Average Error: 30.8 → 0.3

Time: 19.8s

Precision: 64

Internal Precision: 576

\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

↓

\[\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]

Error

The X axis uses an exponential scale — Bits error versus `re`

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Derivation

Initial program 30.8
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
Initial simplification0.3
\[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]
Using strategy rm
Applied *-un-lft-identity0.3
\[\leadsto \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\log base}\]
Applied associate-/l*0.5
\[\leadsto \color{blue}{\frac{1}{\frac{\log base}{\tan^{-1}_* \frac{im}{re}}}}\]
Taylor expanded around -inf 62.8
\[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log -1 - \log \left(\frac{-1}{base}\right)}}\]
Simplified0.3
\[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}}\]
Final simplification0.3
\[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]

Runtime

herbie shell --seed 2018251 
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0)) (+ (* (log base) (log base)) (* 0 0))))