Average Error: 32.2 → 6.8
Time: 44.5s
Precision: 64
Internal Precision: 1344
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -16459979.56349238:\\ \;\;\;\;\frac{\cos \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 2.2407333006580422 \cdot 10^{-07}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\cos \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)} \cdot \sqrt[3]{\left(\sqrt[3]{\cos \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)} \cdot \sqrt[3]{\cos \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)}\right) \cdot \sqrt[3]{\cos \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)}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\cos \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)} \cdot \sqrt[3]{\cos \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)}\right) \cdot \sqrt[3]{\cos \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{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \mathbf{if}\;y.re \le 7.954776863785181 \cdot 10^{+197}:\\ \;\;\;\;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 \cos \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)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \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}}}\\ \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

  1. Inputs

  2. Original Output:

    Herbie Output:

Derivation

  1. Split input into 3 regimes

  2. if y.re < -16459979.56349238 or 7.954776863785181e+197 < y.re

    1. Initial program 34.1

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

    2. Applied simplify12.2

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

    3. Taylor expanded around 0 5.6

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

    if -16459979.56349238 < y.re < 2.2407333006580422e-07

    1. Initial program 33.0

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

    2. Applied simplify6.0

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

    4. Applied pow-exp5.6

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

    6. Applied add-cube-cbrt5.6

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

    8. Applied add-cube-cbrt5.6

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

    10. Applied add-cube-cbrt5.6

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

    if 2.2407333006580422e-07 < y.re < 7.954776863785181e+197

    1. Initial program 19.7

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

Runtime

Time bar (total: 44.5s)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))