Average Error: 30.9 → 21.2
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -2.575195947687393 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\\ \mathbf{elif}\;im \le -2.6606561087441924 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \left(2 \cdot \log re\right)}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\\ \mathbf{elif}\;im \le 5.6769871898150823 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;im \le 8.5731639036405 \cdot 10^{+120}:\\ \;\;\;\;\left(\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \left(\log im \cdot 2\right)}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 5 regimes

  2. if im < -2.575195947687393e-06

    1. Initial program 38.0

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

    3. Applied add-sqr-sqrt38.0

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

    4. Applied pow1/238.0

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

    5. Applied log-pow38.0

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

    6. Applied times-frac38.0

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

    8. Applied add-cube-cbrt38.3

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

    9. Applied sqrt-prod38.3

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

    10. Applied associate-/r*38.2

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

    11. Applied add-cube-cbrt38.0

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

    12. Applied associate-/l*38.0

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

    13. Applied frac-times38.0

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

    14. Simplified37.9

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

    15. Taylor expanded around -inf 14.2

      \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)} \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|}}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\]

    16. Simplified14.2

      \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)} \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|}}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\]

    if -2.575195947687393e-06 < im < -2.6606561087441924e-174

    1. Initial program 17.3

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

    3. Applied add-sqr-sqrt17.3

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

    4. Applied pow1/217.3

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

    5. Applied log-pow17.3

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

    6. Applied times-frac17.3

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

    8. Applied add-cube-cbrt17.8

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

    9. Applied sqrt-prod17.8

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

    10. Applied associate-/r*17.7

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

    11. Applied add-cube-cbrt17.2

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

    12. Applied associate-/l*17.2

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

    13. Applied frac-times17.2

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

    14. Simplified17.2

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

    15. Taylor expanded around 0 39.9

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

    if -2.6606561087441924e-174 < im < 5.6769871898150823e-300

    1. Initial program 30.2

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

    3. Applied add-sqr-sqrt30.2

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

    4. Applied pow1/230.2

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

    5. Applied log-pow30.2

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

    6. Applied times-frac30.2

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

    8. Applied div-inv30.1

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

    9. Applied associate-*r*30.1

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

    10. Taylor expanded around -inf 34.7

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

    11. Simplified34.7

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

    if 5.6769871898150823e-300 < im < 8.5731639036405e+120

    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-sqr-sqrt21.1

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

    4. Applied pow1/221.1

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

    5. Applied log-pow21.1

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

    6. Applied times-frac21.0

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

    8. Applied div-inv20.9

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

    9. Applied associate-*r*20.9

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

    11. Applied *-commutative20.9

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

    12. Applied associate-*l*20.9

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

    if 8.5731639036405e+120 < im

    1. Initial program 53.0

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

    3. Applied add-sqr-sqrt53.0

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

    4. Applied pow1/253.0

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

    5. Applied log-pow53.0

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

    6. Applied times-frac53.0

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

    8. Applied add-cube-cbrt53.1

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

    9. Applied sqrt-prod53.1

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

    10. Applied associate-/r*53.1

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

    11. Applied add-cube-cbrt53.0

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

    12. Applied associate-/l*53.0

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

    13. Applied frac-times53.0

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

    14. Simplified53.0

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

    15. Taylor expanded around inf 7.7

      \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|}}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\]

    16. Simplified7.7

      \[\leadsto \frac{\color{blue}{\left(\log im \cdot 2\right)} \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|}}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification21.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -2.575195947687393 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\\ \mathbf{elif}\;im \le -2.6606561087441924 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \left(2 \cdot \log re\right)}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\\ \mathbf{elif}\;im \le 5.6769871898150823 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;im \le 8.5731639036405 \cdot 10^{+120}:\\ \;\;\;\;\left(\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \left(\log im \cdot 2\right)}{\frac{\sqrt{\log 10}}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt{\sqrt[3]{\log 10}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019072 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))