Average Error: 32.6 → 6.1
Time: 29.2s
Precision: 64
Internal Precision: 2112
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \le 3.914042983393988 \cdot 10^{-43}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) + y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \sin \left((\left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) \cdot y.im + \left(\left(y.im + y.im\right) \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right))_*\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

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)))) < 3.914042983393988e-43

    1. Initial program 1.6

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

    if 3.914042983393988e-43 < (* (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 59.7

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

    2. Initial simplification10.9

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

    3. Taylor expanded around -inf 10.1

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

    5. Applied fma-udef10.1

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

    7. Applied add-cube-cbrt10.1

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

    8. Applied log-prod10.1

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

    9. Applied distribute-lft-in10.1

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

    11. Applied sin-sum10.1

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

    12. Simplified10.1

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \le 3.914042983393988 \cdot 10^{-43}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) + y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \sin \left((\left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) \cdot y.im + \left(\left(y.im + y.im\right) \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right))_*\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\\ \end{array}\]

Runtime

Time bar (total: 29.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes9.06.14.54.563.8%
herbie shell --seed 2018285 +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)))))