Average Error: 32.8 → 7.3
Time: 39.7s
Precision: 64
Internal Precision: 2112
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -0.19398597092510558:\\ \;\;\;\;\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}{\frac{\left(1 + \left({y.im}^{2} \cdot {\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}\right) \cdot \frac{1}{2}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \le 1.0525083366271 \cdot 10^{+105}:\\ \;\;\;\;\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}{\frac{\left(\sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\right) \cdot \sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.im\right)\\ \end{array}\]

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 3 regimes

  2. if y.re < -0.19398597092510558

    1. Initial program 36.2

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

    2. Initial simplification12.0

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

    3. Taylor expanded around 0 5.0

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

    if -0.19398597092510558 < y.re < 1.0525083366271e+105

    1. Initial program 33.4

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

    2. Initial simplification7.3

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

    4. Applied add-cube-cbrt7.3

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

    if 1.0525083366271e+105 < y.re

    1. Initial program 15.5

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -0.19398597092510558:\\ \;\;\;\;\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}{\frac{\left(1 + \left({y.im}^{2} \cdot {\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}\right) \cdot \frac{1}{2}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \le 1.0525083366271 \cdot 10^{+105}:\\ \;\;\;\;\frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}{\frac{\left(\sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\right) \cdot \sqrt[3]{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.im\right)\\ \end{array}\]

Runtime

Time bar (total: 39.7s)Debug logProfile

herbie shell --seed 2018254 +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)))))