Average Error: 31.1 → 0.6
Time: 49.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{\log \left(\sqrt{\sqrt{re^2 + im^2}^*} \cdot \sqrt{\sqrt{re^2 + im^2}^*}\right)}{\sqrt{\log 10}} \cdot \frac{1}{\sqrt{\log 10}}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 31.1

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

  2. Simplified0.6

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

  4. Applied add-sqr-sqrt0.6

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

  5. Applied *-un-lft-identity0.6

    \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re^2 + im^2}^*\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  6. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\sqrt{\log 10}}}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.6

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \frac{\log \color{blue}{\left(\sqrt{\sqrt{re^2 + im^2}^*} \cdot \sqrt{\sqrt{re^2 + im^2}^*}\right)}}{\sqrt{\log 10}}\]

  9. Final simplification0.6

    \[\leadsto \frac{\log \left(\sqrt{\sqrt{re^2 + im^2}^*} \cdot \sqrt{\sqrt{re^2 + im^2}^*}\right)}{\sqrt{\log 10}} \cdot \frac{1}{\sqrt{\log 10}}\]

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))