Average Error: 30.8 → 0.4
Time: 57.9s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log base}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 30.8

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

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log base}}\]
  3. Using strategy rm

  4. Applied pow10.4

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

  5. Applied log-pow0.4

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

  6. Applied *-un-lft-identity0.4

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

  7. Applied times-frac0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

    \[\leadsto \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log base}\]

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))