Average Error: 33.0 → 21.7
Time: 1.3m
Precision: 64
Internal Precision: 2112
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - \log \left(\frac{-1}{x.re}\right) \cdot y.im\right) \le 4.4589135338309 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - \log \left(\frac{-1}{x.re}\right) \cdot y.im\right)\\ \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - \log \left(\frac{-1}{x.re}\right) \cdot y.im\right) \le 2.0210919941077406 \cdot 10^{+265}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{{\left(e^{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{\left(\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (- (* (atan2 x.im x.re) y.re) (* (log (/ -1 x.re)) y.im)))) < 4.4589135338309e-310

    1. Initial program 32.3

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

    2. Taylor expanded around -inf 16.2

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

    if 4.4589135338309e-310 < (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (- (* (atan2 x.im x.re) y.re) (* (log (/ -1 x.re)) y.im)))) < 2.0210919941077406e+265

    1. Initial program 0.5

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

    3. Applied add-cube-cbrt0.5

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

    4. Taylor expanded around inf 47.1

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

    5. Applied simplify0.6

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

    if 2.0210919941077406e+265 < (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (- (* (atan2 x.im x.re) y.re) (* (log (/ -1 x.re)) y.im))))

    1. Initial program 36.4

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

    2. Taylor expanded around inf 27.4

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018166 
(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)))))