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div with rounding modes [+ rounded division] (#33040)
* Re-arrange fld/cld code In preparation for supporting other rounding modes in div, create a three-argument div function that takes a rounding mode as the last argument and make this the fundamental fallback for fld/cld. * Implemented rounded division * add various divrem combinations to avoid overflow * Whitespace/test fixes * Small tweaks to docstrings * Bugfixes * Add the exhaustive test * Tigthen up types for div fallback I think it's better to give a MethodError here than an approximate answer for non-AbstractFloat reals (e.g. custom integer types).
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| Original file line number | Diff line number | Diff line change |
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| # Div is truncating by default | ||
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| """ | ||
| div(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 | ||
| y*round(x/y,r) | ||
| without any intermediate rounding. | ||
| See also: [`fld`](@ref), [`cld`](@ref) which are special cases of this function | ||
| # Examples: | ||
| ```jldoctest | ||
| 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 | ||
| ``` | ||
| """ | ||
| div(x, y, r::RoundingMode) | ||
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| div(a, b) = div(a, b, RoundToZero) | ||
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| """ | ||
| 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). | ||
| """ | ||
| rem(x, y, r::RoundingMode) | ||
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| # 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) | ||
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| """ | ||
| fld(x, y) | ||
| Largest integer less than or equal to `x/y`. Equivalent to `div(x, y, RoundDown)`. | ||
| See also: [`div`](@ref) | ||
| # Examples | ||
| ```jldoctest | ||
| julia> fld(7.3,5.5) | ||
| 1.0 | ||
| ``` | ||
| """ | ||
| fld(a, b) = div(a, b, RoundDown) | ||
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| """ | ||
| cld(x, y) | ||
| Smallest integer larger than or equal to `x/y`. Equivalent to `div(x, y, RoundUp)`. | ||
| See also: [`div`](@ref) | ||
| # Examples | ||
| ```jldoctest | ||
| julia> cld(5.5,2.2) | ||
| 3.0 | ||
| ``` | ||
| """ | ||
| cld(a, b) = div(a, b, RoundUp) | ||
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| # 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 the default | ||
| value of `r`, this call is equivalent to `(x÷y, x%y)`. | ||
| # Examples | ||
| ```jldoctest | ||
| julia> divrem(3,7) | ||
| (0, 3) | ||
| julia> divrem(7,3) | ||
| (2, 1) | ||
| ``` | ||
| """ | ||
| divrem(x, y) = divrem(x, y, RoundToZero) | ||
| divrem(a, b, r::RoundingMode) = (div(a, b, r), rem(a, b, r)) | ||
| 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 | ||
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| """ | ||
| 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))`. | ||
| """ | ||
| fldmod(x,y) = divrem(x, y, RoundDown) | ||
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| # 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 | ||
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| 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 | ||
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| function div(x::Integer, y::Integer, rnd::Union{typeof(RoundNearest), | ||
| typeof(RoundNearestTiesAway), | ||
| typeof(RoundNearestTiesUp)}) | ||
| divrem(x,y,rnd)[1] | ||
| end | ||
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| # 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) | ||
| @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) | ||
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| # 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))) | ||
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| # Promotion | ||
| div(x::Real, y::Real, r::RoundingMode) = div(promote(x, y)..., r) | ||
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| # 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 | ||
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| # 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 | ||
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| # 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)) | ||
| rem(x::T, y::T, ::typeof(RoundUp)) where {T<:AbstractFloat} = convert(T,x-y*ceil(x/y)) |
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