Average Error: 30.8 → 17.5
Time: 30.7s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.8902885864967844 \cdot 10^{+70}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le -8.325291793016466 \cdot 10^{-294}:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)}{\log 10} \cdot 3\\ \mathbf{elif}\;re \le 5.980270498509465 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log 10}{\log im} \cdot \frac{1}{2}}\\ \mathbf{elif}\;re \le 5.575301037732192 \cdot 10^{+100}:\\ \;\;\;\;\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log 10} \cdot \left(\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log 10} \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log 10}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 5 regimes

  2. if re < -4.8902885864967844e+70

    1. Initial program 46.1

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

    2. Taylor expanded around -inf 10.3

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

    3. Simplified10.3

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

    if -4.8902885864967844e+70 < re < -8.325291793016466e-294

    1. Initial program 20.8

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

    3. Applied add-cube-cbrt20.8

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

    5. Applied *-un-lft-identity20.8

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

    6. Applied pow120.8

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

    7. Applied pow220.8

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

    8. Applied pow-prod-up20.8

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

    9. Applied log-pow20.8

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

    10. Applied times-frac20.8

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

    11. Simplified20.8

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

    if -8.325291793016466e-294 < re < 5.980270498509465e-183

    1. Initial program 31.2

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

    3. Applied pow1/231.2

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

    4. Applied log-pow31.2

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

    5. Applied associate-/l*31.2

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

    6. Taylor expanded around 0 33.0

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{1}{2} \cdot \frac{\log 10}{\log im}}}\]

    if 5.980270498509465e-183 < re < 5.575301037732192e+100

    1. Initial program 16.5

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

    3. Applied add-cbrt-cube17.1

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

    4. Applied add-cbrt-cube17.0

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

    5. Applied cbrt-undiv16.6

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

    6. Simplified16.6

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

    if 5.575301037732192e+100 < re

    1. Initial program 49.2

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

    2. Taylor expanded around inf 9.8

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.8902885864967844 \cdot 10^{+70}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le -8.325291793016466 \cdot 10^{-294}:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)}{\log 10} \cdot 3\\ \mathbf{elif}\;re \le 5.980270498509465 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log 10}{\log im} \cdot \frac{1}{2}}\\ \mathbf{elif}\;re \le 5.575301037732192 \cdot 10^{+100}:\\ \;\;\;\;\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log 10} \cdot \left(\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log 10} \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log 10}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))