Average Error: 33.0 → 5.5
Time: 1.4m
Precision: 64
Internal Precision: 2368
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -2.473517843713001 \cdot 10^{-52} \lor \neg \left(y.re \le 4.228632272086652 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{\log_* (1 + (e^{\sin \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)} - 1)^*)}{e^{\log_* (1 + (\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.im \cdot \frac{1}{2}\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right))_*) - y.re \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right))_*\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \end{array}\]

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 2 regimes

  2. if y.re < -2.473517843713001e-52 or 4.228632272086652e-61 < y.re

    1. Initial program 32.1

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

    2. Applied simplify13.0

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

    3. Taylor expanded around 0 9.7

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

    5. Applied add-exp-log9.7

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

    6. Applied pow-exp9.7

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

    7. Applied add-exp-log9.7

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

    8. Applied div-exp6.1

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

    9. Applied simplify4.7

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

    11. Applied log1p-expm1-u4.7

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

    if -2.473517843713001e-52 < y.re < 4.228632272086652e-61

    1. Initial program 33.8

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

    2. Applied simplify5.9

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

    4. Applied add-cube-cbrt6.2

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

  4. Applied simplify5.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;y.re \le -2.473517843713001 \cdot 10^{-52} \lor \neg \left(y.re \le 4.228632272086652 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{\log_* (1 + (e^{\sin \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)} - 1)^*)}{e^{\log_* (1 + (\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.im \cdot \frac{1}{2}\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right))_*) - y.re \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right))_*\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \end{array}}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018193 +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)))))