Average Error: 32.8 → 5.8
Time: 1.2m
Precision: 64
Internal Precision: 2112
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin \left(\left(\sqrt[3]{(\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*} \cdot \sqrt[3]{(\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*}\right) \cdot \sqrt[3]{(\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*}\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}} \le 182.18596573637336:\\ \;\;\;\;\frac{\log_* (1 + (e^{\sin \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)} - 1)^*)}{\frac{e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}^{3}}\right)}^{3}}}}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^{2} \cdot {y.im}^{2}\right)\right)}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \end{array}\]

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 (/ (sin (* (* (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re))) (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re)))) (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re))))) (/ (exp (* y.im (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))) < 182.18596573637336

    1. Initial program 33.0

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

    2. Applied simplify4.5

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

    4. Applied pow-exp3.6

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

    6. Applied add-cbrt-cube3.6

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

    7. Applied simplify3.6

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

    9. Applied log1p-expm1-u3.6

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

    11. Applied add-cbrt-cube3.6

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

    12. Applied simplify3.6

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

    if 182.18596573637336 < (/ (sin (* (* (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re))) (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re)))) (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re))))) (/ (exp (* y.im (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re)))

    1. Initial program 30.0

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

    2. Applied simplify62.8

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

    3. Taylor expanded around 0 31.7

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018178 +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)))))