Average Error: 31.7 → 18.6
Time: 45.1s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.1606882319204325 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -7.220768591101605 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) + \log \left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 5.283325772797005 \cdot 10^{-145}:\\ \;\;\;\;\left(2 \cdot \left({\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}} \cdot \left(\log im \cdot \sqrt{\frac{1}{2}}\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{elif}\;re \le 7.812026660088089 \cdot 10^{+57}:\\ \;\;\;\;\left(\left(\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if re < -4.1606882319204325e+81

    1. Initial program 48.4

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

    2. Initial simplification48.4

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

    4. Applied add-sqr-sqrt48.4

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

    5. Applied pow1/248.4

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

    6. Applied log-pow48.4

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

    7. Applied times-frac48.4

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

    8. Taylor expanded around -inf 9.4

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

    if -4.1606882319204325e+81 < re < -7.220768591101605e-173

    1. Initial program 17.4

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

    2. Initial simplification17.4

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

    4. Applied add-sqr-sqrt17.4

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

    5. Applied pow1/217.4

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

    6. Applied log-pow17.4

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

    7. Applied times-frac17.4

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

    9. Applied add-cube-cbrt17.4

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

    10. Applied log-prod17.4

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

    if -7.220768591101605e-173 < re < 5.283325772797005e-145

    1. Initial program 30.2

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

    2. Initial simplification30.2

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

    4. Applied add-sqr-sqrt30.2

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

    5. Applied pow1/230.2

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

    6. Applied log-pow30.2

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

    7. Applied times-frac30.2

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

    9. Applied add-sqr-sqrt30.2

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

    10. Applied associate-*l*30.2

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

    11. Taylor expanded around 0 34.7

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(2 \cdot \left(\left(\log im \cdot \sqrt{\frac{1}{2}}\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right)}\]

    if 5.283325772797005e-145 < re < 7.812026660088089e+57

    1. Initial program 16.9

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

    2. Initial simplification16.9

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

    4. Applied add-sqr-sqrt16.9

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

    5. Applied pow1/216.9

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

    6. Applied log-pow16.9

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

    7. Applied times-frac16.9

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

    9. Applied add-sqr-sqrt16.9

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

    10. Applied associate-*l*16.8

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

    12. Applied div-inv16.9

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

    13. Applied associate-*r*16.9

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

    if 7.812026660088089e+57 < re

    1. Initial program 44.6

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

    2. Initial simplification44.6

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

    3. Taylor expanded around inf 10.9

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

    4. Simplified10.9

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

  4. Final simplification18.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.1606882319204325 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -7.220768591101605 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) + \log \left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 5.283325772797005 \cdot 10^{-145}:\\ \;\;\;\;\left(2 \cdot \left({\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}} \cdot \left(\log im \cdot \sqrt{\frac{1}{2}}\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{elif}\;re \le 7.812026660088089 \cdot 10^{+57}:\\ \;\;\;\;\left(\left(\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Runtime

Time bar (total: 45.1s)Debug logProfile

herbie shell --seed 2018235 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))