Average Error: 31.5 → 0.0
Time: 1.4s
Precision: 64
Internal Precision: 128
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\log \left(\sqrt{re^2 + im^2}^*\right)\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 31.5

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))