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Reliable quadrant #652

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May 27, 2024
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Reliable quadrant #652

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53 changes: 28 additions & 25 deletions src/intervals/arithmetic/trigonometric.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -4,19 +4,22 @@

# helper functions

function _quadrant(x::AbstractFloat)
# NOTE: this algorithm may be flawed as it relies on `rem2pi(x, RoundNearest)`
# to yield a very tight result. This is not guaranteed by Julia, see e.g.
# https://github.com/JuliaLang/julia/blob/9669eecc99bc4553e28d94d7dd3dc9fd40b3bf3f/base/mpfr.jl#L845-L846
PI_LO, PI_HI = bounds(bareinterval(typeof(x), π))
r = rem2pi(x, RoundNearest)
r2 = 2r # should be exact for floats
r2 ≤ -PI_HI && return 2 # [-π, -π/2)
r2 < -PI_LO && return throw(ArgumentError("could not determine the quadrant, the remainder $r of the division of $x by 2π is lesser or greater than -π/2"))
r2 < 0 && return 3 # [-π/2, 0)
r2 ≤ PI_LO && return 0 # [0, π/2)
r2 < PI_HI && return throw(ArgumentError("could not determine the quadrant, the remainder $r of the division of $x by 2π is lesser or greater than π/2"))
return 1 # [π/2, π]
function _quadrant(f, x::T) where {T<:AbstractFloat}
PI = bareinterval(T, π)
PI_LO, PI_HI = bounds(PI)
if abs(x) ≤ PI_LO # (-π, π)
r2 = 2x # should be exact for floats
r2 ≤ -PI_HI && return 2 # (-π, -π/2)
r2 < -PI_LO && return f(2, 3) # (-π, -π/2) or [-π/2, 0)
r2 < 0 && return 3 # [-π/2, 0)
r2 ≤ PI_LO && return 0 # [0, π/2)
r2 < PI_HI && return f(0, 1) # [0, π/2) or [π/2, π)
return 1 # [π/2, π)
else
k = _unsafe_scale(bareinterval(x) / PI, convert(T, 2))
fk = floor(k)
return f(mod(inf(fk), 4), mod(sup(fk), 4))
end
end

function _quadrantpi(x::AbstractFloat) # used in `sinpi` and `cospi`
Expand Down Expand Up @@ -65,8 +68,8 @@ function Base.sin(x::BareInterval{T}) where {T<:AbstractFloat}

lo, hi = bounds(x)

lo_quadrant = _quadrant(lo)
hi_quadrant = _quadrant(hi)
lo_quadrant = _quadrant(min, lo)
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Why do you need the min and max here?

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_quadrant may return two distinct integers. What min/max aims to do is to "overshoot" the quadrants: for the lowest (resp. largest) value of the interval, we want the smallest (resp. largest) quadrant.

hi_quadrant = _quadrant(max, hi)

if lo_quadrant == hi_quadrant
d ≥ PI_HI && return _unsafe_bareinterval(T, -one(T), one(T))
Expand Down Expand Up @@ -167,8 +170,8 @@ function Base.cos(x::BareInterval{T}) where {T<:AbstractFloat}

lo, hi = bounds(x)

lo_quadrant = _quadrant(lo)
hi_quadrant = _quadrant(hi)
lo_quadrant = _quadrant(min, lo)
hi_quadrant = _quadrant(max, hi)

if lo_quadrant == hi_quadrant
d ≥ PI_HI && return _unsafe_bareinterval(T, -one(T), one(T))
Expand Down Expand Up @@ -269,8 +272,8 @@ function Base.tan(x::BareInterval{T}) where {T<:AbstractFloat}

lo, hi = bounds(x)

lo_quadrant = _quadrant(lo)
hi_quadrant = _quadrant(hi)
lo_quadrant = _quadrant(min, lo)
hi_quadrant = _quadrant(max, hi)
lo_quadrant_mod = mod(lo_quadrant, 2)
hi_quadrant_mod = mod(hi_quadrant, 2)

Expand Down Expand Up @@ -309,8 +312,8 @@ function Base.cot(x::BareInterval{T}) where {T<:AbstractFloat}

lo, hi = bounds(x)

lo_quadrant = _quadrant(lo)
hi_quadrant = _quadrant(hi)
lo_quadrant = _quadrant(min, lo)
hi_quadrant = _quadrant(max, hi)

if (lo_quadrant == 2 || lo_quadrant == 3) && hi == 0
return @round(T, typemin(T), cot(lo)) # singularity from the left
Expand Down Expand Up @@ -341,8 +344,8 @@ function Base.sec(x::BareInterval{T}) where {T<:AbstractFloat}

lo, hi = bounds(x)

lo_quadrant = _quadrant(lo)
hi_quadrant = _quadrant(hi)
lo_quadrant = _quadrant(min, lo)
hi_quadrant = _quadrant(max, hi)

if lo_quadrant == hi_quadrant
(lo_quadrant == 0) | (lo_quadrant == 1) && return @round(T, sec(lo), sec(hi)) # increasing
Expand Down Expand Up @@ -379,8 +382,8 @@ function Base.csc(x::BareInterval{T}) where {T<:AbstractFloat}

lo, hi = bounds(x)

lo_quadrant = _quadrant(lo)
hi_quadrant = _quadrant(hi)
lo_quadrant = _quadrant(min, lo)
hi_quadrant = _quadrant(max, hi)

if (lo_quadrant == 2 || lo_quadrant == 3) && hi == 0
# singularity from the left
Expand Down