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Move `round(T::Type, x)` docstring above `round(z::Complex, ...)` docstring (#50775)

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/base/floatfuncs.jl

# This file is a part of Julia. License is MIT: https://julialang.org/license

## floating-point functions ##

copysign(x::Float64, y::Float64) = copysign_float(x, y)
copysign(x::Float32, y::Float32) = copysign_float(x, y)
copysign(x::Float32, y::Real) = copysign(x, Float32(y))
copysign(x::Float64, y::Real) = copysign(x, Float64(y))

flipsign(x::Float64, y::Float64) = bitcast(Float64, xor_int(bitcast(UInt64, x), and_int(bitcast(UInt64, y), 0x8000000000000000)))
flipsign(x::Float32, y::Float32) = bitcast(Float32, xor_int(bitcast(UInt32, x), and_int(bitcast(UInt32, y), 0x80000000)))
flipsign(x::Float32, y::Real) = flipsign(x, Float32(y))
flipsign(x::Float64, y::Real) = flipsign(x, Float64(y))

signbit(x::Float64) = signbit(bitcast(Int64, x))
signbit(x::Float32) = signbit(bitcast(Int32, x))
signbit(x::Float16) = signbit(bitcast(Int16, x))

"""
    maxintfloat(T=Float64)

The largest consecutive integer-valued floating-point number that is exactly represented in
the given floating-point type `T` (which defaults to `Float64`).

That is, `maxintfloat` returns the smallest positive integer-valued floating-point number
`n` such that `n+1` is *not* exactly representable in the type `T`.

When an `Integer`-type value is needed, use `Integer(maxintfloat(T))`.
"""
maxintfloat(::Type{Float64}) = 9007199254740992.
maxintfloat(::Type{Float32}) = Float32(16777216.)
maxintfloat(::Type{Float16}) = Float16(2048f0)
maxintfloat(x::T) where {T<:AbstractFloat} = maxintfloat(T)

"""
    maxintfloat(T, S)

The largest consecutive integer representable in the given floating-point type `T` that
also does not exceed the maximum integer representable by the integer type `S`.  Equivalently,
it is the minimum of `maxintfloat(T)` and [`typemax(S)`](@ref).
"""
maxintfloat(::Type{S}, ::Type{T}) where {S<:AbstractFloat, T<:Integer} = min(maxintfloat(S), S(typemax(T)))
maxintfloat() = maxintfloat(Float64)

isinteger(x::AbstractFloat) = (x - trunc(x) == 0)

# See rounding.jl for docstring.

function round(::Type{T}, x::AbstractFloat, r::RoundingMode) where {T<:Integer}
    r != RoundToZero && (x = round(x,r))
    trunc(T, x)
end

# NOTE: this relies on the current keyword dispatch behaviour (#9498).
function round(x::Real, r::RoundingMode=RoundNearest;
               digits::Union{Nothing,Integer}=nothing, sigdigits::Union{Nothing,Integer}=nothing, base::Union{Nothing,Integer}=nothing)
    if digits === nothing
        if sigdigits === nothing
            if base === nothing
                # avoid recursive calls
                throw(MethodError(round, (x,r)))
            else
                round(x,r)
                # or throw(ArgumentError("`round` cannot use `base` argument without `digits` or `sigdigits` arguments."))
            end
        else
            isfinite(x) || return float(x)
            _round_sigdigits(x, r, sigdigits, base === nothing ? 10 : base)
        end
    else
        if sigdigits === nothing
            isfinite(x) || return float(x)
            _round_digits(x, r, digits, base === nothing ? 10 : base)
        else
            throw(ArgumentError("`round` cannot use both `digits` and `sigdigits` arguments."))
        end
    end
end

trunc(x::Real; kwargs...) = round(x, RoundToZero; kwargs...)
floor(x::Real; kwargs...) = round(x, RoundDown; kwargs...)
ceil(x::Real; kwargs...)  = round(x, RoundUp; kwargs...)

# fallbacks
trunc(::Type{T}, x::Real; kwargs...) where {T} = round(T, x, RoundToZero; kwargs...)
floor(::Type{T}, x::Real; kwargs...) where {T} = round(T, x, RoundDown; kwargs...)
ceil(::Type{T}, x::Real; kwargs...) where {T} = round(T, x, RoundUp; kwargs...)
round(::Type{T}, x::Real; kwargs...) where {T} = round(T, x, RoundNearest; kwargs...)

round(::Type{T}, x::Real, r::RoundingMode) where {T} = convert(T, round(x, r))

round(x::Integer, r::RoundingMode) = x

# round x to multiples of 1/invstep
function _round_invstep(x, invstep, r::RoundingMode)
    y = round(x * invstep, r) / invstep
    if !isfinite(y)
        return x
    end
    return y
end

# round x to multiples of 1/(invstepsqrt^2)
# Using square root of step prevents overflowing
function _round_invstepsqrt(x, invstepsqrt, r::RoundingMode)
    y = round((x * invstepsqrt) * invstepsqrt, r) / invstepsqrt / invstepsqrt
    if !isfinite(y)
        return x
    end
    return y
end

# round x to multiples of step
function _round_step(x, step, r::RoundingMode)
    # TODO: use div with rounding mode
    y = round(x / step, r) * step
    if !isfinite(y)
        if x > 0
            return (r == RoundUp ? oftype(x, Inf) : zero(x))
        elseif x < 0
            return (r == RoundDown ? -oftype(x, Inf) : -zero(x))
        else
            return x
        end
    end
    return y
end

function _round_digits(x, r::RoundingMode, digits::Integer, base)
    fx = float(x)
    if digits >= 0
        invstep = oftype(fx, base)^digits
        if isfinite(invstep)
            return _round_invstep(fx, invstep, r)
        else
            invstepsqrt = oftype(fx, base)^oftype(fx, digits/2)
            return _round_invstepsqrt(fx, invstepsqrt, r)
        end
    else
        step = oftype(fx, base)^-digits
        return _round_step(fx, step, r)
    end
end

hidigit(x::Integer, base) = ndigits0z(x, base)
function hidigit(x::AbstractFloat, base)
    iszero(x) && return 0
    if base == 10
        return 1 + floor(Int, log10(abs(x)))
    elseif base == 2
        return 1 + exponent(x)
    else
        return 1 + floor(Int, log(base, abs(x)))
    end
end
hidigit(x::Real, base) = hidigit(float(x), base)

function _round_sigdigits(x, r::RoundingMode, sigdigits::Integer, base)
    h = hidigit(x, base)
    _round_digits(x, r, sigdigits-h, base)
end

# C-style round
function round(x::AbstractFloat, ::RoundingMode{:NearestTiesAway})
    y = trunc(x)
    ifelse(x==y,y,trunc(2*x-y))
end
# Java-style round
function round(x::T, ::RoundingMode{:NearestTiesUp}) where {T <: AbstractFloat}
    copysign(floor((x + (T(0.25) - eps(T(0.5)))) + (T(0.25) + eps(T(0.5)))), x)
end

function Base.round(x::AbstractFloat, ::typeof(RoundFromZero))
    signbit(x) ? round(x, RoundDown) : round(x, RoundUp)
end

# isapprox: approximate equality of numbers
"""
    isapprox(x, y; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps, nans::Bool=false[, norm::Function])

Inexact equality comparison. Two numbers compare equal if their relative distance *or* their
absolute distance is within tolerance bounds: `isapprox` returns `true` if
`norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))`. The default `atol` (absolute tolerance) is zero and the
default `rtol` (relative tolerance) depends on the types of `x` and `y`. The keyword argument `nans` determines
whether or not NaN values are considered equal (defaults to false).

For real or complex floating-point values, if an `atol > 0` is not specified, `rtol` defaults to
the square root of [`eps`](@ref) of the type of `x` or `y`, whichever is bigger (least precise).
This corresponds to requiring equality of about half of the significant digits. Otherwise,
e.g. for integer arguments or if an `atol > 0` is supplied, `rtol` defaults to zero.

The `norm` keyword defaults to `abs` for numeric `(x,y)` and to `LinearAlgebra.norm` for
arrays (where an alternative `norm` choice is sometimes useful).
When `x` and `y` are arrays, if `norm(x-y)` is not finite (i.e. `±Inf`
or `NaN`), the comparison falls back to checking whether all elements of `x` and `y` are
approximately equal component-wise.

The binary operator `≈` is equivalent to `isapprox` with the default arguments, and `x ≉ y`
is equivalent to `!isapprox(x,y)`.

Note that `x ≈ 0` (i.e., comparing to zero with the default tolerances) is
equivalent to `x == 0` since the default `atol` is `0`.  In such cases, you should either
supply an appropriate `atol` (or use `norm(x) ≤ atol`) or rearrange your code (e.g.
use `x ≈ y` rather than `x - y ≈ 0`).   It is not possible to pick a nonzero `atol`
automatically because it depends on the overall scaling (the "units") of your problem:
for example, in `x - y ≈ 0`, `atol=1e-9` is an absurdly small tolerance if `x` is the
[radius of the Earth](https://en.wikipedia.org/wiki/Earth_radius) in meters,
but an absurdly large tolerance if `x` is the
[radius of a Hydrogen atom](https://en.wikipedia.org/wiki/Bohr_radius) in meters.

!!! compat "Julia 1.6"
    Passing the `norm` keyword argument when comparing numeric (non-array) arguments
    requires Julia 1.6 or later.

# Examples
```jldoctest
julia> isapprox(0.1, 0.15; atol=0.05)
true

julia> isapprox(0.1, 0.15; rtol=0.34)
true

julia> isapprox(0.1, 0.15; rtol=0.33)
false

julia> 0.1 + 1e-10 ≈ 0.1
true

julia> 1e-10 ≈ 0
false

julia> isapprox(1e-10, 0, atol=1e-8)
true

julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0]) # using `norm`
true
```
"""
function isapprox(x::Number, y::Number;
                  atol::Real=0, rtol::Real=rtoldefault(x,y,atol),
                  nans::Bool=false, norm::Function=abs)
    x == y || (isfinite(x) && isfinite(y) && norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))) || (nans && isnan(x) && isnan(y))
end

"""
    isapprox(x; kwargs...) / ≈(x; kwargs...)

Create a function that compares its argument to `x` using `≈`, i.e. a function equivalent to `y -> y ≈ x`.

The keyword arguments supported here are the same as those in the 2-argument `isapprox`.

!!! compat "Julia 1.5"
    This method requires Julia 1.5 or later.
"""
isapprox(y; kwargs...) = x -> isapprox(x, y; kwargs...)

const ≈ = isapprox
"""
    x ≉ y

This is equivalent to `!isapprox(x,y)` (see [`isapprox`](@ref)).
"""
≉(args...; kws...) = !≈(args...; kws...)

# default tolerance arguments
rtoldefault(::Type{T}) where {T<:AbstractFloat} = sqrt(eps(T))
rtoldefault(::Type{<:Real}) = 0
function rtoldefault(x::Union{T,Type{T}}, y::Union{S,Type{S}}, atol::Real) where {T<:Number,S<:Number}
    rtol = max(rtoldefault(real(T)),rtoldefault(real(S)))
    return atol > 0 ? zero(rtol) : rtol
end

# fused multiply-add

"""
    fma(x, y, z)

Computes `x*y+z` without rounding the intermediate result `x*y`. On some systems this is
significantly more expensive than `x*y+z`. `fma` is used to improve accuracy in certain
algorithms. See [`muladd`](@ref).
"""
function fma end
function fma_emulated(a::Float32, b::Float32, c::Float32)::Float32
    ab = Float64(a) * b
    res = ab+c
    reinterpret(UInt64, res)&0x1fff_ffff!=0x1000_0000 && return res
    # yes error compensation is necessary. It sucks
    reslo = abs(c)>abs(ab) ? ab-(res - c) : c-(res - ab)
    res = iszero(reslo) ? res : (signbit(reslo) ? prevfloat(res) : nextfloat(res))
    return res
end

""" Splits a Float64 into a hi bit and a low bit where the high bit has 27 trailing 0s and the low bit has 26 trailing 0s"""
@inline function splitbits(x::Float64)
    hi = reinterpret(Float64, reinterpret(UInt64, x) & 0xffff_ffff_f800_0000)
    return hi, x-hi
end

function twomul(a::Float64, b::Float64)
    ahi, alo = splitbits(a)
    bhi, blo = splitbits(b)
    abhi = a*b
    blohi, blolo = splitbits(blo)
    ablo = alo*blohi - (((abhi - ahi*bhi) - alo*bhi) - ahi*blo) + blolo*alo
    return abhi, ablo
end

function fma_emulated(a::Float64, b::Float64,c::Float64)
    abhi, ablo = @inline twomul(a,b)
    if !isfinite(abhi+c) || isless(abs(abhi), nextfloat(0x1p-969)) || issubnormal(a) || issubnormal(b)
        aandbfinite = isfinite(a) && isfinite(b)
        if !(isfinite(c) && aandbfinite)
            return aandbfinite ? c : abhi+c
        end
        (iszero(a) || iszero(b)) && return abhi+c
        # The checks above satisfy exponent's nothrow precondition
        bias = Math._exponent_finite_nonzero(a) + Math._exponent_finite_nonzero(b)
        c_denorm = ldexp(c, -bias)
        if isfinite(c_denorm)
            # rescale a and b to [1,2), equivalent to ldexp(a, -exponent(a))
            issubnormal(a) && (a *= 0x1p52)
            issubnormal(b) && (b *= 0x1p52)
            a = reinterpret(Float64, (reinterpret(UInt64, a) & ~Base.exponent_mask(Float64)) | Base.exponent_one(Float64))
            b = reinterpret(Float64, (reinterpret(UInt64, b) & ~Base.exponent_mask(Float64)) | Base.exponent_one(Float64))
            c = c_denorm
            abhi, ablo = twomul(a,b)
            # abhi <= 4 -> isfinite(r)      (α)
            r = abhi+c
            # s ≈ 0                         (β)
            s = (abs(abhi) > abs(c)) ? (abhi-r+c+ablo) : (c-r+abhi+ablo)
            # α ⩓ β -> isfinite(sumhi)      (γ)
            sumhi = r+s
            # If result is subnormal, ldexp will cause double rounding because subnormals have fewer mantisa bits.
            # As such, we need to check whether round to even would lead to double rounding and manually round sumhi to avoid it.
            if issubnormal(ldexp(sumhi, bias))
                sumlo = r-sumhi+s
                # finite: See γ
                # non-zero: If sumhi == ±0., then ldexp(sumhi, bias) == ±0,
                # so we don't take this branch.
                bits_lost = -bias-Math._exponent_finite_nonzero(sumhi)-1022
                sumhiInt = reinterpret(UInt64, sumhi)
                if (bits_lost != 1) ⊻ (sumhiInt&1 == 1)
                    sumhi = nextfloat(sumhi, cmp(sumlo,0))
                end
            end
            return ldexp(sumhi, bias)
        end
        isinf(abhi) && signbit(c) == signbit(a*b) && return abhi
        # fall through
    end
    r = abhi+c
    s = (abs(abhi) > abs(c)) ? (abhi-r+c+ablo) : (c-r+abhi+ablo)
    return r+s
end
fma_llvm(x::Float32, y::Float32, z::Float32) = fma_float(x, y, z)
fma_llvm(x::Float64, y::Float64, z::Float64) = fma_float(x, y, z)

# Disable LLVM's fma if it is incorrect, e.g. because LLVM falls back
# onto a broken system libm; if so, use a software emulated fma
@assume_effects :consistent fma(x::Float32, y::Float32, z::Float32) = Core.Intrinsics.have_fma(Float32) ? fma_llvm(x,y,z) : fma_emulated(x,y,z)
@assume_effects :consistent fma(x::Float64, y::Float64, z::Float64) = Core.Intrinsics.have_fma(Float64) ? fma_llvm(x,y,z) : fma_emulated(x,y,z)

function fma(a::Float16, b::Float16, c::Float16)
    Float16(muladd(Float32(a), Float32(b), Float32(c))) #don't use fma if the hardware doesn't have it.
end

# This is necessary at least on 32-bit Intel Linux, since fma_llvm may
# have called glibc, and some broken glibc fma implementations don't
# properly restore the rounding mode
Rounding.setrounding_raw(Float32, Rounding.JL_FE_TONEAREST)
Rounding.setrounding_raw(Float64, Rounding.JL_FE_TONEAREST)

1	# This file is a part of Julia. License is MIT: https://julialang.org/license
2
3	## floating-point functions ##
4
5	copysign(x::Float64, y::Float64) = copysign_float(x, y)	3,184,897✔
6	copysign(x::Float32, y::Float32) = copysign_float(x, y)	1,549,382✔
7	copysign(x::Float32, y::Real) = copysign(x, Float32(y))	51✔
8	copysign(x::Float64, y::Real) = copysign(x, Float64(y))	104✔
9
10	flipsign(x::Float64, y::Float64) = bitcast(Float64, xor_int(bitcast(UInt64, x), and_int(bitcast(UInt64, y), 0x8000000000000000)))	116,413✔
11	flipsign(x::Float32, y::Float32) = bitcast(Float32, xor_int(bitcast(UInt32, x), and_int(bitcast(UInt32, y), 0x80000000)))	123✔
12	flipsign(x::Float32, y::Real) = flipsign(x, Float32(y))	63✔
13	flipsign(x::Float64, y::Real) = flipsign(x, Float64(y))	251✔
14
15	signbit(x::Float64) = signbit(bitcast(Int64, x))	23,808,369✔
16	signbit(x::Float32) = signbit(bitcast(Int32, x))	17,245,800✔
17	signbit(x::Float16) = signbit(bitcast(Int16, x))	10,304✔
18
19	"""
20	maxintfloat(T=Float64)
21
22	The largest consecutive integer-valued floating-point number that is exactly represented in
23	the given floating-point type `T` (which defaults to `Float64`).
24
25	That is, `maxintfloat` returns the smallest positive integer-valued floating-point number
26	`n` such that `n+1` is not exactly representable in the type `T`.
27
28	When an `Integer`-type value is needed, use `Integer(maxintfloat(T))`.
29	"""
30	maxintfloat(::Type{Float64}) = 9007199254740992.	855,395✔
31	maxintfloat(::Type{Float32}) = Float32(16777216.)	3,125,397✔
32	maxintfloat(::Type{Float16}) = Float16(2048f0)	2,284,972✔
33	maxintfloat(x::T) where {T<:AbstractFloat} = maxintfloat(T)	63✔
34
35	"""
36	maxintfloat(T, S)
37
38	The largest consecutive integer representable in the given floating-point type `T` that
39	also does not exceed the maximum integer representable by the integer type `S`. Equivalently,
40	it is the minimum of `maxintfloat(T)` and [`typemax(S)`](@ref).
41	"""
42	maxintfloat(::Type{S}, ::Type{T}) where {S<:AbstractFloat, T<:Integer} = min(maxintfloat(S), S(typemax(T)))	6,262,364✔
43	maxintfloat() = maxintfloat(Float64)	1✔
44
45	isinteger(x::AbstractFloat) = (x - trunc(x) == 0)	24,004,703✔
46
47	# See rounding.jl for docstring.
48
49	function round(::Type{T}, x::AbstractFloat, r::RoundingMode) where {T<:Integer}	2,180,262✔
50	r != RoundToZero && (x = round(x,r))	11,197,877✔
51	trunc(T, x)	11,197,890✔
52	end
53
54	# NOTE: this relies on the current keyword dispatch behaviour (#9498).
55	function round(x::Real, r::RoundingMode=RoundNearest;	716,783✔
56	digits::Union{Nothing,Integer}=nothing, sigdigits::Union{Nothing,Integer}=nothing, base::Union{Nothing,Integer}=nothing)
57	if digits === nothing	280✔
58	if sigdigits === nothing	47✔
59	if base === nothing	1✔
60	# avoid recursive calls
61	throw(MethodError(round, (x,r)))	×
62	else
63	round(x,r)	1✔
64	# or throw(ArgumentError("`round` cannot use `base` argument without `digits` or `sigdigits` arguments."))
65	end
66	else
67	isfinite(x) \|\| return float(x)	48✔
68	_round_sigdigits(x, r, sigdigits, base === nothing ? 10 : base)	42✔
69	end
70	else
71	if sigdigits === nothing	233✔
72	isfinite(x) \|\| return float(x)	22,654✔
73	_round_digits(x, r, digits, base === nothing ? 10 : base)	22,646✔
74	else
75	throw(ArgumentError("`round` cannot use both `digits` and `sigdigits` arguments."))	×
76	end
77	end
78	end
79
80	trunc(x::Real; kwargs...) = round(x, RoundToZero; kwargs...)	48,222,085✔
81	floor(x::Real; kwargs...) = round(x, RoundDown; kwargs...)	118,469✔
82	ceil(x::Real; kwargs...) = round(x, RoundUp; kwargs...)	17,444✔
83
84	# fallbacks
85	trunc(::Type{T}, x::Real; kwargs...) where {T} = round(T, x, RoundToZero; kwargs...)	309✔
86	floor(::Type{T}, x::Real; kwargs...) where {T} = round(T, x, RoundDown; kwargs...)	436,279✔
87	ceil(::Type{T}, x::Real; kwargs...) where {T} = round(T, x, RoundUp; kwargs...)	1,274,199✔
88	round(::Type{T}, x::Real; kwargs...) where {T} = round(T, x, RoundNearest; kwargs...)	20,658,901✔
89
90	round(::Type{T}, x::Real, r::RoundingMode) where {T} = convert(T, round(x, r))	3✔
91
92	round(x::Integer, r::RoundingMode) = x	387✔
93
94	# round x to multiples of 1/invstep
95	function _round_invstep(x, invstep, r::RoundingMode)	142✔
96	y = round(x * invstep, r) / invstep	22,598✔
97	if !isfinite(y)	22,598✔
98	return x	×
99	end
100	return y	22,598✔
101	end
102
103	# round x to multiples of 1/(invstepsqrt^2)
104	# Using square root of step prevents overflowing
105	function _round_invstepsqrt(x, invstepsqrt, r::RoundingMode)	9✔
106	y = round((x * invstepsqrt) * invstepsqrt, r) / invstepsqrt / invstepsqrt	26✔
107	if !isfinite(y)	26✔
108	return x	16✔
109	end
110	return y	10✔
111	end
112
113	# round x to multiples of step
114	function _round_step(x, step, r::RoundingMode)	6✔
115	# TODO: use div with rounding mode
116	y = round(x / step, r) * step	15✔
117	if !isfinite(y)	15✔
118	if x > 0	12✔
119	return (r == RoundUp ? oftype(x, Inf) : zero(x))	4✔
120	elseif x < 0	8✔
121	return (r == RoundDown ? -oftype(x, Inf) : -zero(x))	4✔
122	else
123	return x	4✔
124	end
125	end
126	return y	3✔
127	end
128
129	function _round_digits(x, r::RoundingMode, digits::Integer, base)	22,639✔
130	fx = float(x)	157✔
131	if digits >= 0	22,639✔
132	invstep = oftype(fx, base)^digits	45,227✔
133	if isfinite(invstep)	22,624✔
134	return _round_invstep(fx, invstep, r)	22,598✔
135	else
136	invstepsqrt = oftype(fx, base)^oftype(fx, digits/2)	26✔
137	return _round_invstepsqrt(fx, invstepsqrt, r)	26✔
138	end
139	else
140	step = oftype(fx, base)^-digits	24✔
141	return _round_step(fx, step, r)	15✔
142	end
143	end
144
145	hidigit(x::Integer, base) = ndigits0z(x, base)	×
146	function hidigit(x::AbstractFloat, base)	42✔
147	iszero(x) && return 0	42✔
148	if base == 10	40✔
149	return 1 + floor(Int, log10(abs(x)))	36✔
150	elseif base == 2	4✔
151	return 1 + exponent(x)	3✔
152	else
153	return 1 + floor(Int, log(base, abs(x)))	1✔
154	end
155	end
156	hidigit(x::Real, base) = hidigit(float(x), base)	3✔
157
158	function _round_sigdigits(x, r::RoundingMode, sigdigits::Integer, base)	42✔
159	h = hidigit(x, base)	42✔
160	_round_digits(x, r, sigdigits-h, base)	42✔
161	end
162
163	# C-style round
164	function round(x::AbstractFloat, ::RoundingMode{:NearestTiesAway})	50,594✔
165	y = trunc(x)	50,594✔
166	ifelse(x==y,y,trunc(2*x-y))	50,594✔
167	end
168	# Java-style round
169	function round(x::T, ::RoundingMode{:NearestTiesUp}) where {T <: AbstractFloat}	50,594✔
170	copysign(floor((x + (T(0.25) - eps(T(0.5)))) + (T(0.25) + eps(T(0.5)))), x)	50,594✔
171	end
172
173	function Base.round(x::AbstractFloat, ::typeof(RoundFromZero))	50,552✔
174	signbit(x) ? round(x, RoundDown) : round(x, RoundUp)	51,076✔
175	end
176
177	# isapprox: approximate equality of numbers
178	"""
179	isapprox(x, y; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps, nans::Bool=false[, norm::Function])
180
181	Inexact equality comparison. Two numbers compare equal if their relative distance or their
182	absolute distance is within tolerance bounds: `isapprox` returns `true` if
183	`norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))`. The default `atol` (absolute tolerance) is zero and the
184	default `rtol` (relative tolerance) depends on the types of `x` and `y`. The keyword argument `nans` determines
185	whether or not NaN values are considered equal (defaults to false).
186
187	For real or complex floating-point values, if an `atol > 0` is not specified, `rtol` defaults to
188	the square root of [`eps`](@ref) of the type of `x` or `y`, whichever is bigger (least precise).
189	This corresponds to requiring equality of about half of the significant digits. Otherwise,
190	e.g. for integer arguments or if an `atol > 0` is supplied, `rtol` defaults to zero.
191
192	The `norm` keyword defaults to `abs` for numeric `(x,y)` and to `LinearAlgebra.norm` for
193	arrays (where an alternative `norm` choice is sometimes useful).
194	When `x` and `y` are arrays, if `norm(x-y)` is not finite (i.e. `±Inf`
195	or `NaN`), the comparison falls back to checking whether all elements of `x` and `y` are
196	approximately equal component-wise.
197
198	The binary operator `≈` is equivalent to `isapprox` with the default arguments, and `x ≉ y`
199	is equivalent to `!isapprox(x,y)`.
200
201	Note that `x ≈ 0` (i.e., comparing to zero with the default tolerances) is
202	equivalent to `x == 0` since the default `atol` is `0`. In such cases, you should either
203	supply an appropriate `atol` (or use `norm(x) ≤ atol`) or rearrange your code (e.g.
204	use `x ≈ y` rather than `x - y ≈ 0`). It is not possible to pick a nonzero `atol`
205	automatically because it depends on the overall scaling (the "units") of your problem:
206	for example, in `x - y ≈ 0`, `atol=1e-9` is an absurdly small tolerance if `x` is the
207	[radius of the Earth](https://en.wikipedia.org/wiki/Earth_radius) in meters,
208	but an absurdly large tolerance if `x` is the
209	[radius of a Hydrogen atom](https://en.wikipedia.org/wiki/Bohr_radius) in meters.
210
211	!!! compat "Julia 1.6"
212	Passing the `norm` keyword argument when comparing numeric (non-array) arguments
213	requires Julia 1.6 or later.
214
215	# Examples
216	```jldoctest
217	julia> isapprox(0.1, 0.15; atol=0.05)
218	true
219
220	julia> isapprox(0.1, 0.15; rtol=0.34)
221	true
222
223	julia> isapprox(0.1, 0.15; rtol=0.33)
224	false
225
226	julia> 0.1 + 1e-10 ≈ 0.1
227	true
228
229	julia> 1e-10 ≈ 0
230	false
231
232	julia> isapprox(1e-10, 0, atol=1e-8)
233	true
234
235	julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0]) # using `norm`
236	true
237	```
238	"""
239	function isapprox(x::Number, y::Number;	730,945✔
240	atol::Real=0, rtol::Real=rtoldefault(x,y,atol),
241	nans::Bool=false, norm::Function=abs)
242	x == y \|\| (isfinite(x) && isfinite(y) && norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))) \|\| (nans && isnan(x) && isnan(y))	379,420✔
243	end
244
245	"""
246	isapprox(x; kwargs...) / ≈(x; kwargs...)
247
248	Create a function that compares its argument to `x` using `≈`, i.e. a function equivalent to `y -> y ≈ x`.
249
250	The keyword arguments supported here are the same as those in the 2-argument `isapprox`.
251
252	!!! compat "Julia 1.5"
253	This method requires Julia 1.5 or later.
254	"""
255	isapprox(y; kwargs...) = x -> isapprox(x, y; kwargs...)	199✔
256
257	const ≈ = isapprox
258	"""
259	x ≉ y
260
261	This is equivalent to `!isapprox(x,y)` (see [`isapprox`](@ref)).
262	"""
263	≉(args...; kws...) = !≈(args...; kws...)	58✔
264
265	# default tolerance arguments
266	rtoldefault(::Type{T}) where {T<:AbstractFloat} = sqrt(eps(T))	776,962✔
267	rtoldefault(::Type{<:Real}) = 0	11,042✔
268	function rtoldefault(x::Union{T,Type{T}}, y::Union{S,Type{S}}, atol::Real) where {T<:Number,S<:Number}	392,178✔
269	rtol = max(rtoldefault(real(T)),rtoldefault(real(S)))	392,178✔
270	return atol > 0 ? zero(rtol) : rtol	394,886✔
271	end
272
273	# fused multiply-add
274
275	"""
276	fma(x, y, z)
277
278	Computes `xy+z` without rounding the intermediate result `xy`. On some systems this is
279	significantly more expensive than `x*y+z`. `fma` is used to improve accuracy in certain
280	algorithms. See [`muladd`](@ref).
281	"""
282	function fma end
283	function fma_emulated(a::Float32, b::Float32, c::Float32)::Float32	262,155✔
284	ab = Float64(a) * b	262,155✔
285	res = ab+c	262,155✔
286	reinterpret(UInt64, res)&0x1fff_ffff!=0x1000_0000 && return res	262,155✔
287	# yes error compensation is necessary. It sucks
288	reslo = abs(c)>abs(ab) ? ab-(res - c) : c-(res - ab)	4✔
289	res = iszero(reslo) ? res : (signbit(reslo) ? prevfloat(res) : nextfloat(res))	3✔
290	return res	2✔
291	end
292
293	""" Splits a Float64 into a hi bit and a low bit where the high bit has 27 trailing 0s and the low bit has 26 trailing 0s"""
294	@inline function splitbits(x::Float64)	×
295	hi = reinterpret(Float64, reinterpret(UInt64, x) & 0xffff_ffff_f800_0000)	923,397✔
296	return hi, x-hi	923,397✔
297	end
298
299	function twomul(a::Float64, b::Float64)	×
300	ahi, alo = splitbits(a)	307,799✔
301	bhi, blo = splitbits(b)	307,799✔
302	abhi = a*b	307,799✔
303	blohi, blolo = splitbits(blo)	307,799✔
304	ablo = aloblohi - (((abhi - ahibhi) - alobhi) - ahiblo) + blolo*alo	307,799✔
305	return abhi, ablo	×
306	end
307
308	function fma_emulated(a::Float64, b::Float64,c::Float64)	262,160✔
309	abhi, ablo = @inline twomul(a,b)	262,160✔
310	if !isfinite(abhi+c) \|\| isless(abs(abhi), nextfloat(0x1p-969)) \|\| issubnormal(a) \|\| issubnormal(b)	491,060✔
311	aandbfinite = isfinite(a) && isfinite(b)	69,652✔
312	if !(isfinite(c) && aandbfinite)	69,652✔
313	return aandbfinite ? c : abhi+c	376✔
314	end
315	(iszero(a) \|\| iszero(b)) && return abhi+c	69,276✔
316	# The checks above satisfy exponent's nothrow precondition
317	bias = Math._exponent_finite_nonzero(a) + Math._exponent_finite_nonzero(b)	69,275✔
318	c_denorm = ldexp(c, -bias)	138,550✔
319	if isfinite(c_denorm)	69,275✔
320	# rescale a and b to [1,2), equivalent to ldexp(a, -exponent(a))
321	issubnormal(a) && (a *= 0x1p52)	45,639✔
322	issubnormal(b) && (b *= 0x1p52)	45,639✔
323	a = reinterpret(Float64, (reinterpret(UInt64, a) & ~Base.exponent_mask(Float64)) \| Base.exponent_one(Float64))	45,639✔
324	b = reinterpret(Float64, (reinterpret(UInt64, b) & ~Base.exponent_mask(Float64)) \| Base.exponent_one(Float64))	45,639✔
325	c = c_denorm	×
326	abhi, ablo = twomul(a,b)	45,639✔
327	# abhi <= 4 -> isfinite(r) (α)
328	r = abhi+c	45,639✔
329	# s ≈ 0 (β)
330	s = (abs(abhi) > abs(c)) ? (abhi-r+c+ablo) : (c-r+abhi+ablo)	58,321✔
331	# α ⩓ β -> isfinite(sumhi) (γ)
332	sumhi = r+s	45,639✔
333	# If result is subnormal, ldexp will cause double rounding because subnormals have fewer mantisa bits.
334	# As such, we need to check whether round to even would lead to double rounding and manually round sumhi to avoid it.
335	if issubnormal(ldexp(sumhi, bias))	91,278✔
336	sumlo = r-sumhi+s	15✔
337	# finite: See γ
338	# non-zero: If sumhi == ±0., then ldexp(sumhi, bias) == ±0,
339	# so we don't take this branch.
340	bits_lost = -bias-Math._exponent_finite_nonzero(sumhi)-1022	15✔
341	sumhiInt = reinterpret(UInt64, sumhi)	15✔
342	if (bits_lost != 1) ⊻ (sumhiInt&1 == 1)	15✔
343	sumhi = nextfloat(sumhi, cmp(sumlo,0))	20✔
344	end
345	end
346	return ldexp(sumhi, bias)	45,639✔
347	end
348	isinf(abhi) && signbit(c) == signbit(a*b) && return abhi	23,636✔
349	# fall through
350	end
351	r = abhi+c	216,144✔
352	s = (abs(abhi) > abs(c)) ? (abhi-r+c+ablo) : (c-r+abhi+ablo)	334,019✔
353	return r+s	216,144✔
354	end
355	fma_llvm(x::Float32, y::Float32, z::Float32) = fma_float(x, y, z)	2,448,598✔
356	fma_llvm(x::Float64, y::Float64, z::Float64) = fma_float(x, y, z)	404,384,773✔
357
358	# Disable LLVM's fma if it is incorrect, e.g. because LLVM falls back
359	# onto a broken system libm; if so, use a software emulated fma
360	@assume_effects :consistent fma(x::Float32, y::Float32, z::Float32) = Core.Intrinsics.have_fma(Float32) ? fma_llvm(x,y,z) : fma_emulated(x,y,z)	2,448,598✔
361	@assume_effects :consistent fma(x::Float64, y::Float64, z::Float64) = Core.Intrinsics.have_fma(Float64) ? fma_llvm(x,y,z) : fma_emulated(x,y,z)	404,384,773✔
362
363	function fma(a::Float16, b::Float16, c::Float16)	499,778✔
364	Float16(muladd(Float32(a), Float32(b), Float32(c))) #don't use fma if the hardware doesn't have it.	499,778✔
365	end
366
367	# This is necessary at least on 32-bit Intel Linux, since fma_llvm may
368	# have called glibc, and some broken glibc fma implementations don't
369	# properly restore the rounding mode
370	Rounding.setrounding_raw(Float32, Rounding.JL_FE_TONEAREST)
371	Rounding.setrounding_raw(Float64, Rounding.JL_FE_TONEAREST)

JuliaLang / julia / #37594

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