
ONE.alloc, ONE.size, ONE.d = 1, 1, pointer(_ONE)
catch ex
Base.showerror_nostdio(ex, "WARNING: Error during initialization of module GMP")
end
# This only works with a patched version of GMP, ignore otherwise
try
ccall((:__gmp_set_alloc_overflow_function, :libgmp), Cvoid,
(Ptr{Cvoid},),
cglobal(:jl_throw_out_of_memory_error))
ALLOC_OVERFLOW_FUNCTION[] = true
catch ex
# ErrorException("ccall: could not find function...")
if typeof(ex) != ErrorException
rethrow()
end
end
end
module MPZ
# wrapping of libgmp functions
# - "output parameters" are labeled x, y, z, and are returned when appropriate
# - constant input parameters are labeled a, b, c
# - a method modifying its input has a "!" appendend to its name, according to Julia's conventions
# - some convenient methods are added (in addition to the pure MPZ ones), e.g. `add(a, b) = add!(BigInt(), a, b)`
# and `add!(x, a) = add!(x, x, a)`.
using ..GMP: BigInt, Limb, BITS_PER_LIMB
const mpz_t = Ref{BigInt}
const bitcnt_t = Culong
gmpz(op::Symbol) = (Symbol(:__gmpz_, op), :libgmp)
init!(x::BigInt) = (ccall((:__gmpz_init, :libgmp), Cvoid, (mpz_t,), x); x)
init2!(x::BigInt, a) = (ccall((:__gmpz_init2, :libgmp), Cvoid, (mpz_t, bitcnt_t), x, a); x)
realloc2!(x, a) = (ccall((:__gmpz_realloc2, :libgmp), Cvoid, (mpz_t, bitcnt_t), x, a); x)
realloc2(a) = realloc2!(BigInt(), a)
sizeinbase(a::BigInt, b) = Int(ccall((:__gmpz_sizeinbase, :libgmp), Csize_t, (mpz_t, Cint), a, b))
for (op, nbits) in (:add => :(BITS_PER_LIMB*(1 + max(abs(a.size), abs(b.size)))),
:sub => :(BITS_PER_LIMB*(1 + max(abs(a.size), abs(b.size)))),
:mul => 0, :fdiv_q => 0, :tdiv_q => 0, :cdiv_q => 0,
:fdiv_r => 0, :tdiv_r => 0, :cdiv_r => 0,
:gcd => 0, :lcm => 0, :and => 0, :ior => 0, :xor => 0)
op! = Symbol(op, :!)
@eval begin
$op!(x::BigInt, a::BigInt, b::BigInt) = (ccall($(gmpz(op)), Cvoid, (mpz_t, mpz_t, mpz_t), x, a, b); x)
$op(a::BigInt, b::BigInt) = $op!(BigInt(nbits=$nbits), a, b)
$op!(x::BigInt, b::BigInt) = $op!(x, x, b)
end
end
invert!(x::BigInt, a::BigInt, b::BigInt) =
ccall((:__gmpz_invert, :libgmp), Cint, (mpz_t, mpz_t, mpz_t), x, a, b)
invert(a::BigInt, b::BigInt) = invert!(BigInt(), a, b)
invert!(x::BigInt, b::BigInt) = invert!(x, x, b)
for op in (:add_ui, :sub_ui, :mul_ui, :mul_2exp, :fdiv_q_2exp, :pow_ui, :bin_ui)
op! = Symbol(op, :!)
@eval begin
$op!(x::BigInt, a::BigInt, b) = (ccall($(gmpz(op)), Cvoid, (mpz_t, mpz_t, Culong), x, a, b); x)
$op(a::BigInt, b) = $op!(BigInt(), a, b)
$op!(x::BigInt, b) = $op!(x, x, b)
end
end
ui_sub!(x::BigInt, a, b::BigInt) = (ccall((:__gmpz_ui_sub, :libgmp), Cvoid, (mpz_t, Culong, mpz_t), x, a, b); x)
ui_sub(a, b::BigInt) = ui_sub!(BigInt(), a, b)
for op in (:scan1, :scan0)
# when there is no meaningful answer, ccall returns typemax(Culong), where Culong can
# be UInt32 (Windows) or UInt64; we return -1 in this case for all architectures
@eval $op(a::BigInt, b) = Int(signed(ccall($(gmpz(op)), Culong, (mpz_t, Culong), a, b)))
end
mul_si!(x::BigInt, a::BigInt, b) = (ccall((:__gmpz_mul_si, :libgmp), Cvoid, (mpz_t, mpz_t, Clong), x, a, b); x)
mul_si(a::BigInt, b) = mul_si!(BigInt(), a, b)
mul_si!(x::BigInt, b) = mul_si!(x, x, b)
for op in (:neg, :com, :sqrt, :set)
op! = Symbol(op, :!)
@eval begin
$op!(x::BigInt, a::BigInt) = (ccall($(gmpz(op)), Cvoid, (mpz_t, mpz_t), x, a); x)
$op(a::BigInt) = $op!(BigInt(), a)
end
op === :set && continue # MPZ.set!(x) would make no sense
@eval $op!(x::BigInt) = $op!(x, x)
end
for (op, T) in ((:fac_ui, Culong), (:set_ui, Culong), (:set_si, Clong), (:set_d, Cdouble))
op! = Symbol(op, :!)
@eval begin
$op!(x::BigInt, a) = (ccall($(gmpz(op)), Cvoid, (mpz_t, $T), x, a); x)
$op(a) = $op!(BigInt(), a)
end
end
popcount(a::BigInt) = Int(signed(ccall((:__gmpz_popcount, :libgmp), Culong, (mpz_t,), a)))
mpn_popcount(d::Ptr{Limb}, s::Integer) = Int(ccall((:__gmpn_popcount, :libgmp), Culong, (Ptr{Limb}, Csize_t), d, s))
mpn_popcount(a::BigInt) = mpn_popcount(a.d, abs(a.size))
function tdiv_qr!(x::BigInt, y::BigInt, a::BigInt, b::BigInt)
ccall((:__gmpz_tdiv_qr, :libgmp), Cvoid, (mpz_t, mpz_t, mpz_t, mpz_t), x, y, a, b)
x, y
end
tdiv_qr(a::BigInt, b::BigInt) = tdiv_qr!(BigInt(), BigInt(), a, b)
powm!(x::BigInt, a::BigInt, b::BigInt, c::BigInt) =
(ccall((:__gmpz_powm, :libgmp), Cvoid, (mpz_t, mpz_t, mpz_t, mpz_t), x, a, b, c); x)
powm(a::BigInt, b::BigInt, c::BigInt) = powm!(BigInt(), a, b, c)
powm!(x::BigInt, b::BigInt, c::BigInt) = powm!(x, x, b, c)
function gcdext!(x::BigInt, y::BigInt, z::BigInt, a::BigInt, b::BigInt)
ccall((:__gmpz_gcdext, :libgmp), Cvoid, (mpz_t, mpz_t, mpz_t, mpz_t, mpz_t), x, y, z, a, b)
x, y, z
end
gcdext(a::BigInt, b::BigInt) = gcdext!(BigInt(), BigInt(), BigInt(), a, b)
cmp(a::BigInt, b::BigInt) = Int(ccall((:__gmpz_cmp, :libgmp), Cint, (mpz_t, mpz_t), a, b))
cmp_si(a::BigInt, b) = Int(ccall((:__gmpz_cmp_si, :libgmp), Cint, (mpz_t, Clong), a, b))
cmp_ui(a::BigInt, b) = Int(ccall((:__gmpz_cmp_ui, :libgmp), Cint, (mpz_t, Culong), a, b))
cmp_d(a::BigInt, b) = Int(ccall((:__gmpz_cmp_d, :libgmp), Cint, (mpz_t, Cdouble), a, b))
mpn_cmp(a::Ptr{Limb}, b::Ptr{Limb}, c) = ccall((:__gmpn_cmp, :libgmp), Cint, (Ptr{Limb}, Ptr{Limb}, Clong), a, b, c)
mpn_cmp(a::BigInt, b::BigInt, c) = mpn_cmp(a.d, b.d, c)
get_str!(x, a, b::BigInt) = (ccall((:__gmpz_get_str,:libgmp), Ptr{Cchar}, (Ptr{Cchar}, Cint, mpz_t), x, a, b); x)
set_str!(x::BigInt, a, b) = Int(ccall((:__gmpz_set_str, :libgmp), Cint, (mpz_t, Ptr{UInt8}, Cint), x, a, b))
get_d(a::BigInt) = ccall((:__gmpz_get_d, :libgmp), Cdouble, (mpz_t,), a)
limbs_write!(x::BigInt, a) = ccall((:__gmpz_limbs_write, :libgmp), Ptr{Limb}, (mpz_t, Clong), x, a)
limbs_finish!(x::BigInt, a) = ccall((:__gmpz_limbs_finish, :libgmp), Cvoid, (mpz_t, Clong), x, a)
import!(x::BigInt, a, b, c, d, e, f) = ccall((:__gmpz_import, :libgmp), Cvoid,
(mpz_t, Csize_t, Cint, Csize_t, Cint, Csize_t, Ptr{Cvoid}), x, a, b, c, d, e, f)
setbit!(x, a) = (ccall((:__gmpz_setbit, :libgmp), Cvoid, (mpz_t, bitcnt_t), x, a); x)
tstbit(a::BigInt, b) = ccall((:__gmpz_tstbit, :libgmp), Cint, (mpz_t, bitcnt_t), a, b) % Bool
end # module MPZ
const ZERO = BigInt()
const ONE = BigInt()
const _ONE = Limb[1]
widen(::Type{Int128}) = BigInt
widen(::Type{UInt128}) = BigInt
widen(::Type{BigInt}) = BigInt
signed(x::BigInt) = x
BigInt(x::BigInt) = x
Signed(x::BigInt) = x
hastypemax(::Type{BigInt}) = false
function tryparse_internal(::Type{BigInt}, s::AbstractString, startpos::Int, endpos::Int, base_::Integer, raise::Bool)
# don't make a copy in the common case where we are parsing a whole String
bstr = startpos == firstindex(s) && endpos == lastindex(s) ? String(s) : String(SubString(s,startpos,endpos))
sgn, base, i = Base.parseint_preamble(true,Int(base_),bstr,firstindex(bstr),lastindex(bstr))
if !(2 <= base <= 62)
raise && throw(ArgumentError("invalid base: base must be 2 ≤ base ≤ 62, got $base"))
return nothing
end
if i == 0
raise && throw(ArgumentError("premature end of integer: $(repr(bstr))"))
return nothing
end
z = BigInt()
if Base.containsnul(bstr)
err = -1 # embedded NUL char (not handled correctly by GMP)
else
err = GC.@preserve bstr MPZ.set_str!(z, pointer(bstr)+(i-firstindex(bstr)), base)
end
if err != 0
raise && throw(ArgumentError("invalid BigInt: $(repr(bstr))"))
return nothing
end
flipsign!(z, sgn)
end
BigInt(x::Union{Clong,Int32}) = MPZ.set_si(x)
BigInt(x::Union{Culong,UInt32}) = MPZ.set_ui(x)
BigInt(x::Bool) = BigInt(UInt(x))
unsafe_trunc(::Type{BigInt}, x::Union{Float16,Float32,Float64}) = MPZ.set_d(x)
function BigInt(x::Float64)
isinteger(x) || throw(InexactError(:BigInt, BigInt, x))
unsafe_trunc(BigInt,x)
end
function trunc(::Type{BigInt}, x::Union{Float16,Float32,Float64})
isfinite(x) || throw(InexactError(:trunc, BigInt, x))
unsafe_trunc(BigInt,x)
end
BigInt(x::Float16) = BigInt(Float64(x))
BigInt(x::Float32) = BigInt(Float64(x))
function BigInt(x::Integer)
# On 64-bit Windows, `Clong` is `Int32`, not `Int64`, so construction of
# `Int64` constants, e.g. `BigInt(3)`, uses this method.
isbits(x) && typemin(Clong) <= x <= typemax(Clong) && return BigInt((x % Clong)::Clong)
nd = ndigits(x, base=2)
z = MPZ.realloc2(nd)
ux = unsigned(x < 0 ? -x : x)
size = 0
limbnbits = sizeof(Limb) << 3
while nd > 0
size += 1
unsafe_store!(z.d, ux % Limb, size)
ux >>= limbnbits
nd -= limbnbits
end
z.size = x < 0 ? -size : size
z
end
rem(x::BigInt, ::Type{Bool}) = !iszero(x) & unsafe_load(x.d) % Bool # never unsafe here
rem(x::BigInt, ::Type{T}) where T<:Union{SLimbMax,ULimbMax} =
iszero(x) ? zero(T) : flipsign(unsafe_load(x.d) % T, x.size)
function rem(x::BigInt, ::Type{T}) where T<:Union{Base.BitUnsigned,Base.BitSigned}
u = zero(T)
for l = 1:min(abs(x.size), cld(sizeof(T), sizeof(Limb)))
u += (unsafe_load(x.d, l) % T) << ((sizeof(Limb)<<3)*(l-1))
end
flipsign(u, x.size)
end
rem(x::Integer, ::Type{BigInt}) = BigInt(x)
isodd(x::BigInt) = MPZ.tstbit(x, 0)
iseven(x::BigInt) = !isodd(x)
function (::Type{T})(x::BigInt) where T<:Base.BitUnsigned
if sizeof(T) < sizeof(Limb)
convert(T, convert(Limb,x))
else
0 <= x.size <= cld(sizeof(T),sizeof(Limb)) || throw(InexactError(nameof(T), T, x))
x % T
end
end
function (::Type{T})(x::BigInt) where T<:Base.BitSigned
n = abs(x.size)
if sizeof(T) < sizeof(Limb)
SLimb = typeof(Signed(one(Limb)))
convert(T, convert(SLimb, x))
else
0 <= n <= cld(sizeof(T),sizeof(Limb)) || throw(InexactError(nameof(T), T, x))
y = x % T
ispos(x) ⊻ (y > 0) && throw(InexactError(nameof(T), T, x)) # catch overflow
y
end
end
Float64(n::BigInt, ::RoundingMode{:ToZero}) = MPZ.get_d(n)
function (::Type{T})(n::BigInt, ::RoundingMode{:ToZero}) where T<:Union{Float16,Float32}
T(Float64(n,RoundToZero),RoundToZero)
end
function (::Type{T})(n::BigInt, ::RoundingMode{:Down}) where T<:CdoubleMax
x = T(n,RoundToZero)
x > n ? prevfloat(x) : x
end
function (::Type{T})(n::BigInt, ::RoundingMode{:Up}) where T<:CdoubleMax
x = T(n,RoundToZero)
x < n ? nextfloat(x) : x
end
function Float64(x::BigInt, ::RoundingMode{:Nearest})
x == 0 && return 0.0
xsize = abs(x.size)
if xsize*BITS_PER_LIMB > 1024
z = Inf64
elseif xsize == 1
z = Float64(unsafe_load(x.d))
elseif Limb == UInt32 && xsize == 2
z = Float64((unsafe_load(x.d, 2) % UInt64) << BITS_PER_LIMB + unsafe_load(x.d))
else
y1 = unsafe_load(x.d, xsize) % UInt64
n = 64 - leading_zeros(y1)
# load first 54(1 + 52 bits of fraction + 1 for rounding)
y = y1 >> (n - (precision(Float64)+1))
if Limb == UInt64
y += n > precision(Float64) ? 0 : (unsafe_load(x.d, xsize-1) >> (10+n))
else
y += (unsafe_load(x.d, xsize-1) % UInt64) >> (n-22)
y += n > (precision(Float64) - 32) ? 0 : (unsafe_load(x.d, xsize-2) >> (10+n))
end
y = (y + 1) >> 1 # round, ties up
y &= ~UInt64(trailing_zeros(x) == (n-54 + (xsize-1)*BITS_PER_LIMB)) # fix last bit to round to even
d = ((n+1021) % UInt64) << 52
z = reinterpret(Float64, d+y)
z = ldexp(z, (xsize-1)*BITS_PER_LIMB)
end
return flipsign(z, x.size)
end
function Float32(x::BigInt, ::RoundingMode{:Nearest})
x == 0 && return 0f0
xsize = abs(x.size)
if xsize*BITS_PER_LIMB > 128
z = Inf32
elseif xsize == 1
z = Float32(unsafe_load(x.d))
else
y1 = unsafe_load(x.d, xsize)
n = BITS_PER_LIMB - leading_zeros(y1)
# load first 25(1 + 23 bits of fraction + 1 for rounding)
y = (y1 >> (n - (precision(Float32)+1))) % UInt32
y += (n > precision(Float32) ? 0 : unsafe_load(x.d, xsize-1) >> (BITS_PER_LIMB - (25-n))) % UInt32
y = (y + one(UInt32)) >> 1 # round, ties up
y &= ~UInt32(trailing_zeros(x) == (n-25 + (xsize-1)*BITS_PER_LIMB)) # fix last bit to round to even
d = ((n+125) % UInt32) << 23
z = reinterpret(Float32, d+y)
z = ldexp(z, (xsize-1)*BITS_PER_LIMB)
end
return flipsign(z, x.size)
end
function Float16(x::BigInt, ::RoundingMode{:Nearest})
x == 0 && return Float16(0.0)
y1 = unsafe_load(x.d)
n = BITS_PER_LIMB - leading_zeros(y1)
if n > 16 || abs(x.size) > 1
z = Inf16
else
# load first 12(1 + 10 bits for fraction + 1 for rounding)
y = (y1 >> (n - (precision(Float16)+1))) % UInt16
y = (y + one(UInt16)) >> 1 # round, ties up
y &= ~UInt16(trailing_zeros(x) == (n-12)) # fix last bit to round to even
d = ((n+13) % UInt16) << 10
z = reinterpret(Float16, d+y)
end
return flipsign(z, x.size)
end
Float64(n::BigInt) = Float64(n, RoundNearest)
Float32(n::BigInt) = Float32(n, RoundNearest)
Float16(n::BigInt) = Float16(n, RoundNearest)
promote_rule(::Type{BigInt}, ::Type{<:Integer}) = BigInt
"""
big(x)
Convert a number to a maximum precision representation (typically [`BigInt`](@ref) or
`BigFloat`). See [`BigFloat`](@ref BigFloat(::Any, rounding::RoundingMode)) for
information about some pitfalls with floating-point numbers.
"""
function big end
big(::Type{<:Integer}) = BigInt
big(::Type{<:Rational}) = Rational{BigInt}
big(n::Integer) = convert(BigInt, n)
# Binary ops
for (fJ, fC) in ((:+, :add), (:-,:sub), (:*, :mul),
(:mod, :fdiv_r), (:rem, :tdiv_r),
(:gcd, :gcd), (:lcm, :lcm),
(:&, :and), (:|, :ior), (:xor, :xor))
@eval begin
($fJ)(x::BigInt, y::BigInt) = MPZ.$fC(x, y)
end
end
for (r, f) in ((RoundToZero, :tdiv_q),
(RoundDown, :fdiv_q),
(RoundUp, :cdiv_q))
@eval div(x::BigInt, y::BigInt, ::typeof($r)) = MPZ.$f(x, y)
end
# For compat only. Remove in 2.0.
div(x::BigInt, y::BigInt) = div(x, y, RoundToZero)
fld(x::BigInt, y::BigInt) = div(x, y, RoundDown)
cld(x::BigInt, y::BigInt) = div(x, y, RoundUp)
/(x::BigInt, y::BigInt) = float(x)/float(y)
function invmod(x::BigInt, y::BigInt)
z = zero(BigInt)
ya = abs(y)
if ya == 1
return z
end
if (y==0 || MPZ.invert!(z, x, ya) == 0)
throw(DomainError(y))
end
# GMP always returns a positive inverse; we instead want to
# normalize such that div(z, y) == 0, i.e. we want a negative z
# when y is negative.
if y < 0
MPZ.add!(z, y)
end
# The postcondition is: mod(z * x, y) == mod(big(1), m) && div(z, y) == 0
return z
end
# More efficient commutative operations
for (fJ, fC) in ((:+, :add), (:*, :mul), (:&, :and), (:|, :ior), (:xor, :xor))
fC! = Symbol(fC, :!)
@eval begin
($fJ)(a::BigInt, b::BigInt, c::BigInt) = MPZ.$fC!(MPZ.$fC(a, b), c)
($fJ)(a::BigInt, b::BigInt, c::BigInt, d::BigInt) = MPZ.$fC!(MPZ.$fC!(MPZ.$fC(a, b), c), d)
($fJ)(a::BigInt, b::BigInt, c::BigInt, d::BigInt, e::BigInt) =
MPZ.$fC!(MPZ.$fC!(MPZ.$fC!(MPZ.$fC(a, b), c), d), e)
end
end
# Basic arithmetic without promotion
+(x::BigInt, c::CulongMax) = MPZ.add_ui(x, c)
+(c::CulongMax, x::BigInt) = x + c
-(x::BigInt, c::CulongMax) = MPZ.sub_ui(x, c)
-(c::CulongMax, x::BigInt) = MPZ.ui_sub(c, x)
+(x::BigInt, c::ClongMax) = c < 0 ? -(x, -(c % Culong)) : x + convert(Culong, c)
+(c::ClongMax, x::BigInt) = c < 0 ? -(x, -(c % Culong)) : x + convert(Culong, c)
-(x::BigInt, c::ClongMax) = c < 0 ? +(x, -(c % Culong)) : -(x, convert(Culong, c))
-(c::ClongMax, x::BigInt) = c < 0 ? -(x + -(c % Culong)) : -(convert(Culong, c), x)
*(x::BigInt, c::CulongMax) = MPZ.mul_ui(x, c)
*(c::CulongMax, x::BigInt) = x * c
*(x::BigInt, c::ClongMax) = MPZ.mul_si(x, c)
*(c::ClongMax, x::BigInt) = x * c
/(x::BigInt, y::Union{ClongMax,CulongMax}) = float(x)/y
/(x::Union{ClongMax,CulongMax}, y::BigInt) = x/float(y)
# unary ops
(-)(x::BigInt) = MPZ.neg(x)
(~)(x::BigInt) = MPZ.com(x)
<<(x::BigInt, c::UInt) = c == 0 ? x : MPZ.mul_2exp(x, c)
>>(x::BigInt, c::UInt) = c == 0 ? x : MPZ.fdiv_q_2exp(x, c)
>>>(x::BigInt, c::UInt) = x >> c
function trailing_zeros(x::BigInt)
c = MPZ.scan1(x, 0)
c == -1 && throw(DomainError(x, "`x` must be non-zero"))
c
end
function trailing_ones(x::BigInt)
c = MPZ.scan0(x, 0)
c == -1 && throw(DomainError(x, "`x` must not be equal to -1"))
c
end
function count_ones(x::BigInt)
c = MPZ.popcount(x)
c == -1 && throw(DomainError(x, "`x` cannot be negative"))
c
end
# generic definition is not used to provide a better error message
function count_zeros(x::BigInt)
c = MPZ.popcount(~x)
c == -1 && throw(DomainError(x, "`x` must be negative"))
c
end
"""
count_ones_abs(x::BigInt)
Number of ones in the binary representation of abs(x).
"""
count_ones_abs(x::BigInt) = iszero(x) ? 0 : MPZ.mpn_popcount(x)
divrem(x::BigInt, y::BigInt) = MPZ.tdiv_qr(x, y)
divrem(x::BigInt, y::Integer) = MPZ.tdiv_qr(x, big(y))
cmp(x::BigInt, y::BigInt) = sign(MPZ.cmp(x, y))
cmp(x::BigInt, y::ClongMax) = sign(MPZ.cmp_si(x, y))
cmp(x::BigInt, y::CulongMax) = sign(MPZ.cmp_ui(x, y))
cmp(x::BigInt, y::Integer) = cmp(x, big(y))
cmp(x::Integer, y::BigInt) = -cmp(y, x)
cmp(x::BigInt, y::CdoubleMax) = isnan(y) ? -1 : sign(MPZ.cmp_d(x, y))
cmp(x::CdoubleMax, y::BigInt) = -cmp(y, x)
isqrt(x::BigInt) = MPZ.sqrt(x)
^(x::BigInt, y::Culong) = MPZ.pow_ui(x, y)
function bigint_pow(x::BigInt, y::Integer)
if y<0; throw(DomainError(y, "`y` cannot be negative.")); end
@noinline throw1(y) =
throw(OverflowError("exponent $y is too large and computation will overflow"))
if x== 1; return x; end
if x==-1; return isodd(y) ? x : -x; end
if y>typemax(Culong)
x==0 && return x
#At this point, x is not 1, 0 or -1 and it is not possible to use
#gmpz_pow_ui to compute the answer. Note that the magnitude of the
#answer is:
#- at least 2^(2^32-1) ≈ 10^(1.3e9) (if Culong === UInt32).
#- at least 2^(2^64-1) ≈ 10^(5.5e18) (if Culong === UInt64).
#
#Assume that the answer will definitely overflow.
throw1(y)
end
return x^convert(Culong, y)
end
^(x::BigInt , y::BigInt ) = bigint_pow(x, y)
^(x::BigInt , y::Bool ) = y ? x : one(x)
^(x::BigInt , y::Integer) = bigint_pow(x, y)
^(x::Integer, y::BigInt ) = bigint_pow(BigInt(x), y)
^(x::Bool , y::BigInt ) = Base.power_by_squaring(x, y)
function powermod(x::BigInt, p::BigInt, m::BigInt)
r = MPZ.powm(x, p, m)
return m < 0 && r > 0 ? MPZ.add!(r, m) : r # choose sign consistent with mod(x^p, m)
end
powermod(x::Integer, p::Integer, m::BigInt) = powermod(big(x), big(p), m)
function gcdx(a::BigInt, b::BigInt)
if iszero(b) # shortcut this to ensure consistent results with gcdx(a,b)
return a < 0 ? (-a,-ONE,b) : (a,one(BigInt),b)
# we don't return the globals ONE and ZERO in case the user wants to
# mutate the result
end
g, s, t = MPZ.gcdext(a, b)
if t == 0
# work around a difference in some versions of GMP
if a == b
return g, t, s
elseif abs(a)==abs(b)
return g, t, -s
end
end
g, s, t
end
+(x::BigInt, y::BigInt, rest::BigInt...) = sum(tuple(x, y, rest...))
sum(arr::Union{AbstractArray{BigInt}, Tuple{BigInt, Vararg{BigInt}}}) =
foldl(MPZ.add!, arr; init=BigInt(0))
function prod(arr::AbstractArray{BigInt})
# compute first the needed number of bits for the result,
# to avoid re-allocations;
# GMP will always request n+m limbs for the result in MPZ.mul!,
# if the arguments have n and m limbs; so we add all the bits
# taken by the array elements, and add BITS_PER_LIMB to that,
# to account for the rounding to limbs in MPZ.mul!
# (BITS_PER_LIMB-1 would typically be enough, to which we add
# 1 for the initial multiplication by init=1 in foldl)
nbits = GC.@preserve arr sum(arr; init=BITS_PER_LIMB) do x
abs(x.size) * BITS_PER_LIMB - leading_zeros(unsafe_load(x.d))
end
init = BigInt(; nbits)
MPZ.set_si!(init, 1)
foldl(MPZ.mul!, arr; init)
end
factorial(x::BigInt) = isneg(x) ? BigInt(0) : MPZ.fac_ui(x)
binomial(n::BigInt, k::UInt) = MPZ.bin_ui(n, k)
binomial(n::BigInt, k::Integer) = k < 0 ? BigInt(0) : binomial(n, UInt(k))
==(x::BigInt, y::BigInt) = cmp(x,y) == 0
==(x::BigInt, i::Integer) = cmp(x,i) == 0
==(i::Integer, x::BigInt) = cmp(x,i) == 0
==(x::BigInt, f::CdoubleMax) = isnan(f) ? false : cmp(x,f) == 0
==(f::CdoubleMax, x::BigInt) = isnan(f) ? false : cmp(x,f) == 0
iszero(x::BigInt) = x.size == 0
isone(x::BigInt) = x == Culong(1)
<=(x::BigInt, y::BigInt) = cmp(x,y) <= 0
<=(x::BigInt, i::Integer) = cmp(x,i) <= 0
<=(i::Integer, x::BigInt) = cmp(x,i) >= 0
<=(x::BigInt, f::CdoubleMax) = isnan(f) ? false : cmp(x,f) <= 0
<=(f::CdoubleMax, x::BigInt) = isnan(f) ? false : cmp(x,f) >= 0
<(x::BigInt, y::BigInt) = cmp(x,y) < 0
<(x::BigInt, i::Integer) = cmp(x,i) < 0
<(i::Integer, x::BigInt) = cmp(x,i) > 0
<(x::BigInt, f::CdoubleMax) = isnan(f) ? false : cmp(x,f) < 0
<(f::CdoubleMax, x::BigInt) = isnan(f) ? false : cmp(x,f) > 0
isneg(x::BigInt) = x.size < 0
ispos(x::BigInt) = x.size > 0
signbit(x::BigInt) = isneg(x)
flipsign!(x::BigInt, y::Integer) = (signbit(y) && (x.size = -x.size); x)
flipsign( x::BigInt, y::Integer) = signbit(y) ? -x : x
flipsign( x::BigInt, y::BigInt) = signbit(y) ? -x : x
# above method to resolving ambiguities with flipsign(::T, ::T) where T<:Signed
function sign(x::BigInt)
isneg(x) && return -one(x)
ispos(x) && return one(x)
return x
end
show(io::IO, x::BigInt) = print(io, string(x))
function string(n::BigInt; base::Integer = 10, pad::Integer = 1)
base < 0 && return Base._base(Int(base), n, pad, (base>0) & (n.size<0))
2 <= base <= 62 || throw(ArgumentError("base must be 2 ≤ base ≤ 62, got $base"))
iszero(n) && pad < 1 && return ""
nd1 = ndigits(n, base=base)
nd = max(nd1, pad)
sv = Base.StringVector(nd + isneg(n))
GC.@preserve sv MPZ.get_str!(pointer(sv) + nd - nd1, base, n)
@inbounds for i = (1:nd-nd1) .+ isneg(n)
sv[i] = '0' % UInt8
end
isneg(n) && (sv[1] = '-' % UInt8)
String(sv)
end
function digits!(a::AbstractVector{T}, n::BigInt; base::Integer = 10) where {T<:Integer}
if 2 ≤ base ≤ 62
s = codeunits(string(n; base))
i, j = firstindex(a)-1, length(s)+1
lasti = min(lastindex(a), firstindex(a) + length(s)-1 - isneg(n))
while i < lasti
# base ≤ 36: 0-9, plus a-z for 10-35
# base > 36: 0-9, plus A-Z for 10-35 and a-z for 36..61
x = s[j -= 1]
a[i += 1] = base ≤ 36 ? (x>0x39 ? x-0x57 : x-0x30) : (x>0x39 ? (x>0x60 ? x-0x3d : x-0x37) : x-0x30)
end
lasti = lastindex(a)
while i < lasti; a[i+=1] = zero(T); end
return isneg(n) ? map!(-,a,a) : a
end
return invoke(digits!, Tuple{typeof(a), Integer}, a, n; base) # slow generic fallback
end
function ndigits0zpb(x::BigInt, b::Integer)
b < 2 && throw(DomainError(b, "`b` cannot be less than 2."))
x.size == 0 && return 0 # for consistency with other ndigits0z methods
if ispow2(b) && 2 <= b <= 62 # GMP assumes b is in this range
MPZ.sizeinbase(x, b)
else
# non-base 2 mpz_sizeinbase might return an answer 1 too big
# use property that log(b, x) < ndigits(x, base=b) <= log(b, x) + 1
n = MPZ.sizeinbase(x, 2)
lb = log2(b) # assumed accurate to <1ulp (true for openlibm)
q,r = divrem(n,lb)
iq = Int(q)
maxerr = q*eps(lb) # maximum error in remainder
if r-1.0 < maxerr
abs(x) >= big(b)^iq ? iq+1 : iq
elseif lb-r < maxerr
abs(x) >= big(b)^(iq+1) ? iq+2 : iq+1
else
iq+1
end
end
end
# Fast paths for nextpow(2, x::BigInt)
# below, ONE is always left-shifted by at least one digit, so a new BigInt is
# allocated, which can be safely mutated
_prevpow2(x::BigInt) = -2 <= x <= 2 ? x : flipsign!(ONE << (ndigits(x, base=2) - 1), x)
_nextpow2(x::BigInt) = count_ones_abs(x) <= 1 ? x : flipsign!(ONE << ndigits(x, base=2), x)
Base.checked_abs(x::BigInt) = abs(x)
Base.checked_neg(x::BigInt) = -x
Base.checked_add(a::BigInt, b::BigInt) = a + b
Base.checked_sub(a::BigInt, b::BigInt) = a - b
Base.checked_mul(a::BigInt, b::BigInt) = a * b
Base.checked_div(a::BigInt, b::BigInt) = div(a, b)
Base.checked_rem(a::BigInt, b::BigInt) = rem(a, b)
Base.checked_fld(a::BigInt, b::BigInt) = fld(a, b)
Base.checked_mod(a::BigInt, b::BigInt) = mod(a, b)
Base.checked_cld(a::BigInt, b::BigInt) = cld(a, b)
Base.add_with_overflow(a::BigInt, b::BigInt) = a + b, false
Base.sub_with_overflow(a::BigInt, b::BigInt) = a - b, false
Base.mul_with_overflow(a::BigInt, b::BigInt) = a * b, false
Base.deepcopy_internal(x::BigInt, stackdict::IdDict) = get!(() -> MPZ.set(x), stackdict, x)
## streamlined hashing for BigInt, by avoiding allocation from shifts ##
if Limb === UInt
# this condition is true most (all?) of the time, and in this case we can define
# an optimized version for BigInt of hash_integer (used e.g. for Rational{BigInt}),
# and of hash
using .Base: hash_uint
function hash_integer(n::BigInt, h::UInt)
GC.@preserve n begin
s = n.size
s == 0 && return hash_integer(0, h)
p = convert(Ptr{UInt}, n.d)
b = unsafe_load(p)
h ⊻= hash_uint(ifelse(s < 0, -b, b) ⊻ h)
for k = 2:abs(s)
h ⊻= hash_uint(unsafe_load(p, k) ⊻ h)
end
return h
end
end
_divLimb(n) = UInt === UInt64 ? n >>> 6 : n >>> 5
_modLimb(n) = UInt === UInt64 ? n & 63 : n & 31
function hash(x::BigInt, h::UInt)
GC.@preserve x begin
sz = x.size
sz == 0 && return hash(0, h)
ptr = Ptr{UInt}(x.d)
if sz == 1
return hash(unsafe_load(ptr), h)
elseif sz == -1
limb = unsafe_load(ptr)
limb <= typemin(Int) % UInt && return hash(-(limb % Int), h)
end
pow = trailing_zeros(x)
nd = Base.ndigits0z(x, 2)
idx = _divLimb(pow) + 1
shift = _modLimb(pow) % UInt
upshift = BITS_PER_LIMB - shift
asz = abs(sz)
if shift == 0
limb = unsafe_load(ptr, idx)
else
limb1 = unsafe_load(ptr, idx)
limb2 = idx < asz ? unsafe_load(ptr, idx+1) : UInt(0)
limb = limb2 << upshift | limb1 >> shift
end
if nd <= 1024 && nd - pow <= 53
return hash(ldexp(flipsign(Float64(limb), sz), pow), h)
end
h = hash_integer(1, h)
h = hash_integer(pow, h)
h ⊻= hash_uint(flipsign(limb, sz) ⊻ h)
for idx = idx+1:asz
if shift == 0
limb = unsafe_load(ptr, idx)
else
limb1 = limb2
if idx == asz
limb = limb1 >> shift
limb == 0 && break # don't hash leading zeros
else
limb2 = unsafe_load(ptr, idx+1)
limb = limb2 << upshift | limb1 >> shift
end
end
h ⊻= hash_uint(limb ⊻ h)
end
return h
end
end
end
module MPQ
# Rational{BigInt}
import .Base: unsafe_rational, __throw_rational_argerror_zero
import ..GMP: BigInt, MPZ, Limb, isneg
gmpq(op::Symbol) = (Symbol(:__gmpq_, op), :libgmp)
mutable struct _MPQ
num_alloc::Cint
num_size::Cint
num_d::Ptr{Limb}
den_alloc::Cint
den_size::Cint
den_d::Ptr{Limb}
# to prevent GC
rat::Rational{BigInt}
end
const mpq_t = Ref{_MPQ}
_MPQ(x::BigInt,y::BigInt) = _MPQ(x.alloc, x.size, x.d,
y.alloc, y.size, y.d,
unsafe_rational(BigInt, x, y))
_MPQ() = _MPQ(BigInt(), BigInt())
_MPQ(x::Rational{BigInt}) = _MPQ(x.num, x.den)
function sync_rational!(xq::_MPQ)
xq.rat.num.alloc = xq.num_alloc
xq.rat.num.size = xq.num_size
xq.rat.num.d = xq.num_d
xq.rat.den.alloc = xq.den_alloc
xq.rat.den.size = xq.den_size
xq.rat.den.d = xq.den_d
return xq.rat
end
function Rational{BigInt}(num::BigInt, den::BigInt)
if iszero(den)
iszero(num) && __throw_rational_argerror_zero(BigInt)
return set_si(flipsign(1, num), 0)
end
xq = _MPQ(MPZ.set(num), MPZ.set(den))
ccall((:__gmpq_canonicalize, :libgmp), Cvoid, (mpq_t,), xq)
return sync_rational!(xq)
end
# define set, set_ui, set_si, set_z, and their inplace versions
function set!(z::Rational{BigInt}, x::Rational{BigInt})
zq = _MPQ(z)
ccall((:__gmpq_set, :libgmp), Cvoid, (mpq_t, mpq_t), zq, _MPQ(x))
return sync_rational!(zq)
end
function set_z!(z::Rational{BigInt}, x::BigInt)
zq = _MPQ(z)
ccall((:__gmpq_set_z, :libgmp), Cvoid, (mpq_t, MPZ.mpz_t), zq, x)
return sync_rational!(zq)
end
for (op, T) in ((:set, Rational{BigInt}), (:set_z, BigInt))
op! = Symbol(op, :!)
@eval $op(a::$T) = $op!(unsafe_rational(BigInt(), BigInt()), a)
end
# note that rationals returned from set_ui and set_si are not checked,
# set_ui(0, 0) will return 0//0 without errors, just like unsafe_rational
for (op, T1, T2) in ((:set_ui, Culong, Culong), (:set_si, Clong, Culong))
op! = Symbol(op, :!)
@eval begin
function $op!(z::Rational{BigInt}, a, b)
zq = _MPQ(z)
ccall($(gmpq(op)), Cvoid, (mpq_t, $T1, $T2), zq, a, b)
return sync_rational!(zq)
end
$op(a, b) = $op!(unsafe_rational(BigInt(), BigInt()), a, b)
end
end
# define add, sub, mul, div, and their inplace versions
function add!(z::Rational{BigInt}, x::Rational{BigInt}, y::Rational{BigInt})
if iszero(x.den) || iszero(y.den)
if iszero(x.den) && iszero(y.den) && isneg(x.num) != isneg(y.num)
throw(DivideError())
end
return set!(z, iszero(x.den) ? x : y)
end
zq = _MPQ(z)
ccall((:__gmpq_add, :libgmp), Cvoid,
(mpq_t,mpq_t,mpq_t), zq, _MPQ(x), _MPQ(y))
return sync_rational!(zq)
end
function sub!(z::Rational{BigInt}, x::Rational{BigInt}, y::Rational{BigInt})
if iszero(x.den) || iszero(y.den)
if iszero(x.den) && iszero(y.den) && isneg(x.num) == isneg(y.num)
throw(DivideError())
end
iszero(x.den) && return set!(z, x)
return set_si!(z, flipsign(-1, y.num), 0)
end
zq = _MPQ(z)
ccall((:__gmpq_sub, :libgmp), Cvoid,
(mpq_t,mpq_t,mpq_t), zq, _MPQ(x), _MPQ(y))
return sync_rational!(zq)
end
function mul!(z::Rational{BigInt}, x::Rational{BigInt}, y::Rational{BigInt})
if iszero(x.den) || iszero(y.den)
if iszero(x.num) || iszero(y.num)
throw(DivideError())
end
return set_si!(z, ifelse(xor(isneg(x.num), isneg(y.num)), -1, 1), 0)
end
zq = _MPQ(z)
ccall((:__gmpq_mul, :libgmp), Cvoid,
(mpq_t,mpq_t,mpq_t), zq, _MPQ(x), _MPQ(y))
return sync_rational!(zq)
end
function div!(z::Rational{BigInt}, x::Rational{BigInt}, y::Rational{BigInt})
if iszero(x.den)
if iszero(y.den)
throw(DivideError())
end
isneg(y.num) || return set!(z, x)
return set_si!(z, flipsign(-1, x.num), 0)
elseif iszero(y.den)
return set_si!(z, 0, 1)
elseif iszero(y.num)
if iszero(x.num)
throw(DivideError())
end
return set_si!(z, flipsign(1, x.num), 0)
end
zq = _MPQ(z)
ccall((:__gmpq_div, :libgmp), Cvoid,
(mpq_t,mpq_t,mpq_t), zq, _MPQ(x), _MPQ(y))
return sync_rational!(zq)
end
for (fJ, fC) in ((:+, :add), (:-, :sub), (:*, :mul), (://, :div))
fC! = Symbol(fC, :!)
@eval begin
($fC!)(x::Rational{BigInt}, y::Rational{BigInt}) = $fC!(x, x, y)
(Base.$fJ)(x::Rational{BigInt}, y::Rational{BigInt}) = $fC!(unsafe_rational(BigInt(), BigInt()), x, y)
end
end
function Base.cmp(x::Rational{BigInt}, y::Rational{BigInt})
Int(ccall((:__gmpq_cmp, :libgmp), Cint, (mpq_t, mpq_t), _MPQ(x), _MPQ(y)))
end
end # MPQ module
end # module