math.log/2 on complex, imaginary part

Average Error: 32.2 → 0.4

Time: 5.6s

Precision: binary64

Math

\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

↓

\[\frac{-\tan^{-1}_* \frac{im}{re}}{2 \cdot \log \left(\sqrt[3]{\frac{1}{base}}\right) + \log \left(\sqrt[3]{\frac{1}{base}}\right)}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 32.2

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

2. Taylor expanded around 0 0.3

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

3. Simplified 0.3

   \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\frac{0 + \log base}{1}}}\]
4. Using strategy rm

5. Applied frac-2neg 0.3

   \[\leadsto \color{blue}{\frac{-\tan^{-1}_* \frac{im}{re}}{-\frac{0 + \log base}{1}}}\]

6. Simplified 0.3

   \[\leadsto \frac{-\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left(\frac{1}{base}\right)}}\]
7. Using strategy rm

8. Applied add-cube-cbrt 0.3

   \[\leadsto \frac{-\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left(\left(\sqrt[3]{\frac{1}{base}} \cdot \sqrt[3]{\frac{1}{base}}\right) \cdot \sqrt[3]{\frac{1}{base}}\right)}}\]

9. Applied log-prod 0.4

   \[\leadsto \frac{-\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left(\sqrt[3]{\frac{1}{base}} \cdot \sqrt[3]{\frac{1}{base}}\right) + \log \left(\sqrt[3]{\frac{1}{base}}\right)}}\]

10. Simplified 0.4

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

11. Final simplification 0.4

    \[\leadsto \frac{-\tan^{-1}_* \frac{im}{re}}{2 \cdot \log \left(\sqrt[3]{\frac{1}{base}}\right) + \log \left(\sqrt[3]{\frac{1}{base}}\right)}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))