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Derivation

1. Split input into 3 regimes

2. if re < -1.98901646800801e+52

   1. Initial program 42.7

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

   3. Applied add-cube-cbrt42.7

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

   5. Applied add-exp-log42.7

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

   6. Taylor expanded around -inf 11.2

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

   7. Simplified11.2

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

   if -1.98901646800801e+52 < re < 2.7167323166120062e+127

   1. Initial program 21.4

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

   3. Applied pow1/221.4

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

   4. Applied log-pow21.4

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

   5. Applied associate-/l*21.4

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

   7. Applied *-un-lft-identity21.4

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

   8. Applied add-sqr-sqrt21.4

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

   9. Applied times-frac21.5

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

   10. Applied associate-/r*21.3

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

   if 2.7167323166120062e+127 < re

   1. Initial program 55.2

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

   3. Applied add-cube-cbrt55.2

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

   4. Taylor expanded around inf 7.6

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

4. Final simplification17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.98901646800801 \cdot 10^{+52}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le 2.7167323166120062 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(im \cdot im + re \cdot re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))

Details

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