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Derivation

1. Initial program 30.8

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

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-frac0.6

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\sqrt{\log 10}}}\]
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-cube-cbrt0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)} \cdot \frac{1}{\sqrt{\log 10}}\right)\]
12. Using strategy rm

13. Applied add-cube-cbrt0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}}\right)} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\]

14. Final simplification0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}}\right)\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\]

Reproduce

herbie shell --seed 2019002 +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: 13.0s)Debug log