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Improve power of complex intervals#745

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OlivierHnt wants to merge 4 commits intoJuliaIntervals:masterfrom
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Improve power of complex intervals#745
OlivierHnt wants to merge 4 commits intoJuliaIntervals:masterfrom
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@OlivierHnt
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@OlivierHnt OlivierHnt commented Mar 9, 2026

Since we use the formula exp(y * log(x)) to evaluate x^y when x::Complex{<:Interval}, y::Complex{<:Interval}, we get the following undesirable overestimation:

x = interval(-0.5) + interval(im) * interval(-1e-14, 1e-14)
x^1 # [-0.5, 0.5]_com + im*[-0.5, 0.5]_com
x^2 # [-0.25, 0.25]_com + im*[-0.25, 0.25]_com

yet

x*x # [0.25, 0.25]_com + im*[-1.0e-14, 1.0e-14]_com

This PR implements the following definition for complex powers:

  • If $y = [a, b]$ ($a,b$ finite) with $b-a \le 2$, then there exist $n \in \mathbb{Z}$, $\delta \subset [-1, 1]$ such that $y = n + \delta$.
    We compute
$$x^y = \underbrace{x \cdot \ldots \cdot x}_{n-\text{times}} \cdot \exp(\delta \, \ln(x)).$$
  • If $b-a &gt; 2$, then we can write $y = \cup_{j=1}^d y_j$ where $y_j = n_j + \delta_j$ with $n_j \in \mathbb{Z}$, $\delta_1 \subset [-1, 0.5]$, $\delta_d \subset [-0.5, 1]$, and $\delta_j = [-0.5, 0.5]$ for $j=2, \dots, d-1$.
    We compute
$$x^y = \cup_{j=1}^d x^{y_j} = \cup_{j=1}^d \underbrace{x \cdot \ldots \cdot x}_{n_j-\text{times}} \cdot \exp(\delta_j \, \ln(x)).$$
  • If $y$ is not in the above two forms (i.e., $y$ is real and unbounded, or $y$ has a non-zero imaginary part), then $x^y = \exp(y \, \ln(x))$.

With this PR

x = interval(-0.5) + interval(im) * interval(-1e-14, 1e-14)
x^1 # [-0.5, -0.5]_com + im*[-1.0e-14, 1.0e-14]_com
x^2 # [0.25, 0.25]_com + im*[-1.0e-14, 1.0e-14]_com

@OlivierHnt OlivierHnt requested a review from Kolaru March 9, 2026 12:47
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github-actions bot commented Mar 9, 2026 •
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Benchmark Results (Julia v1)

Time benchmarks
master e87aac7... master / e87aac7...
BigFloat MPFI/basics/* 24.7 ± 4 μs 25.8 ± 4 μs 0.958 ± 0.22
BigFloat MPFI/basics/+ 13.2 ± 3.9 μs 12.5 ± 4.1 μs 1.05 ± 0.47
BigFloat MPFI/basics/- 17.5 ± 4.1 μs 17.1 ± 3.9 μs 1.03 ± 0.34
BigFloat MPFI/basics// 19 ± 4 μs 19.3 ± 4 μs 0.988 ± 0.29
BigFloat MPFI/basics/abs 10.3 ± 3.9 μs 10.2 ± 3.9 μs 1 ± 0.54
BigFloat MPFI/basics/acos 0.67 ± 0.0084 ms 0.674 ± 0.0083 ms 0.994 ± 0.017
BigFloat MPFI/basics/asin 0.646 ± 0.0095 ms 0.649 ± 0.0083 ms 0.994 ± 0.019
BigFloat MPFI/basics/atan 2.06 ± 0.0072 ms 2.07 ± 0.0076 ms 0.998 ± 0.005
BigFloat MPFI/basics/cos 0.355 ± 0.009 ms 0.367 ± 0.0092 ms 0.966 ± 0.034
BigFloat MPFI/basics/cosh 0.426 ± 0.011 ms 0.425 ± 0.011 ms 1 ± 0.036
BigFloat MPFI/basics/exp 0.5 ± 0.011 ms 0.499 ± 0.01 ms 1 ± 0.03
BigFloat MPFI/basics/inv 18.9 ± 3.9 μs 19.3 ± 3.7 μs 0.979 ± 0.28
BigFloat MPFI/basics/log 0.465 ± 0.0089 ms 0.465 ± 0.0089 ms 0.999 ± 0.027
BigFloat MPFI/basics/sin 0.457 ± 0.0099 ms 0.471 ± 0.0098 ms 0.969 ± 0.029
BigFloat MPFI/basics/sinh 0.566 ± 0.0098 ms 0.567 ± 0.0095 ms 0.998 ± 0.024
BigFloat MPFI/basics/sqrt 29 ± 8.2 μs 28.5 ± 7.8 μs 1.02 ± 0.4
BigFloat MPFI/basics/tan 0.38 ± 0.0093 ms 0.405 ± 0.0092 ms 0.941 ± 0.031
BigFloat MPFI/basics/tanh 0.579 ± 0.0099 ms 0.577 ± 0.01 ms 1 ± 0.024
BigFloat bareinterval/basics/* 0.0426 ± 0.0025 ms 0.0425 ± 0.0029 ms 1 ± 0.089
BigFloat bareinterval/basics/+ 24.3 ± 3.8 μs 23.6 ± 3.7 μs 1.03 ± 0.23
BigFloat bareinterval/basics/- 23.5 ± 3.5 μs 23.7 ± 3.7 μs 0.99 ± 0.22
BigFloat bareinterval/basics// 20.7 ± 5.6 μs 20.7 ± 5.6 μs 0.998 ± 0.38
BigFloat bareinterval/basics/abs 16.7 ± 3.5 μs 17.3 ± 3.6 μs 0.966 ± 0.29
BigFloat bareinterval/basics/acos 0.705 ± 0.03 ms 0.71 ± 0.029 ms 0.992 ± 0.059
BigFloat bareinterval/basics/asin 0.689 ± 0.026 ms 0.691 ± 0.025 ms 0.997 ± 0.052
BigFloat bareinterval/basics/atan 2.07 ± 0.0094 ms 2.07 ± 0.0063 ms 1 ± 0.0055
BigFloat bareinterval/basics/cos 0.338 ± 0.099 ms 0.336 ± 0.1 ms 1 ± 0.42
BigFloat bareinterval/basics/cosh 0.468 ± 0.028 ms 0.47 ± 0.027 ms 0.998 ± 0.082
BigFloat bareinterval/basics/exp 0.507 ± 0.01 ms 0.505 ± 0.01 ms 1 ± 0.029
BigFloat bareinterval/basics/inv 26.5 ± 3.6 μs 26.7 ± 3.7 μs 0.994 ± 0.19
BigFloat bareinterval/basics/log 0.515 ± 0.034 ms 0.516 ± 0.034 ms 0.998 ± 0.094
BigFloat bareinterval/basics/sin 0.472 ± 0.1 ms 0.471 ± 0.1 ms 1 ± 0.31
BigFloat bareinterval/basics/sinh 0.575 ± 0.011 ms 0.574 ± 0.01 ms 1 ± 0.027
BigFloat bareinterval/basics/sqrt 0.0383 ± 0.0029 ms 0.0384 ± 0.0029 ms 0.996 ± 0.11
BigFloat bareinterval/basics/tan 0.332 ± 0.065 ms 0.334 ± 0.065 ms 0.996 ± 0.27
BigFloat bareinterval/basics/tanh 0.591 ± 0.011 ms 0.588 ± 0.011 ms 1 ± 0.027
BigFloat interval/basics/* 0.0459 ± 0.0021 ms 0.0462 ± 0.0021 ms 0.993 ± 0.065
BigFloat interval/basics/+ 27.3 ± 9.6 μs 27 ± 9.4 μs 1.01 ± 0.5
BigFloat interval/basics/- 27.1 ± 9.5 μs 26.5 ± 9.5 μs 1.02 ± 0.52
BigFloat interval/basics// 25.5 ± 5.5 μs 25.7 ± 5.6 μs 0.993 ± 0.3
BigFloat interval/basics/abs 16.4 ± 9 μs 16.3 ± 9.1 μs 1.01 ± 0.79
BigFloat interval/basics/acos 0.697 ± 0.051 ms 0.696 ± 0.049 ms 1 ± 0.1
BigFloat interval/basics/asin 0.678 ± 0.041 ms 0.678 ± 0.039 ms 1 ± 0.084
BigFloat interval/basics/atan 2.08 ± 0.0069 ms 2.08 ± 0.0048 ms 1 ± 0.004
BigFloat interval/basics/cos 0.334 ± 0.012 ms 0.334 ± 0.012 ms 0.999 ± 0.052
BigFloat interval/basics/cosh 0.461 ± 0.026 ms 0.463 ± 0.025 ms 0.996 ± 0.078
BigFloat interval/basics/exp 0.514 ± 0.01 ms 0.515 ± 0.0095 ms 0.999 ± 0.027
BigFloat interval/basics/inv 30.8 ± 8.6 μs 31.4 ± 8.9 μs 0.981 ± 0.39
BigFloat interval/basics/log 0.504 ± 0.049 ms 0.506 ± 0.05 ms 0.997 ± 0.14
BigFloat interval/basics/sin 0.481 ± 0.016 ms 0.478 ± 0.018 ms 1.01 ± 0.05
BigFloat interval/basics/sinh 0.58 ± 0.01 ms 0.581 ± 0.0096 ms 0.998 ± 0.024
BigFloat interval/basics/sqrt 0.042 ± 0.036 ms 0.0425 ± 0.035 ms 0.987 ± 1.2
BigFloat interval/basics/tan 0.281 ± 0.013 ms 0.28 ± 0.012 ms 1 ± 0.062
BigFloat interval/basics/tanh 0.598 ± 0.01 ms 0.597 ± 0.011 ms 1 ± 0.025
bareinterval/basics/* 1.96 ± 0.031 μs 1.94 ± 0.049 μs 1.01 ± 0.03
bareinterval/basics/+ 0.541 ± 0.009 μs 0.531 ± 0.002 μs 1.02 ± 0.017
bareinterval/basics/- 0.551 ± 0.01 μs 0.541 ± 0.01 μs 1.02 ± 0.026
bareinterval/basics// 1.22 ± 0.011 μs 1.23 ± 0.03 μs 0.992 ± 0.026
bareinterval/basics/abs 0.22 ± 0.01 μs 0.21 ± 0.01 μs 1.05 ± 0.069
bareinterval/basics/acos 2.89 ± 0.031 μs 2.94 ± 0.04 μs 0.983 ± 0.017
bareinterval/basics/asin 3.17 ± 0.021 μs 3.15 ± 0.022 μs 1.01 ± 0.0097
bareinterval/basics/atan 13.3 ± 0.2 μs 12.9 ± 0.21 μs 1.03 ± 0.023
bareinterval/basics/cos 0.0505 ± 0.0076 ms 0.052 ± 0.00095 ms 0.972 ± 0.15
bareinterval/basics/cosh 7.26 ± 0.03 μs 7.29 ± 0.031 μs 0.996 ± 0.0059
bareinterval/basics/exp 4.3 ± 0.04 μs 4.32 ± 0.05 μs 0.995 ± 0.015
bareinterval/basics/inv 1.06 ± 0.02 μs 1.05 ± 0.03 μs 1.01 ± 0.034
bareinterval/basics/log 2.12 ± 0.02 μs 6.47 ± 0.03 μs 0.328 ± 0.0034
bareinterval/basics/sin 0.0494 ± 0.021 ms 0.0509 ± 0.02 ms 0.97 ± 0.57
bareinterval/basics/sinh 9.98 ± 0.051 μs 9.97 ± 0.08 μs 1 ± 0.0095
bareinterval/basics/sqrt 1.18 ± 0.02 μs 1.18 ± 0.021 μs 1 ± 0.025
bareinterval/basics/tan 28.7 ± 0.26 μs 29.2 ± 0.31 μs 0.983 ± 0.014
bareinterval/basics/tanh 0.29 ± 0.0083 ms 0.29 ± 0.0081 ms 1 ± 0.04
interval/basics/* 3.17 ± 0.08 μs 3.41 ± 0.069 μs 0.93 ± 0.03
interval/basics/+ 1.48 ± 0.071 μs 1.65 ± 0.07 μs 0.897 ± 0.057
interval/basics/- 1.52 ± 0.09 μs 1.65 ± 0.06 μs 0.921 ± 0.064
interval/basics// 2.45 ± 0.079 μs 2.64 ± 0.06 μs 0.931 ± 0.037
interval/basics/abs 0.932 ± 0.13 μs 0.922 ± 0.12 μs 1.01 ± 0.19
interval/basics/acos 5 ± 0.079 μs 5.18 ± 0.07 μs 0.965 ± 0.02
interval/basics/asin 5.14 ± 0.07 μs 5.28 ± 0.07 μs 0.973 ± 0.019
interval/basics/atan 14.6 ± 0.2 μs 15 ± 0.24 μs 0.975 ± 0.021
interval/basics/cos 0.0704 ± 0.023 ms 0.0711 ± 0.022 ms 0.991 ± 0.45
interval/basics/cosh 10.3 ± 0.07 μs 10.3 ± 0.07 μs 0.998 ± 0.0096
interval/basics/exp 6.62 ± 0.091 μs 6.63 ± 0.09 μs 0.998 ± 0.019
interval/basics/inv 2.04 ± 0.08 μs 2.03 ± 0.08 μs 1 ± 0.056
interval/basics/log 4.95 ± 0.06 μs 9.09 ± 0.07 μs 0.545 ± 0.0078
interval/basics/sin 0.069 ± 0.022 ms 0.0705 ± 0.022 ms 0.979 ± 0.44
interval/basics/sinh 13.2 ± 0.08 μs 13.3 ± 0.081 μs 0.992 ± 0.0085
interval/basics/sqrt 2.12 ± 0.1 μs 2.12 ± 0.08 μs 1 ± 0.06
interval/basics/tan 0.04 ± 0.0024 ms 0.0405 ± 0.0021 ms 0.987 ± 0.078
interval/basics/tanh 0.305 ± 0.0088 ms 0.303 ± 0.0084 ms 1.01 ± 0.04
time_to_load 0.104 ± 0.00089 s 0.107 ± 0.0021 s 0.969 ± 0.021
Memory benchmarks
master e87aac7... master / e87aac7...
BigFloat MPFI/basics/* 0.424 k allocs: 21.3 kB 0.424 k allocs: 21.3 kB 1
BigFloat MPFI/basics/+ 0.302 k allocs: 16.5 kB 0.302 k allocs: 16.5 kB 1
BigFloat MPFI/basics/- 0.402 k allocs: 20.4 kB 0.402 k allocs: 20.4 kB 1
BigFloat MPFI/basics// 0.345 k allocs: 18.2 kB 0.345 k allocs: 18.2 kB 1
BigFloat MPFI/basics/abs 0.324 k allocs: 17.4 kB 0.324 k allocs: 17.4 kB 1
BigFloat MPFI/basics/acos 1.24 k allocs: 0.0445 MB 1.24 k allocs: 0.0445 MB 1
BigFloat MPFI/basics/asin 1.08 k allocs: 0.0355 MB 1.08 k allocs: 0.0355 MB 1
BigFloat MPFI/basics/atan 2.14 k allocs: 0.0543 MB 2.14 k allocs: 0.0543 MB 1
BigFloat MPFI/basics/cos 2.9 k allocs: 0.0975 MB 2.9 k allocs: 0.0975 MB 1
BigFloat MPFI/basics/cosh 0.781 k allocs: 0.0322 MB 0.781 k allocs: 0.0322 MB 1
BigFloat MPFI/basics/exp 0.902 k allocs: 0.036 MB 0.902 k allocs: 0.036 MB 1
BigFloat MPFI/basics/inv 0.351 k allocs: 18.4 kB 0.351 k allocs: 18.4 kB 1
BigFloat MPFI/basics/log 0.407 k allocs: 20.6 kB 0.407 k allocs: 20.6 kB 1
BigFloat MPFI/basics/sin 3.18 k allocs: 0.106 MB 3.18 k allocs: 0.106 MB 1
BigFloat MPFI/basics/sinh 0.915 k allocs: 0.0369 MB 0.915 k allocs: 0.0369 MB 1
BigFloat MPFI/basics/sqrt 0.302 k allocs: 16.5 kB 0.302 k allocs: 16.5 kB 1
BigFloat MPFI/basics/tan 2.8 k allocs: 0.098 MB 2.8 k allocs: 0.098 MB 1
BigFloat MPFI/basics/tanh 0.908 k allocs: 0.0362 MB 0.908 k allocs: 0.0362 MB 1
BigFloat bareinterval/basics/* 0.262 k allocs: 26 kB 0.262 k allocs: 26 kB 1
BigFloat bareinterval/basics/+ 0.202 k allocs: 20.4 kB 0.202 k allocs: 20.4 kB 1
BigFloat bareinterval/basics/- 0.202 k allocs: 20.4 kB 0.202 k allocs: 20.4 kB 1
BigFloat bareinterval/basics// 0.1 k allocs: 10.8 kB 0.1 k allocs: 10.8 kB 1
BigFloat bareinterval/basics/abs 0.199 k allocs: 20.1 kB 0.199 k allocs: 20.1 kB 1
BigFloat bareinterval/basics/acos 1.33 k allocs: 0.0741 MB 1.33 k allocs: 0.0741 MB 1
BigFloat bareinterval/basics/asin 1.23 k allocs: 0.0647 MB 1.23 k allocs: 0.0647 MB 1
BigFloat bareinterval/basics/atan 2.04 k allocs: 0.08 MB 2.04 k allocs: 0.08 MB 1
BigFloat bareinterval/basics/cos 2.45 k allocs: 0.185 MB 2.45 k allocs: 0.185 MB 1
BigFloat bareinterval/basics/cosh 1.01 k allocs: 0.0671 MB 1.01 k allocs: 0.0671 MB 1
BigFloat bareinterval/basics/exp 0.802 k allocs: 0.0397 MB 0.802 k allocs: 0.0397 MB 1
BigFloat bareinterval/basics/inv 0.198 k allocs: 20 kB 0.198 k allocs: 20 kB 1
BigFloat bareinterval/basics/log 0.804 k allocs: 0.0694 MB 0.804 k allocs: 0.0694 MB 1
BigFloat bareinterval/basics/sin 2.9 k allocs: 0.197 MB 2.9 k allocs: 0.197 MB 1
BigFloat bareinterval/basics/sinh 0.815 k allocs: 0.0407 MB 0.815 k allocs: 0.0407 MB 1
BigFloat bareinterval/basics/sqrt 0.304 k allocs: 29.9 kB 0.304 k allocs: 29.9 kB 1
BigFloat bareinterval/basics/tan 1.67 k allocs: 0.124 MB 1.67 k allocs: 0.124 MB 1
BigFloat bareinterval/basics/tanh 0.808 k allocs: 0.0399 MB 0.808 k allocs: 0.0399 MB 1
BigFloat interval/basics/* 0.263 k allocs: 26.8 kB 0.263 k allocs: 26.8 kB 1
BigFloat interval/basics/+ 0.203 k allocs: 21.2 kB 0.203 k allocs: 21.2 kB 1
BigFloat interval/basics/- 0.203 k allocs: 21.2 kB 0.203 k allocs: 21.2 kB 1
BigFloat interval/basics// 0.101 k allocs: 11.6 kB 0.101 k allocs: 11.6 kB 1
BigFloat interval/basics/abs 0.2 k allocs: 20.9 kB 0.2 k allocs: 20.9 kB 1
BigFloat interval/basics/acos 1.77 k allocs: 0.115 MB 1.77 k allocs: 0.115 MB 1
BigFloat interval/basics/asin 1.53 k allocs: 0.0931 MB 1.53 k allocs: 0.0931 MB 1
BigFloat interval/basics/atan 2.04 k allocs: 0.0812 MB 2.04 k allocs: 0.0812 MB 0.999
BigFloat interval/basics/cos 2.45 k allocs: 0.186 MB 2.45 k allocs: 0.186 MB 1
BigFloat interval/basics/cosh 1.01 k allocs: 0.068 MB 1.01 k allocs: 0.068 MB 0.999
BigFloat interval/basics/exp 0.803 k allocs: 0.0407 MB 0.803 k allocs: 0.0409 MB 0.997
BigFloat interval/basics/inv 0.199 k allocs: 20.8 kB 0.199 k allocs: 20.8 kB 1
BigFloat interval/basics/log 1.1 k allocs: 0.0977 MB 1.1 k allocs: 0.0977 MB 1
BigFloat interval/basics/sin 2.9 k allocs: 0.198 MB 2.9 k allocs: 0.198 MB 1
BigFloat interval/basics/sinh 0.816 k allocs: 0.0419 MB 0.816 k allocs: 0.0417 MB 1
BigFloat interval/basics/sqrt 0.758 k allocs: 0.0715 MB 0.758 k allocs: 0.0715 MB 1
BigFloat interval/basics/tan 1.67 k allocs: 0.125 MB 1.67 k allocs: 0.125 MB 1
BigFloat interval/basics/tanh 0.809 k allocs: 0.0409 MB 0.809 k allocs: 0.0409 MB 0.999
bareinterval/basics/* 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/+ 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/- 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics// 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/abs 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/acos 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/asin 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/atan 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/cos 0.56 k allocs: 0.0423 MB 0.56 k allocs: 0.0423 MB 1
bareinterval/basics/cosh 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/exp 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/inv 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/log 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/sin 0.56 k allocs: 0.0423 MB 0.56 k allocs: 0.0423 MB 1
bareinterval/basics/sinh 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/sqrt 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/tan 0.29 k allocs: 23 kB 0.29 k allocs: 23 kB 1
bareinterval/basics/tanh 1.03 k allocs: 0.0359 MB 1.03 k allocs: 0.0359 MB 1
interval/basics/* 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/+ 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/- 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics// 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/abs 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/acos 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/asin 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/atan 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/cos 0.561 k allocs: 0.0431 MB 0.561 k allocs: 0.0431 MB 1
interval/basics/cosh 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/exp 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/inv 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/log 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/sin 0.561 k allocs: 0.0431 MB 0.561 k allocs: 0.0431 MB 1
interval/basics/sinh 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/sqrt 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/tan 0.291 k allocs: 24.4 kB 0.291 k allocs: 24.4 kB 1
interval/basics/tanh 1.03 k allocs: 0.0367 MB 1.03 k allocs: 0.0367 MB 1
time_to_load 0.145 k allocs: 11 kB 0.145 k allocs: 11 kB 1

@Kolaru
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Kolaru commented Mar 10, 2026

This looks good to me. I would just add

  • A comment to the new ^ function saying "See Improve power of complex intervals #745 for a description of the algorithm."
  • Tests checking that x^y is a subset of exp(y * log(x))
  • Tests checking that fastpown(x, n) is a subset of ^(x, interval(n))

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Codecov Report

❌ Patch coverage is 71.79487% with 11 lines in your changes missing coverage. Please review.
✅ Project coverage is 76.18%. Comparing base (b3f2756) to head (813d65a).
⚠️ Report is 3 commits behind head on master.

Files with missing lines Patch % Lines
src/intervals/arithmetic/power.jl 71.79% 11 Missing ⚠️
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@@            Coverage Diff             @@
##           master     #745      +/-   ##
==========================================
- Coverage   76.37%   76.18%   -0.19%     
==========================================
  Files          31       31              
  Lines        2988     3019      +31     
==========================================
+ Hits         2282     2300      +18     
- Misses        706      719      +13

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@OlivierHnt
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OlivierHnt commented Mar 18, 2026 •
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I could not simply have a test that x^y is a subset of exp(y*log(x)) since the new algorithm uses power by squaring which can accumulates a bit more rounding error than simply performing the exp/log version.

Instead, I check that we do have an improved real part:

x = interval(-0.5) + interval(im) * interval(-1e-14, 1e-14)
y = interval(1.5, 2.5)
res = x^y
ref = exp(y*log(x))
# tighter on the real part (right half-plane)
@test inf(real(ref)) < -0.3 < -1e-13 < inf(real(res))

Let me know if you have any ideas for tests before we merge the PR.

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