swh:1:snp:a72e953ecd624a7df6e6196bbdd05851996c5e40
Tip revision: a04cde9bc98a2bfc0e22ffbc4391ed1ea2127dda authored by Kristoffer on 16 February 2024, 17:04:49 UTC
implement a julia specific debug mode with `--debug`
implement a julia specific debug mode with `--debug`
Tip revision: a04cde9
div.jl
# This file is a part of Julia. License is MIT: https://julialang.org/license
# Div is truncating by default
"""
div(x, y, r::RoundingMode=RoundToZero)
The quotient from Euclidean (integer) division. Computes `x / y`, rounded to
an integer according to the rounding mode `r`. In other words, the quantity
round(x / y, r)
without any intermediate rounding.
!!! compat "Julia 1.4"
The three-argument method taking a `RoundingMode` requires Julia 1.4 or later.
See also [`fld`](@ref) and [`cld`](@ref), which are special cases of this function.
!!! compat "Julia 1.9"
`RoundFromZero` requires at least Julia 1.9.
# Examples:
```jldoctest
julia> div(4, 3, RoundToZero) # Matches div(4, 3)
1
julia> div(4, 3, RoundDown) # Matches fld(4, 3)
1
julia> div(4, 3, RoundUp) # Matches cld(4, 3)
2
julia> div(5, 2, RoundNearest)
2
julia> div(5, 2, RoundNearestTiesAway)
3
julia> div(-5, 2, RoundNearest)
-2
julia> div(-5, 2, RoundNearestTiesAway)
-3
julia> div(-5, 2, RoundNearestTiesUp)
-2
julia> div(4, 3, RoundFromZero)
2
julia> div(-4, 3, RoundFromZero)
-2
```
"""
div(x, y, r::RoundingMode)
div(a, b) = div(a, b, RoundToZero)
"""
rem(x, y, r::RoundingMode=RoundToZero)
Compute the remainder of `x` after integer division by `y`, with the quotient rounded
according to the rounding mode `r`. In other words, the quantity
x - y * round(x / y, r)
without any intermediate rounding.
- if `r == RoundNearest`, then the result is exact, and in the interval
``[-|y| / 2, |y| / 2]``. See also [`RoundNearest`](@ref).
- if `r == RoundToZero` (default), then the result is exact, and in the interval
``[0, |y|)`` if `x` is positive, or ``(-|y|, 0]`` otherwise. See also [`RoundToZero`](@ref).
- if `r == RoundDown`, then the result is in the interval ``[0, y)`` if `y` is positive, or
``(y, 0]`` otherwise. The result may not be exact if `x` and `y` have different signs, and
`abs(x) < abs(y)`. See also [`RoundDown`](@ref).
- if `r == RoundUp`, then the result is in the interval ``(-y, 0]`` if `y` is positive, or
``[0, -y)`` otherwise. The result may not be exact if `x` and `y` have the same sign, and
`abs(x) < abs(y)`. See also [`RoundUp`](@ref).
- if `r == RoundFromZero`, then the result is in the interval ``(-y, 0]`` if `y` is positive, or
``[0, -y)`` otherwise. The result may not be exact if `x` and `y` have the same sign, and
`abs(x) < abs(y)`. See also [`RoundFromZero`](@ref).
!!! compat "Julia 1.9"
`RoundFromZero` requires at least Julia 1.9.
# Examples:
```jldoctest
julia> x = 9; y = 4;
julia> x % y # same as rem(x, y)
1
julia> x ÷ y # same as div(x, y)
2
julia> x == div(x, y) * y + rem(x, y)
true
```
"""
rem(x, y, r::RoundingMode)
# TODO: Make these primitive and have the two-argument version call these
rem(x, y, ::RoundingMode{:ToZero}) = rem(x, y)
rem(x, y, ::RoundingMode{:Down}) = mod(x, y)
rem(x, y, ::RoundingMode{:Up}) = mod(x, -y)
rem(x, y, r::RoundingMode{:Nearest}) = x - y * div(x, y, r)
rem(x::Integer, y::Integer, r::RoundingMode{:Nearest}) = divrem(x, y, r)[2]
function rem(x, y, ::typeof(RoundFromZero))
signbit(x) == signbit(y) ? rem(x, y, RoundUp) : rem(x, y, RoundDown)
end
"""
fld(x, y)
Largest integer less than or equal to `x / y`. Equivalent to `div(x, y, RoundDown)`.
See also [`div`](@ref), [`cld`](@ref), [`fld1`](@ref).
# Examples
```jldoctest
julia> fld(7.3, 5.5)
1.0
julia> fld.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
-2 -2 -1 -1 -1 0 0 0 1 1 1
```
Because `fld(x, y)` implements strictly correct floored rounding based on the true
value of floating-point numbers, unintuitive situations can arise. For example:
```jldoctest
julia> fld(6.0, 0.1)
59.0
julia> 6.0 / 0.1
60.0
julia> 6.0 / big(0.1)
59.99999999999999666933092612453056361837965690217069245739573412231113406246995
```
What is happening here is that the true value of the floating-point number written
as `0.1` is slightly larger than the numerical value 1/10 while `6.0` represents
the number 6 precisely. Therefore the true value of `6.0 / 0.1` is slightly less
than 60. When doing division, this is rounded to precisely `60.0`, but
`fld(6.0, 0.1)` always takes the floor of the true value, so the result is `59.0`.
"""
fld(a, b) = div(a, b, RoundDown)
"""
cld(x, y)
Smallest integer larger than or equal to `x / y`. Equivalent to `div(x, y, RoundUp)`.
See also [`div`](@ref), [`fld`](@ref).
# Examples
```jldoctest
julia> cld(5.5, 2.2)
3.0
julia> cld.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
-1 -1 -1 0 0 0 1 1 1 2 2
```
"""
cld(a, b) = div(a, b, RoundUp)
# divrem
"""
divrem(x, y, r::RoundingMode=RoundToZero)
The quotient and remainder from Euclidean division.
Equivalent to `(div(x, y, r), rem(x, y, r))`. Equivalently, with the default
value of `r`, this call is equivalent to `(x ÷ y, x % y)`.
See also: [`fldmod`](@ref), [`cld`](@ref).
# Examples
```jldoctest
julia> divrem(3, 7)
(0, 3)
julia> divrem(7, 3)
(2, 1)
```
"""
divrem(x, y) = divrem(x, y, RoundToZero)
function divrem(a, b, r::RoundingMode)
if r === RoundToZero
# For compat. Remove in 2.0.
(div(a, b), rem(a, b))
elseif r === RoundDown
# For compat. Remove in 2.0.
(fld(a, b), mod(a, b))
else
(div(a, b, r), rem(a, b, r))
end
end
# avoids calling rem for Integers-Integers (all modes),
# a - d * b not precise for Floats - AbstractFloat, AbstractIrrational.
# Rationals are still slower
function divrem(a::Integer, b::Integer, r::Union{typeof(RoundUp),
typeof(RoundDown),
typeof(RoundToZero)})
if r === RoundToZero
# For compat. Remove in 2.0.
d = div(a, b)
(d, a - d * b)
elseif r === RoundDown
# For compat. Remove in 2.0.
d = fld(a, b)
(d, a - d * b)
elseif r === RoundUp
# For compat. Remove in 2.0.
d = div(a, b, r)
(d, a - d * b)
end
end
function divrem(x::Integer, y::Integer, rnd::typeof(RoundNearest))
(q, r) = divrem(x, y)
if x >= 0
if y >= 0
r >= (y÷2) + (isodd(y) | iseven(q)) ? (q+true, r-y) : (q, r)
else
r >= -(y÷2) + (isodd(y) | iseven(q)) ? (q-true, r+y) : (q, r)
end
else
if y >= 0
r <= -signed(y÷2) - (isodd(y) | iseven(q)) ? (q-true, r+y) : (q, r)
else
r <= (y÷2) - (isodd(y) | iseven(q)) ? (q+true, r-y) : (q, r)
end
end
end
function divrem(x::Integer, y::Integer, rnd:: typeof(RoundNearestTiesAway))
(q, r) = divrem(x, y)
if x >= 0
if y >= 0
r >= (y÷2) + isodd(y) ? (q+true, r-y) : (q, r)
else
r >= -(y÷2) + isodd(y) ? (q-true, r+y) : (q, r)
end
else
if y >= 0
r <= -signed(y÷2) - isodd(y) ? (q-true, r+y) : (q, r)
else
r <= (y÷2) - isodd(y) ? (q+true, r-y) : (q, r)
end
end
end
function divrem(x::Integer, y::Integer, rnd::typeof(RoundNearestTiesUp))
(q, r) = divrem(x, y)
if x >= 0
if y >= 0
r >= (y÷2) + isodd(y) ? (q+true, r-y) : (q, r)
else
r >= -(y÷2) + true ? (q-true, r+y) : (q, r)
end
else
if y >= 0
r <= -signed(y÷2) - true ? (q-true, r+y) : (q, r)
else
r <= (y÷2) - isodd(y) ? (q+true, r-y) : (q, r)
end
end
end
function divrem(x, y, ::typeof(RoundFromZero))
signbit(x) == signbit(y) ? divrem(x, y, RoundUp) : divrem(x, y, RoundDown)
end
"""
fldmod(x, y)
The floored quotient and modulus after division. A convenience wrapper for
`divrem(x, y, RoundDown)`. Equivalent to `(fld(x, y), mod(x, y))`.
See also: [`fld`](@ref), [`cld`](@ref), [`fldmod1`](@ref).
"""
fldmod(x, y) = divrem(x, y, RoundDown)
# We definite generic rounding methods for other rounding modes in terms of
# RoundToZero.
function div(x::Signed, y::Unsigned, ::typeof(RoundDown))
(q, r) = divrem(x, y)
q - (signbit(x) & (r != 0))
end
function div(x::Unsigned, y::Signed, ::typeof(RoundDown))
(q, r) = divrem(x, y)
q - (signbit(y) & (r != 0))
end
function div(x::Signed, y::Unsigned, ::typeof(RoundUp))
(q, r) = divrem(x, y)
q + (!signbit(x) & (r != 0))
end
function div(x::Unsigned, y::Signed, ::typeof(RoundUp))
(q, r) = divrem(x, y)
q + (!signbit(y) & (r != 0))
end
function div(x::Integer, y::Integer, rnd::Union{typeof(RoundNearest),
typeof(RoundNearestTiesAway),
typeof(RoundNearestTiesUp)})
divrem(x, y, rnd)[1]
end
function div(x::Integer, y::Integer, ::typeof(RoundFromZero))
signbit(x) == signbit(y) ? div(x, y, RoundUp) : div(x, y, RoundDown)
end
# For bootstrapping purposes, we define div for integers directly. Provide the
# generic signature also
div(a::T, b::T, ::typeof(RoundToZero)) where {T<:Union{BitSigned, BitUnsigned64}} = div(a, b)
div(a::Bool, b::Bool, r::RoundingMode) = div(a, b)
# Prevent ambiguities
for rm in (RoundUp, RoundDown, RoundToZero, RoundFromZero)
@eval div(a::Bool, b::Bool, r::$(typeof(rm))) = div(a, b)
end
function div(x::Bool, y::Bool, rnd::Union{typeof(RoundNearest),
typeof(RoundNearestTiesAway),
typeof(RoundNearestTiesUp)})
div(x, y)
end
fld(a::T, b::T) where {T<:Union{Integer,AbstractFloat}} = div(a, b, RoundDown)
cld(a::T, b::T) where {T<:Union{Integer,AbstractFloat}} = div(a, b, RoundUp)
div(a::Int128, b::Int128, ::typeof(RoundToZero)) = div(a, b)
div(a::UInt128, b::UInt128, ::typeof(RoundToZero)) = div(a, b)
rem(a::Int128, b::Int128, ::typeof(RoundToZero)) = rem(a, b)
rem(a::UInt128, b::UInt128, ::typeof(RoundToZero)) = rem(a, b)
# These are kept for compatibility with external packages overriding fld / cld.
# In 2.0, packages should extend div(a, b, r) instead, in which case, these can
# be removed.
fld(x::Real, y::Real) = div(promote(x, y)..., RoundDown)
cld(x::Real, y::Real) = div(promote(x, y)..., RoundUp)
fld(x::Signed, y::Unsigned) = div(x, y, RoundDown)
fld(x::Unsigned, y::Signed) = div(x, y, RoundDown)
cld(x::Signed, y::Unsigned) = div(x, y, RoundUp)
cld(x::Unsigned, y::Signed) = div(x, y, RoundUp)
fld(x::T, y::T) where {T<:Real} = throw(MethodError(div, (x, y, RoundDown)))
cld(x::T, y::T) where {T<:Real} = throw(MethodError(div, (x, y, RoundUp)))
# Promotion
function div(x::Real, y::Real, r::RoundingMode)
typeof(x) === typeof(y) && throw(MethodError(div, (x, y, r)))
if r === RoundToZero
# For compat. Remove in 2.0.
div(promote(x, y)...)
else
div(promote(x, y)..., r)
end
end
# Integers
# fld(x, y) == div(x, y) - ((x >= 0) != (y >= 0) && rem(x, y) != 0 ? 1 : 0)
div(x::T, y::T, ::typeof(RoundDown)) where {T<:Unsigned} = div(x, y)
function div(x::T, y::T, ::typeof(RoundDown)) where T<:Integer
d = div(x, y, RoundToZero)
return d - (signbit(x ⊻ y) & (d * y != x))
end
# cld(x, y) = div(x, y) + ((x > 0) == (y > 0) && rem(x, y) != 0 ? 1 : 0)
function div(x::T, y::T, ::typeof(RoundUp)) where T<:Unsigned
d = div(x, y, RoundToZero)
return d + (d * y != x)
end
function div(x::T, y::T, ::typeof(RoundUp)) where T<:Integer
d = div(x, y, RoundToZero)
return d + (((x > 0) == (y > 0)) & (d * y != x))
end
# Real
# NOTE: C89 fmod() and x87 FPREM implicitly provide truncating float division,
# so it is used here as the basis of float div().
div(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat} = convert(T, round((x - rem(x, y, r)) / y))