Average Error: 33.0 → 8.2
Time: 44.9s
Precision: 64
Internal Precision: 2176
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -4.653987828294037 \cdot 10^{+132}:\\ \;\;\;\;\frac{\sin \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2} \cdot {y.im}^{2}\right)\right)}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \mathbf{if}\;y.re \le 7.652756721494006 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sin \left(\left(\sqrt[3]{(\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*} \cdot \sqrt[3]{(\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*}\right) \cdot \sqrt[3]{(\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*}\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

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 < -4.653987828294037e+132

    1. Initial program 35.2

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

    2. Applied simplify12.3

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

    3. Taylor expanded around 0 3.5

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

    if -4.653987828294037e+132 < y.re < 7.652756721494006e+20

    1. Initial program 34.3

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

    2. Applied simplify6.9

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

    4. Applied add-cube-cbrt7.5

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

    if 7.652756721494006e+20 < y.re

    1. Initial program 20.5

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

Runtime

Time bar (total: 44.9s)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' +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)))))