Average Error: 33.0 → 23.6
Time: 42.4s
Precision: 64
Internal Precision: 2112
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -84314231.87036043:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;x.re \le 5.288631064984682 \cdot 10^{-262}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log x.re + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x.re < -84314231.87036043

    1. Initial program 40.9

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

    2. Taylor expanded around -inf 22.9

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

    3. Simplified22.9

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

    if -84314231.87036043 < x.re < 5.288631064984682e-262

    1. Initial program 23.2

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

    3. Applied add-cube-cbrt23.2

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

    if 5.288631064984682e-262 < x.re

    1. Initial program 34.8

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

    2. Taylor expanded around inf 24.1

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

  4. Final simplification23.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -84314231.87036043:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;x.re \le 5.288631064984682 \cdot 10^{-262}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \left(\sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}} \cdot \sqrt[3]{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log x.re + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Runtime

Time bar (total: 42.4s)Debug logProfile

herbie shell --seed 2018252 
(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)))))