Average Error: 30.9 → 17.6
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.1099195122858345 \cdot 10^{+97}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{if}\;-re \le -6.776184305828227 \cdot 10^{-289}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right)}^{3}}}\\ \mathbf{if}\;-re \le 1.02074510006257 \cdot 10^{-285}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{if}\;-re \le 4.3212256399051484 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt[3]{\frac{\frac{\sqrt{\log 10} \cdot \log 10}{\log \left(im \cdot im + re \cdot re\right)}}{\log \left(im \cdot im + re \cdot re\right) \cdot \log \left(im \cdot im + re \cdot re\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if (- re) < -1.1099195122858345e+97

    1. Initial program 49.6

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

    2. Taylor expanded around inf 9.2

      \[\leadsto \frac{\log \color{blue}{re}}{\log 10}\]

    if -1.1099195122858345e+97 < (- re) < -6.776184305828227e-289

    1. Initial program 21.9

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

    3. Applied clear-num21.9

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

    5. Applied add-cbrt-cube21.9

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

    6. Applied add-cbrt-cube22.3

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

    7. Applied cbrt-undiv21.9

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

    8. Applied simplify21.9

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

    if -6.776184305828227e-289 < (- re) < 1.02074510006257e-285

    1. Initial program 29.1

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

    2. Taylor expanded around 0 34.9

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

    if 1.02074510006257e-285 < (- re) < 4.3212256399051484e+85

    1. Initial program 20.7

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

    3. Applied clear-num20.7

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

    5. Applied pow1/220.7

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

    6. Applied log-pow20.7

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

    7. Applied add-sqr-sqrt20.7

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

    8. Applied times-frac20.8

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

    9. Applied associate-/r*20.6

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

    10. Applied simplify20.6

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

    12. Applied add-cbrt-cube20.7

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

    13. Applied add-cbrt-cube20.7

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

    14. Applied cbrt-undiv20.7

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

    15. Applied simplify20.7

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

    if 4.3212256399051484e+85 < (- re)

    1. Initial program 47.1

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

    2. Taylor expanded around -inf 8.9

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

    3. Applied simplify8.9

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018166 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))