Average Error: 33.1 → 6.7
Time: 38.1s
Precision: 64
Internal Precision: 1856
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \le 0.15654816171539115:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sin \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) \cdot \cos \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) + \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot \left(y.im + y.im\right)\right) \cdot \cos \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))) < 0.15654816171539115

    1. Initial program 1.7

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    if 0.15654816171539115 < (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re))))

    1. Initial program 62.7

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Initial simplification11.3

      \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]
    3. Using strategy rm

    4. Applied fma-udef11.3

      \[\leadsto \frac{\sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]

    5. Applied sin-sum11.3

      \[\leadsto \frac{\color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt11.3

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)}\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]

    8. Applied log-prod11.4

      \[\leadsto \frac{\sin \left(y.im \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) + \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right)}\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]

    9. Applied distribute-rgt-in11.4

      \[\leadsto \frac{\sin \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im + \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right)} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]

    10. Applied sin-sum11.4

      \[\leadsto \frac{\color{blue}{\left(\sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) \cdot \cos \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) + \cos \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) \cdot \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right)\right)} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]

    11. Simplified11.4

      \[\leadsto \frac{\left(\color{blue}{\cos \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) \cdot \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot \left(y.im + y.im\right)\right)} + \cos \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) \cdot \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \le 0.15654816171539115:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sin \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) \cdot \cos \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) + \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot \left(y.im + y.im\right)\right) \cdot \cos \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\\ \end{array}\]

Runtime

Time bar (total: 38.1s)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))