Average Error: 31.7 → 17.8
Time: 11.1min
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.7849277489671732 \cdot 10^{122}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{-re}\right)}}\\ \mathbf{elif}\;re \le -1.82707962304541887 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;re \le 4.10205923095853832 \cdot 10^{-190}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{im}\right)}}\\ \mathbf{elif}\;re \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 5 regimes

  2. if re < -3.7849277489671732e122

    1. Initial program 55.4

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

    3. Applied add-cube-cbrt55.4

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

    5. Applied pow355.4

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

    6. Applied log-pow55.4

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

    7. Applied associate-/l*55.4

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

    8. Taylor expanded around -inf 8.2

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

    9. Simplified8.2

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

    if -3.7849277489671732e122 < re < -1.82707962304541887e-251

    1. Initial program 19.4

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

    3. Applied add-cube-cbrt19.4

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

    5. Applied pow319.4

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

    6. Applied log-pow19.4

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

    7. Applied associate-/l*19.4

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

    9. Applied pow1/319.6

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

    10. Applied log-pow19.6

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

    11. Applied add-sqr-sqrt19.6

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

    12. Applied times-frac20.0

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

    13. Applied associate-/r*19.9

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

    14. Simplified19.4

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

    if -1.82707962304541887e-251 < re < 4.10205923095853832e-190

    1. Initial program 30.6

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

    3. Applied add-cube-cbrt30.6

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

    5. Applied pow330.6

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

    6. Applied log-pow30.6

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

    7. Applied associate-/l*30.6

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

    8. Taylor expanded around 0 33.5

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

    if 4.10205923095853832e-190 < re < 3.46617861303372194e36

    1. Initial program 18.6

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

    3. Applied add-cube-cbrt18.6

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

    5. Applied pow318.6

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

    6. Applied log-pow18.6

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

    7. Applied associate-/l*18.6

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

    9. Applied add-cube-cbrt18.6

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

    if 3.46617861303372194e36 < re

    1. Initial program 44.1

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

    2. Taylor expanded around inf 11.3

      \[\leadsto \frac{\log \color{blue}{re}}{\log 10}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.7849277489671732 \cdot 10^{122}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{-re}\right)}}\\ \mathbf{elif}\;re \le -1.82707962304541887 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;re \le 4.10205923095853832 \cdot 10^{-190}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{im}\right)}}\\ \mathbf{elif}\;re \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))