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Speed up sin and cos by a factor 6.5 #117

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Apr 11, 2018
Merged

Speed up sin and cos by a factor 6.5 #117

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f167c50
Speed up trig by a few times
dpsanders Apr 9, 2018
69d783e
Fix sin(x::BigFloat)
dpsanders Apr 10, 2018
564d060
Fix sin for big interval
dpsanders Apr 10, 2018
776e419
Revert tan to use previous find_quadrants
dpsanders Apr 11, 2018
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41 changes: 34 additions & 7 deletions src/intervals/trigonometric.jl
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Original file line number Diff line number Diff line change
@@ -1,9 +1,21 @@
# This file is part of the IntervalArithmetic.jl package; MIT licensed

half_pi(::Type{T}) where T = pi_interval(T) / 2
half_pi(x::T) where T<:AbstractFloat = half_pi(T)

two_pi(::Type{T}) where T = pi_interval(T) * 2
# hard code this for efficiency:
const one_over_half_pi_interval = inv(0.5 * pi_interval(Float64))

"""
Multiply an interval by a positive constant.
For efficiency, does not check that the constant is positive.
"""
multiply_by_positive_constant(α, x::Interval) = @round(α*x.lo, α*x.hi)
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Perhaps instead of defining multiply_by_positive_constant we should rather define the methods *(a::T, x::Interval) and *(x::Interval, a::T) (in arithmetic.jl), where T<:Real (or maybe <:Number), or even unspecified, as it is here. Currently we use promotion to an interval and rounding, which may slow down things.

Question: Below you use this function for to define half_pi(::Type{Float64}), but not for other types; can you explain why?

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I use it for two_pi as well.

Actually the half_pi function is no longer used, I believe.

We may well want to define explicit methods for *(constant, Interval). I've opened an issue: #120

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That was actually the suggestion.

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Right, but I'd prefer to leave it for a separate PR.
Also, if you actually know that the constant is positive then this specialised version is even slightly quicker.


half_pi(::Type{Float64}) = multiply_by_positive_constant(0.5, pi_interval(Float64))
half_pi(::Type{T}) where {T} = 0.5 * pi_interval(T)
half_pi(x::T) where {T<:AbstractFloat} = half_pi(T)

two_pi(::Type{Float64}) = multiply_by_positive_constant(2.0, pi_interval(Float64))
two_pi(::Type{T}) where {T} = 2 * pi_interval(T)

range_atan2(::Type{T}) where {T<:Real} = Interval(-(pi_interval(T).hi), pi_interval(T).hi)
half_range_atan2(::Type{T}) where {T} = (temp = half_pi(T); Interval(-(temp.hi), temp.hi) )
Expand All @@ -19,9 +31,17 @@ This is a rather indirect way to determine if π/2 and 3π/2 are contained
in the interval; cf. the formula for sine of an interval in
Tucker, *Validated Numerics*."""

function find_quadrants(x::T) where {T<:AbstractFloat}
function find_quadrants(x::T) where {T}
temp = atomic(Interval{T}, x) / half_pi(x)
(floor(temp.lo), floor(temp.hi))

return SVector(floor(temp.lo), floor(temp.hi))
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What about returning a tuple here, instead of a SVector? (Same comment applies in line 44.)

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It used to be a tuple; changing it to an SVector had a ridiculously big impact on the speed due to the minimum and maximum below that are not specialised for tuples!

In any case all of this code can be simplified a lot when #119 is possible, I believe.

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Fair enough.

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OK, I found an alternative solution: return a tuple and do

max(tuple...)

Too late...

end

function find_quadrants(x::Float64)
temp = multiply_by_positive_constant(x, one_over_half_pi_interval)
# x / half_pi(Float64)

return SVector(floor(temp.lo), floor(temp.hi))
end

function sin(a::Interval{T}) where T
Expand All @@ -31,6 +51,8 @@ function sin(a::Interval{T}) where T

diam(a) > two_pi(T).lo && return whole_range

# The following is equiavlent to doing temp = a / half_pi and
# taking floor(a.lo), floor(a.hi)
lo_quadrant = minimum(find_quadrants(a.lo))
hi_quadrant = maximum(find_quadrants(a.hi))

Expand Down Expand Up @@ -109,14 +131,19 @@ function cos(a::Interval{T}) where T
end
end

function find_quadrants_tan(x::T) where {T}
temp = atomic(Interval{T}, x) / half_pi(x)

return SVector(floor(temp.lo), floor(temp.hi))
end

function tan(a::Interval{T}) where T
isempty(a) && return a

diam(a) > pi_interval(T).lo && return entireinterval(a)

lo_quadrant = minimum(find_quadrants(a.lo))
hi_quadrant = maximum(find_quadrants(a.hi))
lo_quadrant = minimum(find_quadrants_tan(a.lo))
hi_quadrant = maximum(find_quadrants_tan(a.hi))

lo_quadrant_mod = mod(lo_quadrant, 2)
hi_quadrant_mod = mod(hi_quadrant, 2)
Expand Down