Average Error: 31.5 → 0.6
Time: 1.2m
Precision: 64
Internal Precision: 320
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(\sqrt{re^2 + im^2}^*\right)}}\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  2. Applied simplify0.6

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

  4. Applied clear-num0.6

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

  6. Applied pow10.6

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

  7. Applied log-pow0.6

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

  8. Applied add-sqr-sqrt0.6

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

  9. Applied times-frac0.8

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

  10. Applied associate-/r*0.6

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

  11. Applied simplify0.6

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))