Average Error: 31.3 → 0.6
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log 10}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 31.3

    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log 10}}\]

  3. Final simplification0.6

    \[\leadsto \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log 10}\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))