Average Error: 30.4 → 0.4
Time: 34.5s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\frac{\log \left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{re^2 + im^2}^*}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{re^2 + im^2}^*}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{re^2 + im^2}^*}}\right)\right)}{\log base}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 30.4

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

  2. Applied simplify0.4

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

  4. Applied add-cube-cbrt0.4

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

  6. Applied add-cube-cbrt0.4

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

Runtime

Time bar (total: 34.5s)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' +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))))