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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 pow10.6

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

6. Applied log-pow0.6

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

7. Applied times-frac0.5

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

9. Applied div-inv0.4

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

11. Applied add-sqr-sqrt0.4

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

12. Applied associate-*l*0.5

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

14. Applied associate-*r*0.5

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

15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))

Details

Time bar (total: 17.8s)Debug log