Average Error: 32.7 → 4.0
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{(\left((\left(y.im \cdot \frac{1}{2}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + 1)_*} \le 3.99075374980974 \cdot 10^{-310}:\\ \;\;\;\;\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{(\left((\left(y.im \cdot \frac{1}{2}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + 1)_*}\\ \mathbf{if}\;\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{(\left((\left(y.im \cdot \frac{1}{2}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + 1)_*} \le 0.9999405082819539:\\ \;\;\;\;\frac{\cos \left(\left(\sqrt[3]{(\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*} \cdot \sqrt[3]{(\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*}\right) \cdot \sqrt[3]{(\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*}\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left((\left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{\log_* (1 + (\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.im \cdot \frac{1}{2}\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right))_*) - \log \left(\sqrt{x.im^2 + x.re^2}^*\right) \cdot y.re}}\\ \end{array}\]

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 3 regimes

  2. if (/ (pow (hypot x.im x.re) y.re) (fma (fma (* y.im 1/2) (atan2 x.im x.re) 1) (* (atan2 x.im x.re) y.im) 1)) < 3.99075374980974e-310

    1. Initial program 31.7

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

    2. Applied simplify8.0

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

    3. Taylor expanded around 0 2.1

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

    4. Taylor expanded around 0 1.8

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

    5. Applied simplify0

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

    if 3.99075374980974e-310 < (/ (pow (hypot x.im x.re) y.re) (fma (fma (* y.im 1/2) (atan2 x.im x.re) 1) (* (atan2 x.im x.re) y.im) 1)) < 0.9999405082819539

    1. Initial program 31.2

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

    2. Applied simplify1.5

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

    4. Applied add-cube-cbrt1.5

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

    if 0.9999405082819539 < (/ (pow (hypot x.im x.re) y.re) (fma (fma (* y.im 1/2) (atan2 x.im x.re) 1) (* (atan2 x.im x.re) y.im) 1))

    1. Initial program 34.0

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

    2. Applied simplify12.0

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

    3. Taylor expanded around 0 11.8

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

    5. Applied add-exp-log11.9

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

    6. Applied pow-exp11.9

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

    7. Applied add-exp-log11.9

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

    8. Applied div-exp8.1

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

    9. Applied simplify8.0

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

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, real part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))