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Derivation

1. Split input into 3 regimes

2. if re < -2.1579456935513238e+106

   1. Initial program 51.5

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

   2. Taylor expanded around -inf 8.1

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10}\]

   3. Simplified8.1

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)}}{\log 10}\]

   if -2.1579456935513238e+106 < re < 7.774313934961458e+95

   1. Initial program 21.1

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt21.0

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

   4. Applied sqrt-prod21.0

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

   5. Simplified21.0

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

   7. Applied add-exp-log21.0

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

   8. Applied add-exp-log21.1

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

   9. Applied prod-exp21.1

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

   11. Applied add-sqr-sqrt21.1

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

   12. Applied *-un-lft-identity21.1

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

   13. Applied times-frac21.0

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

   14. Simplified21.0

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

   if 7.774313934961458e+95 < re

   1. Initial program 50.4

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

   2. Taylor expanded around inf 10.1

      \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\frac{1}{re}\right)}{\log 10}}\]

   3. Simplified10.1

      \[\leadsto \color{blue}{\frac{\log re}{\log 10}}\]
3. Recombined 3 regimes into one program.

4. Final simplification17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.1579456935513238 \cdot 10^{+106}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le 7.774313934961458 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{im \cdot im + re \cdot re}\right|\right)}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))

Details

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