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Use rem2pi and rem with BigFloats in _quadrant #633

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Use rem2pi and rem with BigFloats in _quadrant #633

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f2742ed
Use re2pi and rem with BigFloat and RoundNearest
lbenet Mar 10, 2024
3b212a9
Add tests for _quadrant involving worst Float64 cases
lbenet Mar 10, 2024
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23 changes: 17 additions & 6 deletions src/intervals/arithmetic/trigonometric.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -3,13 +3,23 @@
# in Section 10.5.3

# helper functions
# NOTE: According to J-M Muller, "Elementary Functions: Algorithms and
# implementations", Birkhäuser, 3rd ed (2005), chap. 11, pp 203,
# "The first solution consists of using multiple-precision arithmetic,
# but this may make the computation significantly slower. Moreover,
# it is not that easy to predict the accuracy with which the computation
# should be performed." Later (pp 210), it estimates that the precision needed
# to compute correctly the worst case of the range reduction algorithm
# for Float64 is about 61 "guard digits". Below we use `rem2pi` for
# BigFloats, with at least 200 bits of precision.
# (As noted before, `rem2pi(x, RoundNearest)`, with Float64 indeed may have inaccuracies.)
#
# For consistency, `_quadrantpi` uses `rem(big(x), 2, RoundNearest)`

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)
@assert precision(BigFloat) > 200
PI_LO, PI_HI = bounds(bareinterval(BigFloat, π))
r = rem2pi(big(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"))
Expand All @@ -20,7 +30,8 @@ function _quadrant(x::AbstractFloat)
end

function _quadrantpi(x::AbstractFloat) # used in `sinpi` and `cospi`
r = rem(x, 2) # [-2, 2], should be exact for floats
@assert precision(BigFloat) > 200
r = rem(big(x), 2, RoundNearest) # [-2, 2], should be exact for floats
2r < -3 && return 0 # [-2π, -3π/2)
r < -1 && return 1 # [-3π/2, -π)
2r < -1 && return 2 # [-π, -π/2)
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12 changes: 12 additions & 0 deletions test/interval_tests/trigonometric.jl
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Expand Up @@ -307,4 +307,16 @@ end
@test isequal_interval(sin(x), interval(-1, 1))
@test isequal_interval(cos(x), interval(-1, 1))
@test isequal_interval(tan(x), interval(-Inf, Inf))

setprecision(100) do
y = 6381956970095103 * 2.0^797 # worst case for C = pi/2 for Float64
yb = 6381956970095103 * big(2.0)^797
@test_throws AssertionError IntervalArithmetic._quadrant(y) == IntervalArithmetic._quadrant(yb)
end
setprecision(1000) do
y = 6381956970095103 * 2.0^797 # worst case for C = pi/2 for Float64
yb = 6381956970095103 * big(2.0)^797
@test IntervalArithmetic._quadrant(y) == IntervalArithmetic._quadrant(yb)
end

end
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