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Base: fix floating-point div #49561

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Base: fix floating-point div #49561

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67 changes: 63 additions & 4 deletions base/div.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -364,7 +364,66 @@ function div(x::T, y::T, ::typeof(RoundUp)) where T<:Integer
return d + (((x > 0) == (y > 0)) & (d * y != x))
end

# Real
# NOTE: C89 fmod() and x87 FPREM implicitly provide truncating float division,
# so it is used here as the basis of float div().
div(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat} = convert(T, round((x - rem(x, y, r)) / y))
function mw_div(x::NTuple{2,T}, y::T) where {T<:AbstractFloat}
(x_hi, x_lo) = x

# "DWDivFP3" AKA "Algorithm 15" from a paper by Joldes, Muller and
# Popescu: https://doi.org/10.1145/3121432
hi = x_hi / y
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I would be interested in if this speeds up and retains most of the accuracy if you replay x_hi / y with x_hi * inv(y) doing so would let you save a division on since then you could do the same for lo = δ / y (and since we only care about the high word of this, it could fold with the canonicalize2 into return fma(inv(y), δ, hi) saving another instruction.

π = Base.Math.two_mul(hi, y)
δ_hi = x_hi - first(π) # exact operation
δ_t = δ_hi - last(π) # exact operation
δ = δ_t + x_lo
lo = δ / y
canonicalize2(hi, lo)
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Since we only care about the high part from this, we could save a few clocks by just returning hi+lo which would save ~3 adds.

end

div_impl4(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat} =
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if we're widening, can't we just use round(x/y, r)?

T(div_impl(widen(x), widen(y), r))

div_impl3(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat} =
round(x / y, r)

div_impl2(x::NTuple{2,T}, y::T) where {T<:AbstractFloat} =
round(first(mw_div(x, y)), RoundNearest)

# Approximately rounded x (with respect to y and r)
frac_round(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat} =
add12(x, -rem(x, y, r))

div_impl1(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat} =
div_impl2(frac_round(x, y, r), y)

div_impl0(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat} =
ifelse(
isfinite(abs(x) + abs(y)),
div_impl1(x, y, r),
div_impl3(x, y, r), # try to avoid overflow
)
Comment on lines +381 to +402
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These need better names that actually reflect what they do. (and should probably be collapsed into their callers.


# Other rounding modes are assumed not to have `rem`.
const RoundingModesWithRem = Union{
RoundingMode{:Nearest},
RoundingMode{:Up},
RoundingMode{:Down},
RoundingMode{:FromZero},
RoundingMode{:ToZero},
}

div_impl(x::T, y::T, r::RoundingModesWithRem) where {T<:AbstractFloat} =
div_impl0(x, y, r)
div_impl(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat} =
div_impl3(x, y, r)

# The only way to achieve close-to-perfect results with `Float16` seems
# to be to widen to `Float32`.
div_impl(x::Float16, y::Float16, r::RoundingModesWithRem) =
div_impl4(x, y, r)
div_impl(x::Float16, y::Float16, r::RoundingMode) =
div_impl4(x, y, r)

function div(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat}
isnan(x) && (return x)
isnan(y) && (return y)
div_impl(x, y, r)
end
104 changes: 104 additions & 0 deletions test/numbers.jl
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Expand Up @@ -1702,6 +1702,110 @@ end
@test cld(-1.1, 0.1) == div(-1.1, 0.1, RoundUp) == ceil(big(-1.1)/big(0.1)) == -11.0
@test fld(-1.1, 0.1) == div(-1.1, 0.1, RoundDown) == floor(big(-1.1)/big(0.1)) == -12.0
end

@testset "correctness despite possible numerical issues: $Small, $rounding_mode_with_rem" for
Small ∈ (Float16, Float32, Float64),
rounding_mode_with_rem ∈ (RoundNearest, RoundUp, RoundDown, RoundFromZero, RoundToZero),
xy = (
(4.445, 0.00675),
(2.941, -0.00301),
(-0.1852, 0.0001928),
(4.56, 0.006817),
(-0.178, -0.0001851),
(-6.51e4, -4.06e3),
(0.008255, -1.01e-5),
(-276.0, -0.1678),
(-39.7, -0.02045),
(0.7246, 0.0003948),
(-6.55e4, -29.81),
(-6.55e4, -27.81),
(-6.55e4, -24.25),
(6.55e4, 29.2),
(-6.55e4, -30.14),
(76.8, -0.11804),
(5.252e3, -2.607),
(4.365e4, -22.03),
(229.0, 0.1162),
(1.009, -0.0006504),
(-6.55e4, 29.81),
(6.55e4, -27.08),
(6.55e4, -31.62),
(6.55e4, -21.34),
(6.55e4, -31.27),
(6.307e4, 2.4e4),
(2.804e3, 3.502),
(-5.786e4, 5.005e4),
(-4.97e4, -2.387e4),
(-2.217, -0.001557),
(6.55e4, 26.6),
(6.55e4, 17.06),
(-6.55e4, -22.73),
(-6.55e4, 29.25),
(6.55e4, 29.28),
(1.169, -0.001584),
(20.94, 0.01427),
(0.1521, 0.0001633),
(2.406e4, -11.78),
(0.0766, 4.73e-5),
(-1.3231137e30, 2.4482906e23),
(-4.8544084e-25, 7.407144e-32),
(3.103397e26, 4.4375036e19),
(-4.7189755e-22, -6.6208687e-29),
(1.4102157e10, 1819.6719),
(1.1742472e13, -1.4370354e6),
(1.483496e-18, 9.9707963e-26),
(-5.016058e24, -4.956325e17),
(4.040856e22, 2.7740254e15),
(3.277167e14, -2.6412792e7),
(-1.849337e32, 1.7634905e25),
(-1.625298e-17, 1.0709875e-24),
(1.1791839e-31, 1.0768432e-38),
(-2.3660772e-30, -1.6153781e-37),
(-1.0070932e29, 7.4530127e21),
(-0.00061855844, -4.1053827e-11),
(2.6379836e12, -173271.67),
(9.8551985e-27, -1.48112095e-33),
(1.858432e26, 1.3008654e19),
(4.271524e19, -3.1128782e12),
(9.1970936e29, 1.2572408e23),
(-36199.926, -0.0023891365),
(2.2035416e20, -1.3350574e13),
(-6.933978e-26, -1.06443724e-32),
(-8.4979565e19, -7.333614e12),
(1.0172350506694424e17, 23.315505841986997),
(-1.47773180470849e-22, -4.0998853221425914e-38),
(-4.816178860121429e-125, 1.5705370217600465e-140),
(1.438033388799865e129, 3.8802573321302744e113),
(5.630449277879664e-200, -2.076135363156452e-215),
(-3.29867858971227e241, -8.079378598133822e225),
(-9.287161662516348e12, -0.00261425024965316),
(-4.017546451947597e-21, -6.334812648091802e-37),
(1.3802702942552817e-103, -1.5665085059273602e-119),
(2.2912868470550055e-76, -2.7514392143441017e-92),
(1.7074328973956706e151, -2.0403029505651566e135),
(-9.290503413404926e-267, -1.0856248919388625e-282),
(1.635940091056537e-143, -3.771680017287185e-159),
(1.1031633123212033e-289, 1.4605623613009784e-305),
(-5.984102559690793e-25, 1.3308245167101786e-40),
(-2.89964579258566e-97, -4.34310409713307e-113),
(-4.895423533487687e-224, -5.77559167362022e-240),
(3.2050860779440005e-66, 4.23835027226126e-82),
(-2.046347683435319e-194, 5.5954131552084545e-210),
(-9.223282007712624e-73, 1.0588268730479068e-88),
(-6.996263999884589e-36, 1.848582153492107e-51),
(-2.42983435949937e178, -6.557998479200877e162),
(3.402679467861051e45, -5.068403522082035e29),
(-1.9402396639482086e35, 2.7453450266399908e19),
(7.114570134851155e264, 1.6742521824981587e249),
)
ab = map(Small, xy)
AB = map(BigFloat, ab)
tested = div(ab..., rounding_mode_with_rem)
correct = div(AB..., rounding_mode_with_rem)
correct_small = Small(correct)
# Only test when the correct result is representable
(correct == correct_small) && @test tested == correct_small
end
end
@testset "return types" begin
for T in (Int8,Int16,Int32,Int64,Int128, UInt8,UInt16,UInt32,UInt64,UInt128)
Expand Down