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math.jl
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# This file is a part of Julia. License is MIT: https://julialang.org/license
module Math
export sin, cos, sincos, tan, sinh, cosh, tanh, asin, acos, atan,
asinh, acosh, atanh, sec, csc, cot, asec, acsc, acot,
sech, csch, coth, asech, acsch, acoth,
sinpi, cospi, sincospi, tanpi, sinc, cosc,
cosd, cotd, cscd, secd, sind, tand, sincosd,
acosd, acotd, acscd, asecd, asind, atand,
rad2deg, deg2rad,
log, log2, log10, log1p, exponent, exp, exp2, exp10, expm1,
cbrt, sqrt, fourthroot, significand,
hypot, max, min, minmax, ldexp, frexp,
clamp, clamp!, modf, ^, mod2pi, rem2pi,
@evalpoly, evalpoly
import .Base: log, exp, sin, cos, tan, sinh, cosh, tanh, asin,
acos, atan, asinh, acosh, atanh, sqrt, log2, log10,
max, min, minmax, ^, exp2, muladd, rem,
exp10, expm1, log1p, @constprop, @assume_effects
using .Base: sign_mask, exponent_mask, exponent_one,
exponent_half, uinttype, significand_mask,
significand_bits, exponent_bits, exponent_bias,
exponent_max, exponent_raw_max
using Core.Intrinsics: sqrt_llvm
using .Base: IEEEFloat
@noinline function throw_complex_domainerror(f::Symbol, x)
throw(DomainError(x,
LazyString(f," was called with a negative real argument but will only return a complex result if called with a complex argument. Try ", f,"(Complex(x)).")))
end
@noinline function throw_complex_domainerror_neg1(f::Symbol, x)
throw(DomainError(x,
LazyString(f," was called with a real argument < -1 but will only return a complex result if called with a complex argument. Try ", f,"(Complex(x)).")))
end
@noinline function throw_exp_domainerror(x)
throw(DomainError(x, LazyString(
"Exponentiation yielding a complex result requires a ",
"complex argument.\nReplace x^y with (x+0im)^y, ",
"Complex(x)^y, or similar.")))
end
# non-type specific math functions
function two_mul(x::T, y::T) where {T<:Number}
xy = x*y
xy, fma(x, y, -xy)
end
@assume_effects :consistent @inline function two_mul(x::Float64, y::Float64)
if Core.Intrinsics.have_fma(Float64)
xy = x*y
return xy, fma(x, y, -xy)
end
return Base.twomul(x,y)
end
@assume_effects :consistent @inline function two_mul(x::T, y::T) where T<: Union{Float16, Float32}
if Core.Intrinsics.have_fma(T)
xy = x*y
return xy, fma(x, y, -xy)
end
xy = widen(x)*y
Txy = T(xy)
return Txy, T(xy-Txy)
end
"""
clamp(x, lo, hi)
Return `x` if `lo <= x <= hi`. If `x > hi`, return `hi`. If `x < lo`, return `lo`. Arguments
are promoted to a common type.
See also [`clamp!`](@ref), [`min`](@ref), [`max`](@ref).
!!! compat "Julia 1.3"
`missing` as the first argument requires at least Julia 1.3.
# Examples
```jldoctest
julia> clamp.([pi, 1.0, big(10)], 2.0, 9.0)
3-element Vector{BigFloat}:
3.141592653589793238462643383279502884197169399375105820974944592307816406286198
2.0
9.0
julia> clamp.([11, 8, 5], 10, 6) # an example where lo > hi
3-element Vector{Int64}:
6
6
10
```
"""
clamp(x::X, lo::L, hi::H) where {X,L,H} =
ifelse(x > hi, convert(promote_type(X,L,H), hi),
ifelse(x < lo,
convert(promote_type(X,L,H), lo),
convert(promote_type(X,L,H), x)))
"""
clamp(x, T)::T
Clamp `x` between `typemin(T)` and `typemax(T)` and convert the result to type `T`.
See also [`trunc`](@ref).
# Examples
```jldoctest
julia> clamp(200, Int8)
127
julia> clamp(-200, Int8)
-128
julia> trunc(Int, 4pi^2)
39
```
"""
clamp(x, ::Type{T}) where {T<:Integer} = clamp(x, typemin(T), typemax(T)) % T
"""
clamp!(array::AbstractArray, lo, hi)
Restrict values in `array` to the specified range, in-place.
See also [`clamp`](@ref).
!!! compat "Julia 1.3"
`missing` entries in `array` require at least Julia 1.3.
# Examples
```jldoctest
julia> row = collect(-4:4)';
julia> clamp!(row, 0, Inf)
1×9 adjoint(::Vector{Int64}) with eltype Int64:
0 0 0 0 0 1 2 3 4
julia> clamp.((-4:4)', 0, Inf)
1×9 Matrix{Float64}:
0.0 0.0 0.0 0.0 0.0 1.0 2.0 3.0 4.0
```
"""
function clamp!(x::AbstractArray, lo, hi)
@inbounds for i in eachindex(x)
x[i] = clamp(x[i], lo, hi)
end
x
end
"""
clamp(x::Integer, r::AbstractUnitRange)
Clamp `x` to lie within range `r`.
!!! compat "Julia 1.6"
This method requires at least Julia 1.6.
"""
clamp(x::Integer, r::AbstractUnitRange{<:Integer}) = clamp(x, first(r), last(r))
"""
evalpoly(x, p)
Evaluate the polynomial ``\\sum_k x^{k-1} p[k]`` for the coefficients `p[1]`, `p[2]`, ...;
that is, the coefficients are given in ascending order by power of `x`.
Loops are unrolled at compile time if the number of coefficients is statically known, i.e.
when `p` is a `Tuple`.
This function generates efficient code using Horner's method if `x` is real, or using
a Goertzel-like [^DK62] algorithm if `x` is complex.
[^DK62]: Donald Knuth, Art of Computer Programming, Volume 2: Seminumerical Algorithms, Sec. 4.6.4.
!!! compat "Julia 1.4"
This function requires Julia 1.4 or later.
# Examples
```jldoctest
julia> evalpoly(2, (1, 2, 3))
17
```
"""
function evalpoly(x, p::Tuple)
if @generated
N = length(p.parameters::Core.SimpleVector)
ex = :(p[end])
for i in N-1:-1:1
ex = :(muladd(x, $ex, p[$i]))
end
ex
else
_evalpoly(x, p)
end
end
evalpoly(x, p::AbstractVector) = _evalpoly(x, p)
function _evalpoly(x, p)
Base.require_one_based_indexing(p)
N = length(p)
ex = p[end]
for i in N-1:-1:1
ex = muladd(x, ex, p[i])
end
ex
end
function evalpoly(z::Complex, p::Tuple)
if @generated
N = length(p.parameters)
a = :(p[end])
b = :(p[end-1])
as = []
for i in N-2:-1:1
ai = Symbol("a", i)
push!(as, :($ai = $a))
a = :(muladd(r, $ai, $b))
b = :(muladd(-s, $ai, p[$i]))
end
ai = :a0
push!(as, :($ai = $a))
C = Expr(:block,
:(x = real(z)),
:(y = imag(z)),
:(r = x + x),
:(s = muladd(x, x, y*y)),
as...,
:(muladd($ai, z, $b)))
else
_evalpoly(z, p)
end
end
evalpoly(z::Complex, p::Tuple{<:Any}) = p[1]
evalpoly(z::Complex, p::AbstractVector) = _evalpoly(z, p)
function _evalpoly(z::Complex, p)
Base.require_one_based_indexing(p)
length(p) == 1 && return p[1]
N = length(p)
a = p[end]
b = p[end-1]
x = real(z)
y = imag(z)
r = 2x
s = muladd(x, x, y*y)
for i in N-2:-1:1
ai = a
a = muladd(r, ai, b)
b = muladd(-s, ai, p[i])
end
ai = a
muladd(ai, z, b)
end
"""
@horner(x, p...)
Evaluate `p[1] + x * (p[2] + x * (....))`, i.e. a polynomial via Horner's rule.
See also [`@evalpoly`](@ref), [`evalpoly`](@ref).
"""
macro horner(x, p...)
xesc, pesc = esc(x), esc.(p)
:(invoke(evalpoly, Tuple{Any, Tuple}, $xesc, ($(pesc...),)))
end
# Evaluate p[1] + z*p[2] + z^2*p[3] + ... + z^(n-1)*p[n]. This uses
# Horner's method if z is real, but for complex z it uses a more
# efficient algorithm described in Knuth, TAOCP vol. 2, section 4.6.4,
# equation (3).
"""
@evalpoly(z, c...)
Evaluate the polynomial ``\\sum_k z^{k-1} c[k]`` for the coefficients `c[1]`, `c[2]`, ...;
that is, the coefficients are given in ascending order by power of `z`. This macro expands
to efficient inline code that uses either Horner's method or, for complex `z`, a more
efficient Goertzel-like algorithm.
See also [`evalpoly`](@ref).
# Examples
```jldoctest
julia> @evalpoly(3, 1, 0, 1)
10
julia> @evalpoly(2, 1, 0, 1)
5
julia> @evalpoly(2, 1, 1, 1)
7
```
"""
macro evalpoly(z, p...)
zesc, pesc = esc(z), esc.(p)
:(evalpoly($zesc, ($(pesc...),)))
end
# polynomial evaluation using compensated summation.
# much more accurate, especially when lo can be combined with other rounding errors
@inline function exthorner(x::T, p::Tuple{T,T,T}) where T<:Union{Float32,Float64}
hi, lo = p[lastindex(p)], zero(x)
hi, lo = _exthorner(2, x, p, hi, lo)
hi, lo = _exthorner(1, x, p, hi, lo)
return hi, lo
end
@inline function _exthorner(i::Int, x::T, p::Tuple{T,T,T}, hi::T, lo::T) where T<:Union{Float32,Float64}
i == 2 || i == 1 || error("unexpected index")
pi = p[i]
prod, err = two_mul(hi,x)
hi = pi+prod
lo = fma(lo, x, prod - (hi - pi) + err)
return hi, lo
end
"""
rad2deg(x)
Convert `x` from radians to degrees.
See also [`deg2rad`](@ref).
# Examples
```jldoctest
julia> rad2deg(pi)
180.0
```
"""
rad2deg(z::AbstractFloat) = z * (180 / oftype(z, pi))
"""
deg2rad(x)
Convert `x` from degrees to radians.
See also [`rad2deg`](@ref), [`sind`](@ref), [`pi`](@ref).
# Examples
```jldoctest
julia> deg2rad(90)
1.5707963267948966
```
"""
deg2rad(z::AbstractFloat) = z * (oftype(z, pi) / 180)
rad2deg(z::Real) = rad2deg(float(z))
deg2rad(z::Real) = deg2rad(float(z))
rad2deg(z::Number) = (z/pi)*180
deg2rad(z::Number) = (z*pi)/180
log(b::T, x::T) where {T<:Number} = log(x)/log(b)
"""
log(b,x)
Compute the base `b` logarithm of `x`. Throw a [`DomainError`](@ref) for negative
[`Real`](@ref) arguments.
# Examples
```jldoctest; filter = r"Stacktrace:(\\n \\[[0-9]+\\].*)*"
julia> log(4,8)
1.5
julia> log(4,2)
0.5
julia> log(-2, 3)
ERROR: DomainError with -2.0:
log was called with a negative real argument but will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
julia> log(2, -3)
ERROR: DomainError with -3.0:
log was called with a negative real argument but will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
```
!!! note
If `b` is a power of 2 or 10, [`log2`](@ref) or [`log10`](@ref) should be used, as these will
typically be faster and more accurate. For example,
```jldoctest
julia> log(100,1000000)
2.9999999999999996
julia> log10(1000000)/2
3.0
```
"""
log(b::Number, x::Number) = log(promote(b,x)...)
# type specific math functions
const libm = Base.libm_name
# functions with no domain error
"""
sinh(x)
Compute hyperbolic sine of `x`.
See also [`sin`](@ref).
"""
sinh(x::Number)
"""
cosh(x)
Compute hyperbolic cosine of `x`.
See also [`cos`](@ref).
"""
cosh(x::Number)
"""
tanh(x)
Compute hyperbolic tangent of `x`.
See also [`tan`](@ref), [`atanh`](@ref).
# Examples
```jldoctest
julia> tanh.(-3:3f0) # Here 3f0 isa Float32
7-element Vector{Float32}:
-0.9950548
-0.9640276
-0.7615942
0.0
0.7615942
0.9640276
0.9950548
julia> tan.(im .* (1:3))
3-element Vector{ComplexF64}:
0.0 + 0.7615941559557649im
0.0 + 0.9640275800758169im
0.0 + 0.9950547536867306im
```
"""
tanh(x::Number)
"""
atan(y)
atan(y, x)
Compute the inverse tangent of `y` or `y/x`, respectively.
For one real argument, this is the angle in radians between the positive *x*-axis and the point
(1, *y*), returning a value in the interval ``[-\\pi/2, \\pi/2]``.
For two arguments, this is the angle in radians between the positive *x*-axis and the
point (*x*, *y*), returning a value in the interval ``[-\\pi, \\pi]``. This corresponds to a
standard [`atan2`](https://en.wikipedia.org/wiki/Atan2) function. Note that by convention
`atan(0.0,x)` is defined as ``\\pi`` and `atan(-0.0,x)` is defined as ``-\\pi`` when `x < 0`.
See also [`atand`](@ref) for degrees.
# Examples
```jldoctest
julia> rad2deg(atan(-1/√3))
-30.000000000000004
julia> rad2deg(atan(-1, √3))
-30.000000000000004
julia> rad2deg(atan(1, -√3))
150.0
```
"""
atan(x::Number)
"""
asinh(x)
Compute the inverse hyperbolic sine of `x`.
"""
asinh(x::Number)
# utility for converting NaN return to DomainError
# the branch in nan_dom_err prevents its callers from inlining, so be sure to force it
# until the heuristics can be improved
@inline nan_dom_err(out, x) = isnan(out) & !isnan(x) ? throw(DomainError(x, "NaN result for non-NaN input.")) : out
# functions that return NaN on non-NaN argument for domain error
"""
sin(x::T) where {T <: Number} -> float(T)
Compute sine of `x`, where `x` is in radians.
Throw a [`DomainError`](@ref) if `isinf(x)`, return a `T(NaN)` if `isnan(x)`.
See also [`sind`](@ref), [`sinpi`](@ref), [`sincos`](@ref), [`cis`](@ref), [`asin`](@ref).
# Examples
```jldoctest
julia> round.(sin.(range(0, 2pi, length=9)'), digits=3)
1×9 Matrix{Float64}:
0.0 0.707 1.0 0.707 0.0 -0.707 -1.0 -0.707 -0.0
julia> sind(45)
0.7071067811865476
julia> sinpi(1/4)
0.7071067811865475
julia> round.(sincos(pi/6), digits=3)
(0.5, 0.866)
julia> round(cis(pi/6), digits=3)
0.866 + 0.5im
julia> round(exp(im*pi/6), digits=3)
0.866 + 0.5im
```
"""
sin(x::Number)
"""
cos(x::T) where {T <: Number} -> float(T)
Compute cosine of `x`, where `x` is in radians.
Throw a [`DomainError`](@ref) if `isinf(x)`, return a `T(NaN)` if `isnan(x)`.
See also [`cosd`](@ref), [`cospi`](@ref), [`sincos`](@ref), [`cis`](@ref).
"""
cos(x::Number)
"""
tan(x::T) where {T <: Number} -> float(T)
Compute tangent of `x`, where `x` is in radians.
Throw a [`DomainError`](@ref) if `isinf(x)`, return a `T(NaN)` if `isnan(x)`.
See also [`tanh`](@ref).
"""
tan(x::Number)
"""
asin(x::T) where {T <: Number} -> float(T)
Compute the inverse sine of `x`, where the output is in radians.
Return a `T(NaN)` if `isnan(x)`.
See also [`asind`](@ref) for output in degrees.
# Examples
```jldoctest
julia> asin.((0, 1/2, 1))
(0.0, 0.5235987755982989, 1.5707963267948966)
julia> asind.((0, 1/2, 1))
(0.0, 30.000000000000004, 90.0)
```
"""
asin(x::Number)
"""
acos(x::T) where {T <: Number} -> float(T)
Compute the inverse cosine of `x`, where the output is in radians
Return a `T(NaN)` if `isnan(x)`.
"""
acos(x::Number)
"""
acosh(x)
Compute the inverse hyperbolic cosine of `x`.
"""
acosh(x::Number)
"""
atanh(x)
Compute the inverse hyperbolic tangent of `x`.
"""
atanh(x::Number)
"""
log(x)
Compute the natural logarithm of `x`.
Throw a [`DomainError`](@ref) for negative [`Real`](@ref) arguments.
Use [`Complex`](@ref) arguments to obtain [`Complex`](@ref) results.
!!! note "Branch cut"
`log` has a branch cut along the negative real axis; `-0.0im` is taken
to be below the axis.
See also [`ℯ`](@ref), [`log1p`](@ref), [`log2`](@ref), [`log10`](@ref).
# Examples
```jldoctest; filter = r"Stacktrace:(\\n \\[[0-9]+\\].*)*"
julia> log(2)
0.6931471805599453
julia> log(-3)
ERROR: DomainError with -3.0:
log was called with a negative real argument but will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
julia> log(-3 + 0im)
1.0986122886681098 + 3.141592653589793im
julia> log(-3 - 0.0im)
1.0986122886681098 - 3.141592653589793im
julia> log.(exp.(-1:1))
3-element Vector{Float64}:
-1.0
0.0
1.0
```
"""
log(x::Number)
"""
log2(x)
Compute the logarithm of `x` to base 2. Throw a [`DomainError`](@ref) for negative
[`Real`](@ref) arguments.
See also: [`exp2`](@ref), [`ldexp`](@ref), [`ispow2`](@ref).
# Examples
```jldoctest; filter = r"Stacktrace:(\\n \\[[0-9]+\\].*)*"
julia> log2(4)
2.0
julia> log2(10)
3.321928094887362
julia> log2(-2)
ERROR: DomainError with -2.0:
log2 was called with a negative real argument but will only return a complex result if called with a complex argument. Try log2(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(f::Symbol, x::Float64) at ./math.jl:31
[...]
julia> log2.(2.0 .^ (-1:1))
3-element Vector{Float64}:
-1.0
0.0
1.0
```
"""
log2(x)
"""
log10(x)
Compute the logarithm of `x` to base 10.
Throw a [`DomainError`](@ref) for negative [`Real`](@ref) arguments.
# Examples
```jldoctest; filter = r"Stacktrace:(\\n \\[[0-9]+\\].*)*"
julia> log10(100)
2.0
julia> log10(2)
0.3010299956639812
julia> log10(-2)
ERROR: DomainError with -2.0:
log10 was called with a negative real argument but will only return a complex result if called with a complex argument. Try log10(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(f::Symbol, x::Float64) at ./math.jl:31
[...]
```
"""
log10(x)
"""
log1p(x)
Accurate natural logarithm of `1+x`. Throw a [`DomainError`](@ref) for [`Real`](@ref)
arguments less than -1.
# Examples
```jldoctest; filter = r"Stacktrace:(\\n \\[[0-9]+\\].*)*"
julia> log1p(-0.5)
-0.6931471805599453
julia> log1p(0)
0.0
julia> log1p(-2)
ERROR: DomainError with -2.0:
log1p was called with a real argument < -1 but will only return a complex result if called with a complex argument. Try log1p(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
```
"""
log1p(x)
@inline function sqrt(x::Union{Float32,Float64})
x < zero(x) && throw_complex_domainerror(:sqrt, x)
sqrt_llvm(x)
end
"""
sqrt(x)
Return ``\\sqrt{x}``.
Throw a [`DomainError`](@ref) for negative [`Real`](@ref) arguments.
Use [`Complex`](@ref) negative arguments instead to obtain a [`Complex`](@ref) result.
The prefix operator `√` is equivalent to `sqrt`.
!!! note "Branch cut"
`sqrt` has a branch cut along the negative real axis; `-0.0im` is taken
to be below the axis.
See also: [`hypot`](@ref).
# Examples
```jldoctest; filter = r"Stacktrace:(\\n \\[[0-9]+\\].*)*"
julia> sqrt(big(81))
9.0
julia> sqrt(big(-81))
ERROR: DomainError with -81.0:
NaN result for non-NaN input.
Stacktrace:
[1] sqrt(::BigFloat) at ./mpfr.jl:501
[...]
julia> sqrt(big(complex(-81)))
0.0 + 9.0im
julia> sqrt(-81 - 0.0im) # -0.0im is below the branch cut
0.0 - 9.0im
julia> .√(1:4)
4-element Vector{Float64}:
1.0
1.4142135623730951
1.7320508075688772
2.0
```
"""
sqrt(x)
"""
fourthroot(x)
Return the fourth root of `x` by applying `sqrt` twice successively.
"""
fourthroot(x::Number) = sqrt(sqrt(x))
"""
hypot(x, y)
Compute the hypotenuse ``\\sqrt{|x|^2+|y|^2}`` avoiding overflow and underflow.
This code is an implementation of the algorithm described in:
An Improved Algorithm for `hypot(a,b)`
by Carlos F. Borges
The article is available online at arXiv at the link
https://arxiv.org/abs/1904.09481
hypot(x...)
Compute the hypotenuse ``\\sqrt{\\sum |x_i|^2}`` avoiding overflow and underflow.
See also `norm` in the [`LinearAlgebra`](@ref man-linalg) standard library.
# Examples
```jldoctest; filter = r"Stacktrace:(\\n \\[[0-9]+\\].*)*"
julia> a = Int64(10)^10;
julia> hypot(a, a)
1.4142135623730951e10
julia> √(a^2 + a^2) # a^2 overflows
ERROR: DomainError with -2.914184810805068e18:
sqrt was called with a negative real argument but will only return a complex result if called with a complex argument. Try sqrt(Complex(x)).
Stacktrace:
[...]
julia> hypot(3, 4im)
5.0
julia> hypot(-5.7)
5.7
julia> hypot(3, 4im, 12.0)
13.0
julia> using LinearAlgebra
julia> norm([a, a, a, a]) == hypot(a, a, a, a)
true
```
"""
hypot(x::Number) = abs(float(x))
hypot(x::Number, y::Number) = _hypot(float.(promote(x, y))...)
hypot(x::Number, y::Number, xs::Number...) = _hypot(float.(promote(x, y, xs...)))
function _hypot(x, y)
# preserves unit
axu = abs(x)
ayu = abs(y)
# unitless
ax = axu / oneunit(axu)
ay = ayu / oneunit(ayu)
# Return Inf if either or both inputs is Inf (Compliance with IEEE754)
if isinf(ax) || isinf(ay)
return typeof(axu)(Inf)
end
# Order the operands
if ay > ax
axu, ayu = ayu, axu
ax, ay = ay, ax
end
# Widely varying operands
if ay <= ax*sqrt(eps(typeof(ax))/2) #Note: This also gets ay == 0
return axu
end
# Operands do not vary widely
scale = eps(typeof(ax))*sqrt(floatmin(ax)) #Rescaling constant
if ax > sqrt(floatmax(ax)/2)
ax = ax*scale
ay = ay*scale
scale = inv(scale)
elseif ay < sqrt(floatmin(ax))
ax = ax/scale
ay = ay/scale
else
scale = oneunit(scale)
end
h = sqrt(muladd(ax, ax, ay*ay))
# This branch is correctly rounded but requires a native hardware fma.
if Core.Intrinsics.have_fma(typeof(h))
hsquared = h*h
axsquared = ax*ax
h -= (fma(-ay, ay, hsquared-axsquared) + fma(h, h,-hsquared) - fma(ax, ax, -axsquared))/(2*h)
# This branch is within one ulp of correctly rounded.
else
if h <= 2*ay
delta = h-ay
h -= muladd(delta, delta-2*(ax-ay), ax*(2*delta - ax))/(2*h)
else
delta = h-ax
h -= muladd(delta, delta, muladd(ay, (4*delta - ay), 2*delta*(ax - 2*ay)))/(2*h)
end
end
return h*scale*oneunit(axu)
end
@inline function _hypot(x::Float32, y::Float32)
if isinf(x) || isinf(y)
return Inf32
end
_x, _y = Float64(x), Float64(y)
return Float32(sqrt(muladd(_x, _x, _y*_y)))
end
@inline function _hypot(x::Float16, y::Float16)
if isinf(x) || isinf(y)
return Inf16
end
_x, _y = Float32(x), Float32(y)
return Float16(sqrt(muladd(_x, _x, _y*_y)))
end
_hypot(x::ComplexF16, y::ComplexF16) = Float16(_hypot(ComplexF32(x), ComplexF32(y)))
function _hypot(x::NTuple{N,<:Number}) where {N}
maxabs = maximum(abs, x)
if isnan(maxabs) && any(isinf, x)
return typeof(maxabs)(Inf)
elseif (iszero(maxabs) || isinf(maxabs))
return maxabs
else
return maxabs * sqrt(sum(y -> abs2(y / maxabs), x))
end
end
function _hypot(x::NTuple{N,<:IEEEFloat}) where {N}
T = eltype(x)
infT = convert(T, Inf)
x = abs.(x) # doesn't change result but enables computational shortcuts
# note: any() was causing this to not inline for N=3 but mapreduce() was not
mapreduce(==(infT), |, x) && return infT # return Inf even if an argument is NaN
maxabs = reinterpret(T, maximum(z -> reinterpret(Signed, z), x)) # for abs(::IEEEFloat) values, a ::BitInteger cast does not change the result
maxabs > zero(T) || return maxabs # catch NaN before the @fastmath below, but also shortcut 0 since we can (remove if no more @fastmath below)
scale,invscale = scaleinv(maxabs)
# @fastmath(+) to allow reassociation (see #48129)
add_fast(x, y) = Core.Intrinsics.add_float_fast(x, y) # @fastmath is not available during bootstrap
return scale * sqrt(mapreduce(y -> abs2(y * invscale), add_fast, x))
end
atan(y::Real, x::Real) = atan(promote(float(y),float(x))...)
atan(y::T, x::T) where {T<:AbstractFloat} = Base.no_op_err("atan", T)
_isless(x::T, y::T) where {T<:AbstractFloat} = (x < y) || (signbit(x) > signbit(y))
min(x::T, y::T) where {T<:AbstractFloat} = isnan(x) || ~isnan(y) && _isless(x, y) ? x : y
max(x::T, y::T) where {T<:AbstractFloat} = isnan(x) || ~isnan(y) && _isless(y, x) ? x : y
minmax(x::T, y::T) where {T<:AbstractFloat} = min(x, y), max(x, y)
_isless(x::Float16, y::Float16) = signbit(widen(x) - widen(y))
const has_native_fminmax = Sys.ARCH === :aarch64
@static if has_native_fminmax
@eval begin
Base.@assume_effects :total @inline llvm_min(x::Float64, y::Float64) = ccall("llvm.minimum.f64", llvmcall, Float64, (Float64, Float64), x, y)
Base.@assume_effects :total @inline llvm_min(x::Float32, y::Float32) = ccall("llvm.minimum.f32", llvmcall, Float32, (Float32, Float32), x, y)
Base.@assume_effects :total @inline llvm_max(x::Float64, y::Float64) = ccall("llvm.maximum.f64", llvmcall, Float64, (Float64, Float64), x, y)
Base.@assume_effects :total @inline llvm_max(x::Float32, y::Float32) = ccall("llvm.maximum.f32", llvmcall, Float32, (Float32, Float32), x, y)
end
end
function min(x::T, y::T) where {T<:Union{Float32,Float64}}
@static if has_native_fminmax
return llvm_min(x,y)
end
diff = x - y
argmin = ifelse(signbit(diff), x, y)
anynan = isnan(x)|isnan(y)
return ifelse(anynan, diff, argmin)
end
function max(x::T, y::T) where {T<:Union{Float32,Float64}}
@static if has_native_fminmax
return llvm_max(x,y)
end
diff = x - y
argmax = ifelse(signbit(diff), y, x)
anynan = isnan(x)|isnan(y)
return ifelse(anynan, diff, argmax)
end
function minmax(x::T, y::T) where {T<:Union{Float32,Float64}}
@static if has_native_fminmax
return llvm_min(x, y), llvm_max(x, y)
end
diff = x - y
sdiff = signbit(diff)
min, max = ifelse(sdiff, x, y), ifelse(sdiff, y, x)
anynan = isnan(x)|isnan(y)
return ifelse(anynan, diff, min), ifelse(anynan, diff, max)
end
"""
ldexp(x, n)
Compute ``x \\times 2^n``.
See also [`frexp`](@ref), [`exponent`](@ref).
# Examples
```jldoctest
julia> ldexp(5.0, 2)
20.0
```
"""
function ldexp(x::T, e::Integer) where T<:IEEEFloat
xu = reinterpret(Unsigned, x)
xs = xu & ~sign_mask(T)
xs >= exponent_mask(T) && return x # NaN or Inf
k = (xs >> significand_bits(T)) % Int
if k == 0 # x is subnormal
xs == 0 && return x # +-0
m = leading_zeros(xs) - exponent_bits(T)
ys = xs << unsigned(m)
xu = ys | (xu & sign_mask(T))
k = 1 - m
# underflow, otherwise may have integer underflow in the following n + k
e < -50000 && return flipsign(T(0.0), x)
end
# For cases where e of an Integer larger than Int make sure we properly
# overflow/underflow; this is optimized away otherwise.
if e > typemax(Int)
return flipsign(T(Inf), x)
elseif e < typemin(Int)