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arb.jl
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arb.jl
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###############################################################################
#
# arb.jl : Arb real numbers
#
# Copyright (C) 2015 Tommy Hofmann
# Copyright (C) 2015 Fredrik Johansson
#
###############################################################################
import Base: ceil, isinteger
export add_error!, ball, radius, midpoint, contains, contains_zero, contains_negative,
contains_positive, contains_nonnegative, contains_nonpositive, convert,
iszero, is_nonzero, is_exact, is_positive, isfinite, is_nonnegative,
is_negative, is_nonpositive, add!, mul!, sub!, div!, overlaps,
unique_integer, accuracy_bits, trim, ldexp, setunion, setintersection,
const_pi, const_e, const_log2, const_log10, const_euler, const_catalan,
const_khinchin, const_glaisher, floor, ceil, hypot, rsqrt, sqrt1pm1,
sqrtpos, root, log, log1p, expm1, sin, cos, sinpi, cospi, tan, cot,
tanpi, cotpi, sinh, cosh, tanh, coth, atan, asin, acos, atanh, asinh,
acosh, gamma, lgamma, rgamma, digamma, gamma_regularized, gamma_lower,
gamma_lower_regularized, zeta, sincos, sincospi, sinhcosh, atan2, agm,
factorial, binomial, fibonacci, bernoulli, rising_factorial,
rising_factorial2, polylog, chebyshev_t, chebyshev_t2, chebyshev_u,
chebyshev_u2, bell, numpart, lindep, airy_ai, airy_bi, airy_ai_prime,
airy_bi_prime, canonical_unit, simplest_rational_inside
###############################################################################
#
# Basic manipulation
#
###############################################################################
elem_type(::Type{ArbField}) = arb
parent_type(::Type{arb}) = ArbField
base_ring(R::ArbField) = Union{}
base_ring(x::arb) = Union{}
parent(x::arb) = x.parent
is_domain_type(::Type{arb}) = true
is_exact_type(::Type{arb}) = false
zero(R::ArbField) = R(0)
one(R::ArbField) = R(1)
# TODO: Add hash (and document under arb basic functionality)
@doc raw"""
accuracy_bits(x::arb)
Return the relative accuracy of $x$ measured in bits, capped between
`typemax(Int)` and `-typemax(Int)`.
"""
function accuracy_bits(x::arb)
return ccall((:arb_rel_accuracy_bits, libarb), Int, (Ref{arb},), x)
end
function deepcopy_internal(a::arb, dict::IdDict)
b = parent(a)()
ccall((:arb_set, libarb), Nothing, (Ref{arb}, Ref{arb}), b, a)
return b
end
function canonical_unit(x::arb)
return x
end
function check_parent(a::arb, b::arb)
parent(a) != parent(b) &&
error("Incompatible arb elements")
end
characteristic(::ArbField) = 0
################################################################################
#
# Conversions
#
################################################################################
@doc raw"""
Float64(x::arb, round::RoundingMode=RoundNearest)
Converts $x$ to a `Float64`, rounded in the direction specified by $round$.
For `RoundNearest` the return value approximates the midpoint of $x$. For
`RoundDown` or `RoundUp` the return value is a lower bound or upper bound for
all values in $x$.
"""
function Float64(x::arb, round::RoundingMode=RoundNearest)
t = _arb_get_arf(x, round)
return _arf_get_d(t, round)
end
@doc raw"""
BigFloat(x::arb, round::RoundingMode=RoundNearest)
Converts $x$ to a `BigFloat` of the currently used precision, rounded in the
direction specified by $round$. For `RoundNearest` the return value
approximates the midpoint of $x$. For `RoundDown` or `RoundUp` the return
value is a lower bound or upper bound for all values in $x$.
"""
function BigFloat(x::arb, round::RoundingMode=RoundNearest)
t = _arb_get_arf(x, round)
return _arf_get_mpfr(t, round)
end
function _arb_get_arf(x::arb, ::RoundingMode{:Nearest})
t = arf_struct()
GC.@preserve x begin
t1 = ccall((:arb_mid_ptr, libarb), Ptr{arf_struct},
(Ref{arb}, ),
x)
ccall((:arf_set, libarb), Nothing,
(Ref{arf_struct}, Ptr{arf_struct}),
t, t1)
end
return t
end
for (b, f) in ((RoundingMode{:Down}, :arb_get_lbound_arf),
(RoundingMode{:Up}, :arb_get_ubound_arf))
@eval begin
function _arb_get_arf(x::arb, ::$b)
t = arf_struct()
ccall(($(string(f)), libarb), Nothing,
(Ref{arf_struct}, Ref{arb}, Int),
t, x, parent(x).prec)
return t
end
end
end
for (b, i) in ((RoundingMode{:Down}, 2),
(RoundingMode{:Up}, 3),
(RoundingMode{:Nearest}, 4))
@eval begin
function _arf_get_d(t::arf_struct, ::$b)
d = ccall((:arf_get_d, libarb), Float64,
(Ref{arf_struct}, Int),
t, $i)
return d
end
function _arf_get_mpfr(t::arf_struct, ::$b)
d = BigFloat()
ccall((:arf_get_mpfr, libarb), Int32,
(Ref{BigFloat}, Ref{arf_struct}, Base.MPFR.MPFRRoundingMode),
d, t, $b())
return d
end
end
end
function convert(::Type{Float64}, x::arb)
return Float64(x)
end
function convert(::Type{BigFloat}, x::arb)
return BigFloat(x)
end
@doc raw"""
ZZRingElem(x::arb)
Return $x$ as an `ZZRingElem` if it represents an unique integer, else throws an
error.
"""
function ZZRingElem(x::arb)
if is_exact(x)
ok, z = unique_integer(x)
ok && return z
end
error("Argument must represent a unique integer")
end
BigInt(x::arb) = BigInt(ZZRingElem(x))
function (::Type{T})(x::arb) where {T <: Integer}
typemin(T) <= x <= typemax(T) ||
error("Argument does not fit inside datatype.")
return T(ZZRingElem(x))
end
################################################################################
#
# String I/O
#
################################################################################
function native_string(x::arb)
d = ceil(parent(x).prec * 0.30102999566398119521)
cstr = ccall((:arb_get_str, libarb), Ptr{UInt8},
(Ref{arb}, Int, UInt),
x, Int(d), UInt(0))
res = unsafe_string(cstr)
ccall((:flint_free, libflint), Nothing,
(Ptr{UInt8},),
cstr)
return res
end
function expressify(x::arb; context = nothing)
if is_exact(x) && is_negative(x)
# TODO is_exact does not imply it is printed without radius
return Expr(:call, :-, native_string(-x))
else
return native_string(x)
end
end
function show(io::IO, x::ArbField)
print(io, "Real Field with ")
print(io, precision(x))
print(io, " bits of precision and error bounds")
end
function show(io::IO, x::arb)
print(io, native_string(x))
end
################################################################################
#
# Containment
#
################################################################################
@doc raw"""
overlaps(x::arb, y::arb)
Returns `true` if any part of the ball $x$ overlaps any part of the ball $y$,
otherwise return `false`.
"""
function overlaps(x::arb, y::arb)
r = ccall((:arb_overlaps, libarb), Cint, (Ref{arb}, Ref{arb}), x, y)
return Bool(r)
end
#function contains(x::arb, y::arf)
# r = ccall((:arb_contains_arf, libarb), Cint, (Ref{arb}, Ref{arf}), x, y)
# return Bool(r)
#end
@doc raw"""
contains(x::arb, y::QQFieldElem)
Returns `true` if the ball $x$ contains the given rational value, otherwise
return `false`.
"""
function contains(x::arb, y::QQFieldElem)
r = ccall((:arb_contains_fmpq, libarb), Cint, (Ref{arb}, Ref{QQFieldElem}), x, y)
return Bool(r)
end
@doc raw"""
contains(x::arb, y::ZZRingElem)
Returns `true` if the ball $x$ contains the given integer value, otherwise
return `false`.
"""
function contains(x::arb, y::ZZRingElem)
r = ccall((:arb_contains_fmpz, libarb), Cint, (Ref{arb}, Ref{ZZRingElem}), x, y)
return Bool(r)
end
function contains(x::arb, y::Int)
r = ccall((:arb_contains_si, libarb), Cint, (Ref{arb}, Int), x, y)
return Bool(r)
end
@doc raw"""
contains(x::arb, y::Integer)
Returns `true` if the ball $x$ contains the given integer value, otherwise
return `false`.
"""
contains(x::arb, y::Integer) = contains(x, ZZRingElem(y))
@doc raw"""
contains(x::arb, y::Rational{T}) where {T <: Integer}
Returns `true` if the ball $x$ contains the given rational value, otherwise
return `false`.
"""
contains(x::arb, y::Rational{T}) where {T <: Integer} = contains(x, QQFieldElem(y))
@doc raw"""
contains(x::arb, y::BigFloat)
Returns `true` if the ball $x$ contains the given floating point value,
otherwise return `false`.
"""
function contains(x::arb, y::BigFloat)
r = ccall((:arb_contains_mpfr, libarb), Cint,
(Ref{arb}, Ref{BigFloat}), x, y)
return Bool(r)
end
@doc raw"""
contains(x::arb, y::arb)
Returns `true` if the ball $x$ contains the ball $y$, otherwise return
`false`.
"""
function contains(x::arb, y::arb)
r = ccall((:arb_contains, libarb), Cint, (Ref{arb}, Ref{arb}), x, y)
return Bool(r)
end
@doc raw"""
contains_zero(x::arb)
Returns `true` if the ball $x$ contains zero, otherwise return `false`.
"""
function contains_zero(x::arb)
r = ccall((:arb_contains_zero, libarb), Cint, (Ref{arb}, ), x)
return Bool(r)
end
@doc raw"""
contains_negative(x::arb)
Returns `true` if the ball $x$ contains any negative value, otherwise return
`false`.
"""
function contains_negative(x::arb)
r = ccall((:arb_contains_negative, libarb), Cint, (Ref{arb}, ), x)
return Bool(r)
end
@doc raw"""
contains_positive(x::arb)
Returns `true` if the ball $x$ contains any positive value, otherwise return
`false`.
"""
function contains_positive(x::arb)
r = ccall((:arb_contains_positive, libarb), Cint, (Ref{arb}, ), x)
return Bool(r)
end
@doc raw"""
contains_nonnegative(x::arb)
Returns `true` if the ball $x$ contains any nonnegative value, otherwise
return `false`.
"""
function contains_nonnegative(x::arb)
r = ccall((:arb_contains_nonnegative, libarb), Cint, (Ref{arb}, ), x)
return Bool(r)
end
@doc raw"""
contains_nonpositive(x::arb)
Returns `true` if the ball $x$ contains any nonpositive value, otherwise
return `false`.
"""
function contains_nonpositive(x::arb)
r = ccall((:arb_contains_nonpositive, libarb), Cint, (Ref{arb}, ), x)
return Bool(r)
end
################################################################################
#
# Comparison
#
################################################################################
@doc raw"""
isequal(x::arb, y::arb)
Return `true` if the balls $x$ and $y$ are precisely equal, i.e. have the
same midpoints and radii.
"""
function isequal(x::arb, y::arb)
r = ccall((:arb_equal, libarb), Cint, (Ref{arb}, Ref{arb}), x, y)
return Bool(r)
end
function ==(x::arb, y::arb)
return Bool(ccall((:arb_eq, libarb), Cint, (Ref{arb}, Ref{arb}), x, y))
end
function !=(x::arb, y::arb)
return Bool(ccall((:arb_ne, libarb), Cint, (Ref{arb}, Ref{arb}), x, y))
end
function isless(x::arb, y::arb)
return Bool(ccall((:arb_lt, libarb), Cint, (Ref{arb}, Ref{arb}), x, y))
end
function <=(x::arb, y::arb)
return Bool(ccall((:arb_le, libarb), Cint, (Ref{arb}, Ref{arb}), x, y))
end
==(x::arb, y::Int) = x == arb(y)
!=(x::arb, y::Int) = x != arb(y)
<=(x::arb, y::Int) = x <= arb(y)
<(x::arb, y::Int) = x < arb(y)
==(x::Int, y::arb) = arb(x) == y
!=(x::Int, y::arb) = arb(x) != y
<=(x::Int, y::arb) = arb(x) <= y
<(x::Int, y::arb) = arb(x) < y
==(x::arb, y::ZZRingElem) = x == arb(y)
!=(x::arb, y::ZZRingElem) = x != arb(y)
<=(x::arb, y::ZZRingElem) = x <= arb(y)
<(x::arb, y::ZZRingElem) = x < arb(y)
==(x::ZZRingElem, y::arb) = arb(x) == y
!=(x::ZZRingElem, y::arb) = arb(x) != y
<=(x::ZZRingElem, y::arb) = arb(x) <= y
<(x::ZZRingElem, y::arb) = arb(x) < y
==(x::arb, y::Integer) = x == ZZRingElem(y)
!=(x::arb, y::Integer) = x != ZZRingElem(y)
<=(x::arb, y::Integer) = x <= ZZRingElem(y)
<(x::arb, y::Integer) = x < ZZRingElem(y)
==(x::Integer, y::arb) = ZZRingElem(x) == y
!=(x::Integer, y::arb) = ZZRingElem(x) != y
<=(x::Integer, y::arb) = ZZRingElem(x) <= y
<(x::Integer, y::arb) = ZZRingElem(x) < y
==(x::arb, y::Float64) = x == arb(y)
!=(x::arb, y::Float64) = x != arb(y)
<=(x::arb, y::Float64) = x <= arb(y)
<(x::arb, y::Float64) = x < arb(y)
==(x::Float64, y::arb) = arb(x) == y
!=(x::Float64, y::arb) = arb(x) != y
<=(x::Float64, y::arb) = arb(x) <= y
<(x::Float64, y::arb) = arb(x) < y
==(x::arb, y::BigFloat) = x == arb(y)
!=(x::arb, y::BigFloat) = x != arb(y)
<=(x::arb, y::BigFloat) = x <= arb(y)
<(x::arb, y::BigFloat) = x < arb(y)
==(x::BigFloat, y::arb) = arb(x) == y
!=(x::BigFloat, y::arb) = arb(x) != y
<=(x::BigFloat, y::arb) = arb(x) <= y
<(x::BigFloat, y::arb) = arb(x) < y
==(x::arb, y::QQFieldElem) = x == arb(y, precision(parent(x)))
!=(x::arb, y::QQFieldElem) = x != arb(y, precision(parent(x)))
<=(x::arb, y::QQFieldElem) = x <= arb(y, precision(parent(x)))
<(x::arb, y::QQFieldElem) = x < arb(y, precision(parent(x)))
==(x::QQFieldElem, y::arb) = arb(x, precision(parent(y))) == y
!=(x::QQFieldElem, y::arb) = arb(x, precision(parent(y))) != y
<=(x::QQFieldElem, y::arb) = arb(x, precision(parent(y))) <= y
<(x::QQFieldElem, y::arb) = arb(x, precision(parent(y))) < y
==(x::arb, y::Rational{T}) where {T <: Integer} = x == QQFieldElem(y)
!=(x::arb, y::Rational{T}) where {T <: Integer} = x != QQFieldElem(y)
<=(x::arb, y::Rational{T}) where {T <: Integer} = x <= QQFieldElem(y)
<(x::arb, y::Rational{T}) where {T <: Integer} = x < QQFieldElem(y)
==(x::Rational{T}, y::arb) where {T <: Integer} = QQFieldElem(x) == y
!=(x::Rational{T}, y::arb) where {T <: Integer} = QQFieldElem(x) != y
<=(x::Rational{T}, y::arb) where {T <: Integer} = QQFieldElem(x) <= y
<(x::Rational{T}, y::arb) where {T <: Integer} = QQFieldElem(x) < y
################################################################################
#
# Predicates
#
################################################################################
function is_unit(x::arb)
!iszero(x)
end
@doc raw"""
iszero(x::arb)
Return `true` if $x$ is certainly zero, otherwise return `false`.
"""
function iszero(x::arb)
return Bool(ccall((:arb_is_zero, libarb), Cint, (Ref{arb},), x))
end
@doc raw"""
is_nonzero(x::arb)
Return `true` if $x$ is certainly not equal to zero, otherwise return
`false`.
"""
function is_nonzero(x::arb)
return Bool(ccall((:arb_is_nonzero, libarb), Cint, (Ref{arb},), x))
end
@doc raw"""
isone(x::arb)
Return `true` if $x$ is certainly one, otherwise return `false`.
"""
function isone(x::arb)
return Bool(ccall((:arb_is_one, libarb), Cint, (Ref{arb},), x))
end
@doc raw"""
isfinite(x::arb)
Return `true` if $x$ is finite, i.e. having finite midpoint and radius,
otherwise return `false`.
"""
function isfinite(x::arb)
return Bool(ccall((:arb_is_finite, libarb), Cint, (Ref{arb},), x))
end
@doc raw"""
is_exact(x::arb)
Return `true` if $x$ is exact, i.e. has zero radius, otherwise return
`false`.
"""
function is_exact(x::arb)
return Bool(ccall((:arb_is_exact, libarb), Cint, (Ref{arb},), x))
end
@doc raw"""
isinteger(x::arb)
Return `true` if $x$ is an exact integer, otherwise return `false`.
"""
function isinteger(x::arb)
return Bool(ccall((:arb_is_int, libarb), Cint, (Ref{arb},), x))
end
@doc raw"""
is_positive(x::arb)
Return `true` if $x$ is certainly positive, otherwise return `false`.
"""
function is_positive(x::arb)
return Bool(ccall((:arb_is_positive, libarb), Cint, (Ref{arb},), x))
end
@doc raw"""
is_nonnegative(x::arb)
Return `true` if $x$ is certainly nonnegative, otherwise return `false`.
"""
function is_nonnegative(x::arb)
return Bool(ccall((:arb_is_nonnegative, libarb), Cint, (Ref{arb},), x))
end
@doc raw"""
is_negative(x::arb)
Return `true` if $x$ is certainly negative, otherwise return `false`.
"""
function is_negative(x::arb)
return Bool(ccall((:arb_is_negative, libarb), Cint, (Ref{arb},), x))
end
@doc raw"""
is_nonpositive(x::arb)
Return `true` if $x$ is certainly nonpositive, otherwise return `false`.
"""
function is_nonpositive(x::arb)
return Bool(ccall((:arb_is_nonpositive, libarb), Cint, (Ref{arb},), x))
end
################################################################################
#
# Parts of numbers
#
################################################################################
@doc raw"""
ball(x::arb, y::arb)
Constructs an Arb ball enclosing $x_m \pm (|x_r| + |y_m| + |y_r|)$, given the
pair $(x, y) = (x_m \pm x_r, y_m \pm y_r)$.
"""
function ball(mid::arb, rad::arb)
z = arb(mid, rad)
z.parent = parent(mid)
return z
end
@doc raw"""
radius(x::arb)
Return the radius of the ball $x$ as an Arb ball.
"""
function radius(x::arb)
z = parent(x)()
ccall((:arb_get_rad_arb, libarb), Nothing, (Ref{arb}, Ref{arb}), z, x)
return z
end
@doc raw"""
midpoint(x::arb)
Return the midpoint of the ball $x$ as an Arb ball.
"""
function midpoint(x::arb)
z = parent(x)()
ccall((:arb_get_mid_arb, libarb), Nothing, (Ref{arb}, Ref{arb}), z, x)
return z
end
@doc raw"""
add_error!(x::arb, y::arb)
Adds the absolute values of the midpoint and radius of $y$ to the radius of $x$.
"""
function add_error!(x::arb, y::arb)
ccall((:arb_add_error, libarb), Nothing, (Ref{arb}, Ref{arb}), x, y)
end
################################################################################
#
# Sign
#
################################################################################
function sign(::Type{Int}, x::arb)
if is_positive(x)
return 1
elseif is_negative(x)
return -1
else
error("Could not determine sign")
end
end
################################################################################
#
# Unary operations
#
################################################################################
function -(x::arb)
z = parent(x)()
ccall((:arb_neg, libarb), Nothing, (Ref{arb}, Ref{arb}), z, x)
return z
end
################################################################################
#
# Binary operations
#
################################################################################
for (s,f) in ((:+,"arb_add"), (:*,"arb_mul"), (://, "arb_div"), (:-,"arb_sub"))
@eval begin
function ($s)(x::arb, y::arb)
z = parent(x)()
ccall(($f, libarb), Nothing, (Ref{arb}, Ref{arb}, Ref{arb}, Int),
z, x, y, parent(x).prec)
return z
end
end
end
for (f,s) in ((:+, "add"), (:*, "mul"))
@eval begin
#function ($f)(x::arb, y::arf)
# z = parent(x)()
# ccall(($("arb_"*s*"_arf"), libarb), Nothing,
# (Ref{arb}, Ref{arb}, Ref{arf}, Int),
# z, x, y, parent(x).prec)
# return z
#end
#($f)(x::arf, y::arb) = ($f)(y, x)
function ($f)(x::arb, y::UInt)
z = parent(x)()
ccall(($("arb_"*s*"_ui"), libarb), Nothing,
(Ref{arb}, Ref{arb}, UInt, Int),
z, x, y, parent(x).prec)
return z
end
($f)(x::UInt, y::arb) = ($f)(y, x)
function ($f)(x::arb, y::Int)
z = parent(x)()
ccall(($("arb_"*s*"_si"), libarb), Nothing,
(Ref{arb}, Ref{arb}, Int, Int), z, x, y, parent(x).prec)
return z
end
($f)(x::Int, y::arb) = ($f)(y,x)
function ($f)(x::arb, y::ZZRingElem)
z = parent(x)()
ccall(($("arb_"*s*"_fmpz"), libarb), Nothing,
(Ref{arb}, Ref{arb}, Ref{ZZRingElem}, Int),
z, x, y, parent(x).prec)
return z
end
($f)(x::ZZRingElem, y::arb) = ($f)(y,x)
end
end
#function -(x::arb, y::arf)
# z = parent(x)()
# ccall((:arb_sub_arf, libarb), Nothing,
# (Ref{arb}, Ref{arb}, Ref{arf}, Int), z, x, y, parent(x).prec)
# return z
#end
#-(x::arf, y::arb) = -(y - x)
function -(x::arb, y::UInt)
z = parent(x)()
ccall((:arb_sub_ui, libarb), Nothing,
(Ref{arb}, Ref{arb}, UInt, Int), z, x, y, parent(x).prec)
return z
end
-(x::UInt, y::arb) = -(y - x)
function -(x::arb, y::Int)
z = parent(x)()
ccall((:arb_sub_si, libarb), Nothing,
(Ref{arb}, Ref{arb}, Int, Int), z, x, y, parent(x).prec)
return z
end
-(x::Int, y::arb) = -(y - x)
function -(x::arb, y::ZZRingElem)
z = parent(x)()
ccall((:arb_sub_fmpz, libarb), Nothing,
(Ref{arb}, Ref{arb}, Ref{ZZRingElem}, Int),
z, x, y, parent(x).prec)
return z
end
-(x::ZZRingElem, y::arb) = -(y-x)
+(x::arb, y::Integer) = x + ZZRingElem(y)
-(x::arb, y::Integer) = x - ZZRingElem(y)
*(x::arb, y::Integer) = x*ZZRingElem(y)
//(x::arb, y::Integer) = x//ZZRingElem(y)
+(x::Integer, y::arb) = ZZRingElem(x) + y
-(x::Integer, y::arb) = ZZRingElem(x) - y
*(x::Integer, y::arb) = ZZRingElem(x)*y
//(x::Integer, y::arb) = ZZRingElem(x)//y
#function //(x::arb, y::arf)
# z = parent(x)()
# ccall((:arb_div_arf, libarb), Nothing,
# (Ref{arb}, Ref{arb}, Ref{arf}, Int), z, x, y, parent(x).prec)
# return z
#end
function //(x::arb, y::UInt)
z = parent(x)()
ccall((:arb_div_ui, libarb), Nothing,
(Ref{arb}, Ref{arb}, UInt, Int), z, x, y, parent(x).prec)
return z
end
function //(x::arb, y::Int)
z = parent(x)()
ccall((:arb_div_si, libarb), Nothing,
(Ref{arb}, Ref{arb}, Int, Int), z, x, y, parent(x).prec)
return z
end
function //(x::arb, y::ZZRingElem)
z = parent(x)()
ccall((:arb_div_fmpz, libarb), Nothing,
(Ref{arb}, Ref{arb}, Ref{ZZRingElem}, Int),
z, x, y, parent(x).prec)
return z
end
function //(x::UInt, y::arb)
z = parent(y)()
ccall((:arb_ui_div, libarb), Nothing,
(Ref{arb}, UInt, Ref{arb}, Int), z, x, y, parent(y).prec)
return z
end
function //(x::Int, y::arb)
z = parent(y)()
t = arb(x)
ccall((:arb_div, libarb), Nothing,
(Ref{arb}, Ref{arb}, Ref{arb}, Int), z, t, y, parent(y).prec)
return z
end
function //(x::ZZRingElem, y::arb)
z = parent(y)()
t = arb(x)
ccall((:arb_div, libarb), Nothing,
(Ref{arb}, Ref{arb}, Ref{arb}, Int), z, t, y, parent(y).prec)
return z
end
function ^(x::arb, y::arb)
z = parent(x)()
ccall((:arb_pow, libarb), Nothing,
(Ref{arb}, Ref{arb}, Ref{arb}, Int), z, x, y, parent(x).prec)
return z
end
function ^(x::arb, y::ZZRingElem)
z = parent(x)()
ccall((:arb_pow_fmpz, libarb), Nothing,
(Ref{arb}, Ref{arb}, Ref{ZZRingElem}, Int),
z, x, y, parent(x).prec)
return z
end
^(x::arb, y::Integer) = x^ZZRingElem(y)
function ^(x::arb, y::UInt)
z = parent(x)()
ccall((:arb_pow_ui, libarb), Nothing,
(Ref{arb}, Ref{arb}, UInt, Int), z, x, y, parent(x).prec)
return z
end
function ^(x::arb, y::QQFieldElem)
z = parent(x)()
ccall((:arb_pow_fmpq, libarb), Nothing,
(Ref{arb}, Ref{arb}, Ref{QQFieldElem}, Int),
z, x, y, parent(x).prec)
return z
end
+(x::QQFieldElem, y::arb) = parent(y)(x) + y
+(x::arb, y::QQFieldElem) = x + parent(x)(y)
-(x::QQFieldElem, y::arb) = parent(y)(x) - y
//(x::arb, y::QQFieldElem) = x//parent(x)(y)
//(x::QQFieldElem, y::arb) = parent(y)(x)//y
-(x::arb, y::QQFieldElem) = x - parent(x)(y)
*(x::QQFieldElem, y::arb) = parent(y)(x) * y
*(x::arb, y::QQFieldElem) = x * parent(x)(y)
^(x::QQFieldElem, y::arb) = parent(y)(x) ^ y
+(x::Float64, y::arb) = parent(y)(x) + y
+(x::arb, y::Float64) = x + parent(x)(y)
-(x::Float64, y::arb) = parent(y)(x) - y
//(x::arb, y::Float64) = x//parent(x)(y)
//(x::Float64, y::arb) = parent(y)(x)//y
-(x::arb, y::Float64) = x - parent(x)(y)
*(x::Float64, y::arb) = parent(y)(x) * y
*(x::arb, y::Float64) = x * parent(x)(y)
^(x::Float64, y::arb) = parent(y)(x) ^ y
^(x::arb, y::Float64) = x ^ parent(x)(y)
+(x::BigFloat, y::arb) = parent(y)(x) + y
+(x::arb, y::BigFloat) = x + parent(x)(y)
-(x::BigFloat, y::arb) = parent(y)(x) - y
//(x::arb, y::BigFloat) = x//parent(x)(y)
//(x::BigFloat, y::arb) = parent(y)(x)//y
-(x::arb, y::BigFloat) = x - parent(x)(y)
*(x::BigFloat, y::arb) = parent(y)(x) * y
*(x::arb, y::BigFloat) = x * parent(x)(y)
^(x::BigFloat, y::arb) = parent(y)(x) ^ y
^(x::arb, y::BigFloat) = x ^ parent(x)(y)
+(x::Rational{T}, y::arb) where {T <: Integer} = QQFieldElem(x) + y
+(x::arb, y::Rational{T}) where {T <: Integer} = x + QQFieldElem(y)
-(x::Rational{T}, y::arb) where {T <: Integer} = QQFieldElem(x) - y
-(x::arb, y::Rational{T}) where {T <: Integer} = x - QQFieldElem(y)
//(x::Rational{T}, y::arb) where {T <: Integer} = QQFieldElem(x)//y
//(x::arb, y::Rational{T}) where {T <: Integer} = x//QQFieldElem(y)
*(x::Rational{T}, y::arb) where {T <: Integer} = QQFieldElem(x) * y
*(x::arb, y::Rational{T}) where {T <: Integer} = x * QQFieldElem(y)
^(x::Rational{T}, y::arb) where {T <: Integer} = QQFieldElem(x) ^ y
^(x::arb, y::Rational{T}) where {T <: Integer} = x ^ QQFieldElem(y)
/(x::arb, y::arb) = x // y
/(x::ZZRingElem, y::arb) = x // y
/(x::arb, y::ZZRingElem) = x // y
/(x::Int, y::arb) = x // y
/(x::arb, y::Int) = x // y
/(x::UInt, y::arb) = x // y
/(x::arb, y::UInt) = x // y
/(x::QQFieldElem, y::arb) = x // y
/(x::arb, y::QQFieldElem) = x // y
/(x::Float64, y::arb) = x // y
/(x::arb, y::Float64) = x // y
/(x::BigFloat, y::arb) = x // y
/(x::arb, y::BigFloat) = x // y
/(x::Rational{T}, y::arb) where {T <: Integer} = x // y
/(x::arb, y::Rational{T}) where {T <: Integer} = x // y
divexact(x::arb, y::arb; check::Bool=true) = x // y
divexact(x::ZZRingElem, y::arb; check::Bool=true) = x // y
divexact(x::arb, y::ZZRingElem; check::Bool=true) = x // y
divexact(x::Int, y::arb; check::Bool=true) = x // y
divexact(x::arb, y::Int; check::Bool=true) = x // y
divexact(x::UInt, y::arb; check::Bool=true) = x // y
divexact(x::arb, y::UInt; check::Bool=true) = x // y
divexact(x::QQFieldElem, y::arb; check::Bool=true) = x // y
divexact(x::arb, y::QQFieldElem; check::Bool=true) = x // y
divexact(x::Float64, y::arb; check::Bool=true) = x // y
divexact(x::arb, y::Float64; check::Bool=true) = x // y
divexact(x::BigFloat, y::arb; check::Bool=true) = x // y
divexact(x::arb, y::BigFloat; check::Bool=true) = x // y
divexact(x::Rational{T}, y::arb; check::Bool=true) where {T <: Integer} = x // y
divexact(x::arb, y::Rational{T}; check::Bool=true) where {T <: Integer} = x // y
################################################################################
#
# Absolute value
#
################################################################################
function abs(x::arb)
z = parent(x)()
ccall((:arb_abs, libarb), Nothing, (Ref{arb}, Ref{arb}), z, x)
return z
end
################################################################################
#
# Inverse
#
################################################################################
function inv(x::arb)
z = parent(x)()
ccall((:arb_inv, libarb), Nothing,
(Ref{arb}, Ref{arb}, Int), z, x, parent(x).prec)
return parent(x)(z)
end
################################################################################
#
# Shifting
#
################################################################################
function ldexp(x::arb, y::Int)
z = parent(x)()
ccall((:arb_mul_2exp_si, libarb), Nothing,
(Ref{arb}, Ref{arb}, Int), z, x, y)
return z
end
function ldexp(x::arb, y::ZZRingElem)
z = parent(x)()
ccall((:arb_mul_2exp_fmpz, libarb), Nothing,
(Ref{arb}, Ref{arb}, Ref{ZZRingElem}), z, x, y)
return z
end
################################################################################
#
# Miscellaneous
#
################################################################################
@doc raw"""
trim(x::arb)
Return an `arb` interval containing $x$ but which may be more economical,
by rounding off insignificant bits from the midpoint.
"""
function trim(x::arb)
z = parent(x)()
ccall((:arb_trim, libarb), Nothing, (Ref{arb}, Ref{arb}), z, x)
return z
end
@doc raw"""
unique_integer(x::arb)
Return a pair where the first value is a boolean and the second is an `ZZRingElem`
integer. The boolean indicates whether the interval $x$ contains a unique
integer. If this is the case, the second return value is set to this unique
integer.
"""
function unique_integer(x::arb)
z = ZZRingElem()
unique = ccall((:arb_get_unique_fmpz, libarb), Int,
(Ref{ZZRingElem}, Ref{arb}), z, x)
return (unique != 0, z)