Average Error: 32.8 → 3.5
Time: 29.6s
Precision: 64
Internal Precision: 1344
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{(e^{\log_* (1 + \cos \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right))} - 1)^*}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \left(\sqrt[3]{y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)} \cdot \sqrt[3]{y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}\right) \cdot \sqrt[3]{y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}}}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 32.8

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

  2. Initial simplification3.4

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

  4. Applied add-cube-cbrt3.5

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

  6. Applied expm1-log1p-u3.5

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

  7. Final simplification3.5

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

Runtime

Time bar (total: 29.6s)Debug logProfile

herbie shell --seed 2018234 +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)))))