Average Error: 30.3 → 0.4
Time: 18.5s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\frac{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot e^{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}\right)\right)}{\log base}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.3

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

  2. Initial simplification0.4

    \[\leadsto \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log base}\]
  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-exp-log0.4

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

  7. Final simplification0.4

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

Runtime

Time bar (total: 18.5s)Debug logProfile

herbie shell --seed 2018258 +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))))