Average Error: 31.1 → 0.4
Time: 52.0s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\frac{\log \left((e^{\log_* (1 + \sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*})} - 1)^*\right) + \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log base}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 31.1

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

  2. Simplified0.4

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

  4. Applied 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. Applied log-prod0.4

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

  7. Applied expm1-log1p-u0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019088 +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))))