rounding.jl
# This file is a part of Julia. License is MIT: https://julialang.org/license
module Rounding
let fenv_consts = Vector{Cint}(undef, 9)
ccall(:jl_get_fenv_consts, Cvoid, (Ptr{Cint},), fenv_consts)
global const JL_FE_INEXACT = fenv_consts[1]
global const JL_FE_UNDERFLOW = fenv_consts[2]
global const JL_FE_OVERFLOW = fenv_consts[3]
global const JL_FE_DIVBYZERO = fenv_consts[4]
global const JL_FE_INVALID = fenv_consts[5]
global const JL_FE_TONEAREST = fenv_consts[6]
global const JL_FE_UPWARD = fenv_consts[7]
global const JL_FE_DOWNWARD = fenv_consts[8]
global const JL_FE_TOWARDZERO = fenv_consts[9]
end
export
RoundingMode, RoundNearest, RoundToZero, RoundUp, RoundDown, RoundFromZero,
RoundNearestTiesAway, RoundNearestTiesUp,
rounding, setrounding,
get_zero_subnormals, set_zero_subnormals
## rounding modes ##
"""
RoundingMode
A type used for controlling the rounding mode of floating point operations (via
[`rounding`](@ref)/[`setrounding`](@ref) functions), or as
optional arguments for rounding to the nearest integer (via the [`round`](@ref)
function).
Currently supported rounding modes are:
- [`RoundNearest`](@ref) (default)
- [`RoundNearestTiesAway`](@ref)
- [`RoundNearestTiesUp`](@ref)
- [`RoundToZero`](@ref)
- [`RoundFromZero`](@ref)
- [`RoundUp`](@ref)
- [`RoundDown`](@ref)
!!! compat "Julia 1.9"
`RoundFromZero` requires at least Julia 1.9. Prior versions support
`RoundFromZero` for `BigFloat`s only.
"""
struct RoundingMode{T} end
"""
RoundNearest
The default rounding mode. Rounds to the nearest integer, with ties (fractional values of
0.5) being rounded to the nearest even integer.
"""
const RoundNearest = RoundingMode{:Nearest}()
"""
RoundToZero
[`round`](@ref) using this rounding mode is an alias for [`trunc`](@ref).
"""
const RoundToZero = RoundingMode{:ToZero}()
"""
RoundUp
[`round`](@ref) using this rounding mode is an alias for [`ceil`](@ref).
"""
const RoundUp = RoundingMode{:Up}()
"""
RoundDown
[`round`](@ref) using this rounding mode is an alias for [`floor`](@ref).
"""
const RoundDown = RoundingMode{:Down}()
"""
RoundFromZero
Rounds away from zero.
!!! compat "Julia 1.9"
`RoundFromZero` requires at least Julia 1.9. Prior versions support
`RoundFromZero` for `BigFloat`s only.
# Examples
```jldoctest
julia> BigFloat("1.0000000000000001", 5, RoundFromZero)
1.06
```
"""
const RoundFromZero = RoundingMode{:FromZero}()
"""
RoundNearestTiesAway
Rounds to nearest integer, with ties rounded away from zero (C/C++
[`round`](@ref) behaviour).
"""
const RoundNearestTiesAway = RoundingMode{:NearestTiesAway}()
"""
RoundNearestTiesUp
Rounds to nearest integer, with ties rounded toward positive infinity (Java/JavaScript
[`round`](@ref) behaviour).
"""
const RoundNearestTiesUp = RoundingMode{:NearestTiesUp}()
# Rounding mode predicates. TODO: better names
# Overload these for other rounding modes
rounds_to_nearest(::RoundingMode) = false
rounds_to_nearest(::RoundingMode{:Nearest}) = true
rounds_to_nearest(::RoundingMode{:NearestTiesUp}) = true
rounds_to_nearest(::RoundingMode{:NearestTiesAway}) = true
rounds_away_from_zero(::RoundingMode{:Up}, sign_bit::Bool) = !sign_bit
rounds_away_from_zero(::RoundingMode{:Down}, sign_bit::Bool) = sign_bit
rounds_away_from_zero(::RoundingMode{:FromZero}, ::Bool) = true
rounds_away_from_zero(::RoundingMode{:ToZero}, ::Bool) = false
tie_breaker_is_to_even(::RoundingMode{:Nearest}) = true
tie_breaker_is_to_even(::RoundingMode{:NearestTiesUp}) = false
tie_breaker_is_to_even(::RoundingMode{:NearestTiesAway}) = false
tie_breaker_rounds_away_from_zero(::RoundingMode{:NearestTiesUp}, sign_bit::Bool) = !sign_bit
tie_breaker_rounds_away_from_zero(::RoundingMode{:NearestTiesAway}, ::Bool) = true
rounds_to_nearest(t::Tuple{Any,Bool}) = rounds_to_nearest(first(t))
rounds_away_from_zero(t::Tuple{Any,Bool}) = rounds_away_from_zero(t...)
tie_breaker_is_to_even(t::Tuple{Any,Bool}) = tie_breaker_is_to_even(first(t))
tie_breaker_rounds_away_from_zero(t::Tuple{Any,Bool}) = tie_breaker_rounds_away_from_zero(t...)
struct FinalBit end
struct RoundBit end
struct StickyBit end
function correct_rounding_requires_increment(x, rounding_mode, sign_bit::Bool)
r = (rounding_mode, sign_bit)
f = let y = x
(z::Union{FinalBit,RoundBit,StickyBit}) -> y(z)::Bool
end
if rounds_to_nearest(r)
if f(RoundBit())
if f(StickyBit())
true
else
if tie_breaker_is_to_even(r)
f(FinalBit())
else
tie_breaker_rounds_away_from_zero(r)::Bool
end
end
else
false
end
else
if rounds_away_from_zero(r)
if f(RoundBit())
true
else
f(StickyBit())
end
else
false
end
end::Bool
end
to_fenv(::RoundingMode{:Nearest}) = JL_FE_TONEAREST
to_fenv(::RoundingMode{:ToZero}) = JL_FE_TOWARDZERO
to_fenv(::RoundingMode{:Up}) = JL_FE_UPWARD
to_fenv(::RoundingMode{:Down}) = JL_FE_DOWNWARD
function from_fenv(r::Integer)
if r == JL_FE_TONEAREST
return RoundNearest
elseif r == JL_FE_DOWNWARD
return RoundDown
elseif r == JL_FE_UPWARD
return RoundUp
elseif r == JL_FE_TOWARDZERO
return RoundToZero
else
throw(ArgumentError("invalid rounding mode code: $r"))
end
end
"""
setrounding(T, mode)
Set the rounding mode of floating point type `T`, controlling the rounding of basic
arithmetic functions ([`+`](@ref), [`-`](@ref), [`*`](@ref),
[`/`](@ref) and [`sqrt`](@ref)) and type conversion. Other numerical
functions may give incorrect or invalid values when using rounding modes other than the
default [`RoundNearest`](@ref).
Note that this is currently only supported for `T == BigFloat`.
!!! warning
This function is not thread-safe. It will affect code running on all threads, but
its behavior is undefined if called concurrently with computations that use the
setting.
"""
setrounding(T::Type, mode)
"""
rounding(T)
Get the current floating point rounding mode for type `T`, controlling the rounding of basic
arithmetic functions ([`+`](@ref), [`-`](@ref), [`*`](@ref), [`/`](@ref)
and [`sqrt`](@ref)) and type conversion.
See [`RoundingMode`](@ref) for available modes.
"""
:rounding
setrounding_raw(::Type{<:Union{Float32,Float64}}, i::Integer) = ccall(:jl_set_fenv_rounding, Int32, (Int32,), i)
rounding_raw(::Type{<:Union{Float32,Float64}}) = ccall(:jl_get_fenv_rounding, Int32, ())
rounding(::Type{T}) where {T<:Union{Float32,Float64}} = from_fenv(rounding_raw(T))
"""
setrounding(f::Function, T, mode)
Change the rounding mode of floating point type `T` for the duration of `f`. It is logically
equivalent to:
old = rounding(T)
setrounding(T, mode)
f()
setrounding(T, old)
See [`RoundingMode`](@ref) for available rounding modes.
"""
function setrounding(f::Function, ::Type{T}, rounding::RoundingMode) where T
old_rounding_raw = rounding_raw(T)
setrounding(T,rounding)
try
return f()
finally
setrounding_raw(T,old_rounding_raw)
end
end
function setrounding_raw(f::Function, ::Type{T}, rounding) where T
old_rounding_raw = rounding_raw(T)
setrounding_raw(T,rounding)
try
return f()
finally
setrounding_raw(T,old_rounding_raw)
end
end
# Should be equivalent to:
# setrounding(Float64,r) do
# convert(T,x)
# end
# but explicit checks are currently quicker (~20x).
# Assumes conversion is performed by rounding to nearest value.
# To avoid ambiguous dispatch with methods in mpfr.jl:
(::Type{T})(x::Real, r::RoundingMode) where {T<:AbstractFloat} = _convert_rounding(T,x,r)::T
_convert_rounding(::Type{T}, x::Real, r::RoundingMode{:Nearest}) where {T<:AbstractFloat} = convert(T,x)::T
function _convert_rounding(::Type{T}, x::Real, r::RoundingMode{:Down}) where T<:AbstractFloat
y = convert(T,x)::T
y > x ? prevfloat(y) : y
end
function _convert_rounding(::Type{T}, x::Real, r::RoundingMode{:Up}) where T<:AbstractFloat
y = convert(T,x)::T
y < x ? nextfloat(y) : y
end
function _convert_rounding(::Type{T}, x::Real, r::RoundingMode{:ToZero}) where T<:AbstractFloat
y = convert(T,x)::T
if x > 0.0
y > x ? prevfloat(y) : y
else
y < x ? nextfloat(y) : y
end
end
# Default definitions
"""
set_zero_subnormals(yes::Bool) -> Bool
If `yes` is `false`, subsequent floating-point operations follow rules for IEEE arithmetic
on subnormal values ("denormals"). Otherwise, floating-point operations are permitted (but
not required) to convert subnormal inputs or outputs to zero. Returns `true` unless
`yes==true` but the hardware does not support zeroing of subnormal numbers.
`set_zero_subnormals(true)` can speed up some computations on some hardware. However, it can
break identities such as `(x-y==0) == (x==y)`.
!!! warning
This function only affects the current thread.
"""
set_zero_subnormals(yes::Bool) = ccall(:jl_set_zero_subnormals,Int32,(Int8,),yes)==0
"""
get_zero_subnormals() -> Bool
Return `false` if operations on subnormal floating-point values ("denormals") obey rules
for IEEE arithmetic, and `true` if they might be converted to zeros.
!!! warning
This function only affects the current thread.
"""
get_zero_subnormals() = ccall(:jl_get_zero_subnormals,Int32,())!=0
end #module
using .Rounding
"""
round([T,] x, [r::RoundingMode])
round(x, [r::RoundingMode]; digits::Integer=0, base = 10)
round(x, [r::RoundingMode]; sigdigits::Integer, base = 10)
Rounds the number `x`.
Without keyword arguments, `x` is rounded to an integer value, returning a value of type
`T`, or of the same type of `x` if no `T` is provided. An [`InexactError`](@ref) will be
thrown if the value is not representable by `T`, similar to [`convert`](@ref).
If the `digits` keyword argument is provided, it rounds to the specified number of digits
after the decimal place (or before if negative), in base `base`.
If the `sigdigits` keyword argument is provided, it rounds to the specified number of
significant digits, in base `base`.
The [`RoundingMode`](@ref) `r` controls the direction of the rounding; the default is
[`RoundNearest`](@ref), which rounds to the nearest integer, with ties (fractional values
of 0.5) being rounded to the nearest even integer. Note that `round` may give incorrect
results if the global rounding mode is changed (see [`rounding`](@ref)).
# Examples
```jldoctest
julia> round(1.7)
2.0
julia> round(Int, 1.7)
2
julia> round(1.5)
2.0
julia> round(2.5)
2.0
julia> round(pi; digits=2)
3.14
julia> round(pi; digits=3, base=2)
3.125
julia> round(123.456; sigdigits=2)
120.0
julia> round(357.913; sigdigits=4, base=2)
352.0
```
!!! note
Rounding to specified digits in bases other than 2 can be inexact when
operating on binary floating point numbers. For example, the [`Float64`](@ref)
value represented by `1.15` is actually *less* than 1.15, yet will be
rounded to 1.2. For example:
```jldoctest
julia> x = 1.15
1.15
julia> big(1.15)
1.149999999999999911182158029987476766109466552734375
julia> x < 115//100
true
julia> round(x, digits=1)
1.2
```
# Extensions
To extend `round` to new numeric types, it is typically sufficient to define `Base.round(x::NewType, r::RoundingMode)`.
"""
function round end
"""
trunc([T,] x)
trunc(x; digits::Integer= [, base = 10])
trunc(x; sigdigits::Integer= [, base = 10])
`trunc(x)` returns the nearest integral value of the same type as `x` whose absolute value
is less than or equal to the absolute value of `x`.
`trunc(T, x)` converts the result to type `T`, throwing an `InexactError` if the truncated
value is not representable a `T`.
Keywords `digits`, `sigdigits` and `base` work as for [`round`](@ref).
To support `trunc` for a new type, define `Base.round(x::NewType, ::RoundingMode{:ToZero})`.
See also: [`%`](@ref rem), [`floor`](@ref), [`unsigned`](@ref), [`unsafe_trunc`](@ref).
# Examples
```jldoctest
julia> trunc(2.22)
2.0
julia> trunc(-2.22, digits=1)
-2.2
julia> trunc(Int, -2.22)
-2
```
"""
function trunc end
"""
floor([T,] x)
floor(x; digits::Integer= [, base = 10])
floor(x; sigdigits::Integer= [, base = 10])
`floor(x)` returns the nearest integral value of the same type as `x` that is less than or
equal to `x`.
`floor(T, x)` converts the result to type `T`, throwing an `InexactError` if the floored
value is not representable a `T`.
Keywords `digits`, `sigdigits` and `base` work as for [`round`](@ref).
To support `floor` for a new type, define `Base.round(x::NewType, ::RoundingMode{:Down})`.
"""
function floor end
"""
ceil([T,] x)
ceil(x; digits::Integer= [, base = 10])
ceil(x; sigdigits::Integer= [, base = 10])
`ceil(x)` returns the nearest integral value of the same type as `x` that is greater than or
equal to `x`.
`ceil(T, x)` converts the result to type `T`, throwing an `InexactError` if the ceiled
value is not representable as a `T`.
Keywords `digits`, `sigdigits` and `base` work as for [`round`](@ref).
To support `ceil` for a new type, define `Base.round(x::NewType, ::RoundingMode{:Up})`.
"""
function ceil end
trunc(x; kws...) = round(x, RoundToZero; kws...)
floor(x; kws...) = round(x, RoundDown; kws...)
ceil(x; kws...) = round(x, RoundUp; kws...)
round(x; kws...) = round(x, RoundNearest; kws...)
trunc(::Type{T}, x) where T = round(T, x, RoundToZero)
floor(::Type{T}, x) where T = round(T, x, RoundDown)
ceil(::Type{T}, x) where T = round(T, x, RoundUp)
round(::Type{T}, x) where T = round(T, x, RoundNearest)
round(::Type{T}, x, r::RoundingMode) where T = convert(T, round(x, r))
round(x::Integer, r::RoundingMode) = x