Optimized stolarsky_mean
#2274
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Optimized stolarsky_mean
#2274
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andrewwinters5000
requested changes
Feb 11, 2025
Co-authored-by: Andrew Winters <[email protected]>
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Hendrik made me realize that for integers the For real numbers: For integers So, basically just by preallocating the power functions a 50% speed up is gained. The |
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Results on university machine (Goldstein): new version old version: There's still a roughly 17% improvement. Benchmarks for Goldstein: Integers: |
ranocha
reviewed
Feb 11, 2025
| @@ -412,8 +412,15 @@ Given ε = 1.0e-4, we use the following algorithm. | |||
| c3 = convert(RealT, -1 / 21) * (2 * gamma * (gamma - 2) - 9) * c2 | |||
| return 0.5f0 * (x + y) * @evalpoly(f2, 1, c1, c2, c3) | |||
| else | |||
| return (gamma - 1) / gamma * (y^gamma - x^gamma) / | |||
| (y^(gamma - 1) - x^(gamma - 1)) | |||
| if isinteger(gamma) | |||
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Suggested change
| if isinteger(gamma) | |
| if gamma isa Integer |
isinteger checks also the values of floating-point numbers
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That seems much faster, but I had to remove the specific parametrization. That would imply also to change the struct, so that, for instance, 2 can be accepted, instead of 2.0.
julia> @inline function stolarsky_mean_1(x::RealT, y::RealT, gamma::RealT) where {RealT <: Real}
epsilon_f2 = convert(RealT, 1.0e-4)
f2 = (x * (x - 2 * y) + y * y) / (x * (x + 2 * y) + y * y) # f2 = f^2
if f2 < epsilon_f2
# convenience coefficients
c1 = convert(RealT, 1 / 3) * (gamma - 2)
c2 = convert(RealT, -1 / 15) * (gamma + 1) * (gamma - 3) * c1
c3 = convert(RealT, -1 / 21) * (2 * gamma * (gamma - 2) - 9) * c2
return 0.5f0 * (x + y) * @evalpoly(f2, 1, c1, c2, c3)
else
if isinteger(gamma)
yg = y^(gamma-1)
xg = x^(gamma-1)
else
yg = exp((gamma-1)*log(y))
xg = exp((gamma-1)*log(x))
end
return (gamma - 1) * (yg*y - xg*x) / (gamma * ( yg - xg))
end
end
stolarsky_mean_1 (generic function with 1 method)
julia> @inline function stolarsky_mean_2(x::Real, y::Real, gamma::Real)
epsilon_f2 = convert(Real, 1.0e-4)
f2 = (x * (x - 2 * y) + y * y) / (x * (x + 2 * y) + y * y) # f2 = f^2
if f2 < epsilon_f2
# convenience coefficients
c1 = convert(Real, 1 / 3) * (gamma - 2)
c2 = convert(Real, -1 / 15) * (gamma + 1) * (gamma - 3) * c1
c3 = convert(Real, -1 / 21) * (2 * gamma * (gamma - 2) - 9) * c2
return 0.5f0 * (x + y) * @evalpoly(f2, 1, c1, c2, c3)
else
if gamma isa Integer
yg = y^(gamma-1)
xg = x^(gamma-1)
else
yg = exp((gamma-1)*log(y))
xg = exp((gamma-1)*log(x))
end
return (gamma - 1) * (yg*y - xg*x) / (gamma * (yg - xg) )
end
end
stolarsky_mean_2 (generic function with 1 method)
julia> @benchmark value = stolarsky_mean_1($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(1.7))[])
BenchmarkTools.Trial: 10000 samples with 997 evaluations per sample.
Range (min … max): 19.647 ns … 27.185 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 19.982 ns ┊ GC (median): 0.00%
Time (mean ± σ): 20.086 ns ± 0.438 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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19.6 ns Histogram: log(frequency) by time 22.1 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_2($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(1.7))[])
BenchmarkTools.Trial: 10000 samples with 997 evaluations per sample.
Range (min … max): 19.394 ns … 35.163 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 19.588 ns ┊ GC (median): 0.00%
Time (mean ± σ): 19.620 ns ± 0.326 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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19.4 ns Histogram: frequency by time 20.7 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_1($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(1.0))[])
BenchmarkTools.Trial: 10000 samples with 999 evaluations per sample.
Range (min … max): 8.078 ns … 20.871 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 9.662 ns ┊ GC (median): 0.00%
Time (mean ± σ): 9.584 ns ± 0.374 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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8.08 ns Histogram: log(frequency) by time 10.5 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_2($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(1))[])
BenchmarkTools.Trial: 10000 samples with 1000 evaluations per sample.
Range (min … max): 2.500 ns … 7.277 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 2.536 ns ┊ GC (median): 0.00%
Time (mean ± σ): 2.543 ns ± 0.107 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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2.5 ns Histogram: frequency by time 2.57 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_1($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(2.0))[])
BenchmarkTools.Trial: 10000 samples with 999 evaluations per sample.
Range (min … max): 9.267 ns … 17.994 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 10.538 ns ┊ GC (median): 0.00%
Time (mean ± σ): 10.502 ns ± 0.383 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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9.27 ns Histogram: log(frequency) by time 11.8 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_2($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(2))[])
BenchmarkTools.Trial: 10000 samples with 1000 evaluations per sample.
Range (min … max): 5.276 ns … 11.283 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 5.684 ns ┊ GC (median): 0.00%
Time (mean ± σ): 5.751 ns ± 0.224 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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5.28 ns Histogram: log(frequency) by time 6.24 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
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Can you please change the code
@inline stolarsky_mean(x::Real, y::Real, gamma::Real) = stolarsky_mean(promote(x, y,
gamma)...)a few lines above to
@inline stolarsky_mean(x::Real, y::Real, gamma::Real) = stolarsky_mean(promote(x, y)..., gamma)and then check the gamma isa Integer option?
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I get the following error when trying that. As far as I understood, that creates a loop because the main function is always defined for type Reals, and all the variables are bounded to be of the same type, hence the function calls itself and doesn't switch to the "main" one.
julia> function stolarsky_mean(x::RealT, y::RealT, gamma::RealT) where {RealT <: Real}
epsilon_f2 = convert(RealT, 1.0e-4)
f2 = (x * (x - 2 * y) + y * y) / (x * (x + 2 * y) + y * y) # f2 = f^2
if f2 < epsilon_f2
# convenience coefficients
c1 = convert(RealT, 1 / 3) * (gamma - 2)
c2 = convert(RealT, -1 / 15) * (gamma + 1) * (gamma - 3) * c1
c3 = convert(RealT, -1 / 21) * (2 * gamma * (gamma - 2) - 9) * c2
return 0.5f0 * (x + y) * @evalpoly(f2, 1, c1, c2, c3)
else
if gamma isa Integer
yg = y^(gamma - 1)
xg = x^(gamma - 1)
else
yg = exp((gamma - 1) * log(y)) # equivalent to y^gamma but faster for non-integers
xg = exp((gamma - 1) * log(x)) # equivalent to x^gamma but faster for non-integers
end
return (gamma - 1) / gamma * (yg * y - xg * x) /
(yg - xg)
end
end
stolarsky_mean (generic function with 2 methods)
julia> stolarsky_mean(x::Real, y::Real, gamma::Real) = stolarsky_mean(promote(x, y)..., gamma)
stolarsky_mean (generic function with 2 methods)
julia> @benchmark value = stolarsky_mean($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(2))[])
ERROR: StackOverflowError:
Stacktrace:
[1] stolarsky_mean(x::Float64, y::Float64, gamma::Int64) (repeats 79984 times)
@ Main .\REPL[31]:1
julia> stolarsky_mean(301.2, 421.2, 2)
ERROR: StackOverflowError:
Stacktrace:
[1] stolarsky_mean(x::Float64, y::Float64, gamma::Int64) (repeats 79984 times)
@ Main .\REPL[31]:1
A workaround would be something like:
stolarsky_mean(x::Real, y::Real, gamma::Real) = stolarsky_mean(promote(x, y)..., gamma)
function stolarsky_mean(x::RealT, y::RealT, gamma::Real) where {RealT <: Real}
epsilon_f2 = convert(RealT, 1.0e-4)
f2 = (x * (x - 2 * y) + y * y) / (x * (x + 2 * y) + y * y) # f2 = f^2
if f2 < epsilon_f2
# convenience coefficients
c1 = convert(RealT, 1 / 3) * (gamma - 2)
c2 = convert(RealT, -1 / 15) * (gamma + 1) * (gamma - 3) * c1
c3 = convert(RealT, -1 / 21) * (2 * gamma * (gamma - 2) - 9) * c2
return 0.5f0 * (x + y) * @evalpoly(f2, 1, c1, c2, c3)
else
if gamma isa Integer
yg = y^(gamma - 1)
xg = x^(gamma - 1)
else
yg = exp((gamma - 1) * log(y)) # equivalent to y^gamma but faster for non-integers
xg = exp((gamma - 1) * log(x)) # equivalent to x^gamma but faster for non-integers
end
return (gamma - 1) / gamma * (yg * y - xg * x) /
(yg - xg)
end
end
but again, shouldn't I allow the equation struct to accept different types and not promoting the gamma?
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What I meant was basically the second option:
julia> stolarsky_mean(x::Real, y::Real, gamma::Real) = stolarsky_mean(promote(x, y)..., gamma)
stolarsky_mean (generic function with 1 method)
julia> function stolarsky_mean(x::RealT, y::RealT, gamma::Real) where {RealT <: Real}
epsilon_f2 = convert(RealT, 1.0e-4)
f2 = (x * (x - 2 * y) + y * y) / (x * (x + 2 * y) + y * y) # f2 = f^2
if f2 < epsilon_f2
# convenience coefficients
c1 = convert(RealT, 1 / 3) * (gamma - 2)
c2 = convert(RealT, -1 / 15) * (gamma + 1) * (gamma - 3) * c1
c3 = convert(RealT, -1 / 21) * (2 * gamma * (gamma - 2) - 9) * c2
return 0.5f0 * (x + y) * @evalpoly(f2, 1, c1, c2, c3)
else
if gamma isa Integer
yg = y^(gamma - 1)
xg = x^(gamma - 1)
else
yg = exp((gamma - 1) * log(y)) # equivalent to y^gamma but faster for non-integers
xg = exp((gamma - 1) * log(x)) # equivalent to x^gamma but faster for non-integers
end
return (gamma - 1) * (yg * y - xg * x) / (gamma * (yg - xg))
end
end
stolarsky_mean (generic function with 2 methods)
julia> stolarsky_mean(300.1, 410.7, 2)
355.40000000000003
julia> stolarsky_mean(300.1, 410.7, 2.0)
355.39999999999986
julia> stolarsky_mean(300.1, 410.7f0, 2.0)
355.4000061035154
julia> stolarsky_mean(300.1, 410.7f0, 2)
355.40000610351564Using this approach, gamma can be an Integer and will not be promoted.
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Perfect, thank you! Sorry for the missunderstanding, I will introduce that.
ranocha
reviewed
Feb 11, 2025
ranocha
reviewed
Feb 11, 2025
Comment on lines
+422
to
+423
| return (gamma - 1) / gamma * (yg * y - xg * x) / | ||
| (yg - xg) |
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Suggested change
| return (gamma - 1) / gamma * (yg * y - xg * x) / | |
| (yg - xg) | |
| return (gamma - 1) / gamma * (yg * y - xg * x) / (yg - xg) |
Does this formatting work now?
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And does it make a difference if you avoid an additional division by rewriting this part as something like
Suggested change
| return (gamma - 1) / gamma * (yg * y - xg * x) / | |
| (yg - xg) | |
| return (gamma - 1) * (yg * y - xg * x) / (gamma * (yg - xg)) |
?
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It looks like there's not a big difference. It is slightly faster for reals, but slightly slower for integers.
julia> @inline function stolarsky_mean_1(x::RealT, y::RealT, gamma::RealT) where {RealT <: Real}
epsilon_f2 = convert(RealT, 1.0e-4)
f2 = (x * (x - 2 * y) + y * y) / (x * (x + 2 * y) + y * y) # f2 = f^2
if f2 < epsilon_f2
# convenience coefficients
c1 = convert(RealT, 1 / 3) * (gamma - 2)
c2 = convert(RealT, -1 / 15) * (gamma + 1) * (gamma - 3) * c1
c3 = convert(RealT, -1 / 21) * (2 * gamma * (gamma - 2) - 9) * c2
return 0.5f0 * (x + y) * @evalpoly(f2, 1, c1, c2, c3)
else
if isinteger(gamma)
yg = y^(gamma-1)
xg = x^(gamma-1)
else
yg = exp((gamma-1)*log(y))
xg = exp((gamma-1)*log(x))
end
return (gamma - 1)/gamma * (yg*y - xg*x) / (yg - xg)
end
end
stolarsky_mean_1 (generic function with 1 method)
julia> @inline function stolarsky_mean_2(x::RealT, y::RealT, gamma::RealT) where {RealT <: Real}
epsilon_f2 = convert(RealT, 1.0e-4)
f2 = (x * (x - 2 * y) + y * y) / (x * (x + 2 * y) + y * y) # f2 = f^2
if f2 < epsilon_f2
# convenience coefficients
c1 = convert(RealT, 1 / 3) * (gamma - 2)
c2 = convert(RealT, -1 / 15) * (gamma + 1) * (gamma - 3) * c1
c3 = convert(RealT, -1 / 21) * (2 * gamma * (gamma - 2) - 9) * c2
return 0.5f0 * (x + y) * @evalpoly(f2, 1, c1, c2, c3)
else
if isinteger(gamma)
yg = y^(gamma-1)
xg = x^(gamma-1)
else
yg = exp((gamma-1)*log(y))
xg = exp((gamma-1)*log(x))
end
return (gamma - 1) * (yg*y - xg*x) / (gamma * (yg - xg) )
end
end
stolarsky_mean_2 (generic function with 1 method)
julia> @benchmark value = stolarsky_mean_1($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(2.0))[])
BenchmarkTools.Trial: 10000 samples with 999 evaluations per sample.
Range (min … max): 8.965 ns … 101.965 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 9.815 ns ┊ GC (median): 0.00%
Time (mean ± σ): 9.794 ns ± 1.531 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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8.96 ns Histogram: log(frequency) by time 11.4 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_2($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(2.0))[])
BenchmarkTools.Trial: 10000 samples with 999 evaluations per sample.
Range (min … max): 9.183 ns … 19.411 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 10.492 ns ┊ GC (median): 0.00%
Time (mean ± σ): 10.481 ns ± 0.363 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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9.18 ns Histogram: log(frequency) by time 11.8 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_1($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(1.4))[])
BenchmarkTools.Trial: 10000 samples with 997 evaluations per sample.
Range (min … max): 19.662 ns … 45.032 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 19.761 ns ┊ GC (median): 0.00%
Time (mean ± σ): 19.815 ns ± 0.562 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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19.7 ns Histogram: frequency by time 21.4 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_2($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(1.4))[])
BenchmarkTools.Trial: 10000 samples with 997 evaluations per sample.
Range (min … max): 19.621 ns … 27.873 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 19.987 ns ┊ GC (median): 0.00%
Time (mean ± σ): 20.087 ns ± 0.432 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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19.6 ns Histogram: log(frequency) by time 22.1 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_1($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(1.7))[])
BenchmarkTools.Trial: 10000 samples with 997 evaluations per sample.
Range (min … max): 19.658 ns … 27.584 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 19.765 ns ┊ GC (median): 0.00%
Time (mean ± σ): 19.884 ns ± 0.437 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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19.7 ns Histogram: log(frequency) by time 21.8 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark value = stolarsky_mean_2($(Ref(300.1))[], $(Ref(410.7))[], $(Ref(1.7))[])
BenchmarkTools.Trial: 10000 samples with 997 evaluations per sample.
Range (min … max): 19.642 ns … 42.111 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 19.983 ns ┊ GC (median): 0.00%
Time (mean ± σ): 20.010 ns ± 0.352 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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19.6 ns Histogram: frequency by time 21.2 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
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I am mainly looking at the median results. There, the first option seems to be best all the time?
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I agree with you, I will compare also full runs, to see if there are any noticeable differences.
Co-authored-by: Hendrik Ranocha <[email protected]>
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The stolarsky mean will come in handy in Trixi Atmo. Here a faster version:
Simulation of EC Polytropic Euler with the previous stolarsky mean:
──────────────────────────────────────────────────────────────────────────────────── Trixi.jl Time Allocations ─────────────────────── ──────────────────────── Tot / % measured: 1.98s / 99.3% 2.34MiB / 90.4% Section ncalls time %tot avg alloc %tot avg ──────────────────────────────────────────────────────────────────────────────────── rhs! 1.42k 1.93s 98.1% 1.36ms 5.14KiB 0.2% 3.70B volume integral 1.42k 1.74s 88.4% 1.22ms 0.00B 0.0% 0.00B interface flux 1.42k 163ms 8.3% 115μs 0.00B 0.0% 0.00B surface integral 1.42k 15.3ms 0.8% 10.8μs 0.00B 0.0% 0.00B Jacobian 1.42k 8.10ms 0.4% 5.70μs 0.00B 0.0% 0.00B reset ∂u/∂t 1.42k 3.72ms 0.2% 2.62μs 0.00B 0.0% 0.00B ~rhs!~ 1.42k 1.00ms 0.1% 705ns 5.14KiB 0.2% 3.70B boundary flux 1.42k 46.3μs 0.0% 32.6ns 0.00B 0.0% 0.00B source terms 1.42k 45.5μs 0.0% 32.0ns 0.00B 0.0% 0.00B calculate dt 285 26.2ms 1.3% 92.0μs 0.00B 0.0% 0.00B analyze solution 4 7.33ms 0.4% 1.83ms 314KiB 14.5% 78.5KiB I/O 5 4.19ms 0.2% 838μs 1.81MiB 85.3% 370KiB save solution 4 3.82ms 0.2% 956μs 1.80MiB 84.9% 460KiB ~I/O~ 5 365μs 0.0% 73.1μs 8.83KiB 0.4% 1.77KiB get element variables 4 730ns 0.0% 182ns 0.00B 0.0% 0.00B save mesh 4 608ns 0.0% 152ns 0.00B 0.0% 0.00B get node variables 4 95.0ns 0.0% 23.8ns 0.00B 0.0% 0.00B ────────────────────────────────────────────────────────────────────────────────────Results for the optimized version
──────────────────────────────────────────────────────────────────────────────────── Trixi.jl Time Allocations ─────────────────────── ──────────────────────── Tot / % measured: 1.62s / 99.1% 2.34MiB / 90.4% Section ncalls time %tot avg alloc %tot avg ──────────────────────────────────────────────────────────────────────────────────── rhs! 1.42k 1.57s 97.7% 1.11ms 5.14KiB 0.2% 3.70B volume integral 1.42k 1.41s 87.8% 994μs 0.00B 0.0% 0.00B interface flux 1.42k 131ms 8.2% 92.4μs 0.00B 0.0% 0.00B surface integral 1.42k 14.9ms 0.9% 10.5μs 0.00B 0.0% 0.00B Jacobian 1.42k 7.29ms 0.5% 5.13μs 0.00B 0.0% 0.00B reset ∂u/∂t 1.42k 3.83ms 0.2% 2.70μs 0.00B 0.0% 0.00B ~rhs!~ 1.42k 956μs 0.1% 673ns 5.14KiB 0.2% 3.70B boundary flux 1.42k 61.6μs 0.0% 43.3ns 0.00B 0.0% 0.00B source terms 1.42k 24.5μs 0.0% 17.3ns 0.00B 0.0% 0.00B calculate dt 285 26.0ms 1.6% 91.2μs 0.00B 0.0% 0.00B analyze solution 4 6.36ms 0.4% 1.59ms 314KiB 14.5% 78.6KiB I/O 5 5.35ms 0.3% 1.07ms 1.81MiB 85.3% 370KiB save solution 4 5.00ms 0.3% 1.25ms 1.80MiB 84.9% 461KiB ~I/O~ 5 352μs 0.0% 70.5μs 8.83KiB 0.4% 1.77KiB save mesh 4 755ns 0.0% 189ns 0.00B 0.0% 0.00B get element variables 4 477ns 0.0% 119ns 0.00B 0.0% 0.00B get node variables 4 86.0ns 0.0% 21.5ns 0.00B 0.0% 0.00B ────────────────────────────────────────────────────────────────────────────────────Thus on my machine, there's an 18% improvement.