Average Error: 37.8 → 11.5
Time: 5.6m
Precision: 64
Internal Precision: 1920
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.im \le -5.816106641287622:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.im\right)\right) \cdot \frac{{\left(-x.im\right)}^{y.re}}{{\left(\sqrt[3]{e^{y.im}} \cdot \sqrt[3]{e^{y.im}}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\sqrt[3]{e^{y.im}}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{if}\;x.im \le -1.9388125716537117 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(-x.im\right)\right)}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2} \cdot {y.im}^{2}\right)\right)}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}}}\\ \mathbf{if}\;x.im \le -2.246750664079573 \cdot 10^{-309}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(y.im \cdot \left(\sqrt[3]{\log \left(-x.im\right)} \cdot \sqrt[3]{\log \left(-x.im\right)}\right)\right) \cdot \sqrt[3]{\log \left(-x.im\right)}\right) \cdot \frac{{\left(-x.im\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}}}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 4 regimes

  2. if x.im < -5.816106641287622

    1. Initial program 40.3

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

    2. Applied simplify40.2

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

    3. Taylor expanded around -inf 22.5

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

    4. Applied simplify22.5

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

    5. Taylor expanded around -inf 0.6

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

    6. Applied simplify0.6

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

    8. Applied add-cube-cbrt0.6

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

    9. Applied unpow-prod-down0.6

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

    if -5.816106641287622 < x.im < -1.9388125716537117e-148

    1. Initial program 34.5

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

    2. Applied simplify36.8

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

    3. Taylor expanded around -inf 27.4

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

    4. Applied simplify27.4

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

    5. Taylor expanded around 0 18.3

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

    if -1.9388125716537117e-148 < x.im < -2.246750664079573e-309

    1. Initial program 46.6

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

    2. Applied simplify44.7

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

    3. Taylor expanded around -inf 28.0

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

    4. Applied simplify28.0

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

    5. Taylor expanded around -inf 3.0

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

    6. Applied simplify3.0

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

    8. Applied add-cube-cbrt3.2

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

    9. Applied associate-*r*3.2

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

    if -2.246750664079573e-309 < x.im

    1. Initial program 35.4

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

    2. Applied simplify34.8

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

    3. Taylor expanded around inf 16.7

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{x.im} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 5.6m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(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)))))