Average Error: 31.2 → 17.9
Time: 29.0s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.739157930024531 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -2.6401746271055864 \cdot 10^{-185}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^{3} \cdot \left(\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\right)}\\ \mathbf{elif}\;re \le 1.6510598051339403 \cdot 10^{-137}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log im\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 8.578878492887041 \cdot 10^{+100}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^{3} \cdot \left(\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

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 4 regimes

  2. if re < -8.739157930024531e+82

    1. Initial program 48.3

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

    2. Initial simplification48.3

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

    4. Applied add-sqr-sqrt48.3

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

    5. Applied *-un-lft-identity48.3

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

    6. Applied times-frac48.3

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

    8. Applied div-inv48.2

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

    9. Applied associate-*r*48.2

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

    10. Taylor expanded around -inf 9.8

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

    11. Simplified9.8

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

    if -8.739157930024531e+82 < re < -2.6401746271055864e-185 or 1.6510598051339403e-137 < re < 8.578878492887041e+100

    1. Initial program 16.3

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

    2. Initial simplification16.3

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

    4. Applied add-sqr-sqrt16.3

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

    5. Applied *-un-lft-identity16.3

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

    6. Applied times-frac16.3

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

    8. Applied div-inv16.1

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

    9. Applied associate-*r*16.1

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

    11. Applied add-cbrt-cube16.1

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

    12. Applied add-cbrt-cube16.3

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

    13. Applied add-cbrt-cube16.3

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

    14. Applied cbrt-unprod16.3

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

    15. Applied cbrt-unprod16.3

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

    16. Simplified16.3

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

    if -2.6401746271055864e-185 < re < 1.6510598051339403e-137

    1. Initial program 30.1

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

    2. Initial simplification30.1

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

    4. Applied add-sqr-sqrt30.1

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

    5. Applied *-un-lft-identity30.1

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

    6. Applied times-frac30.1

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

    8. Applied div-inv30.0

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

    9. Applied associate-*r*30.0

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

    10. Taylor expanded around 0 33.7

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

    if 8.578878492887041e+100 < re

    1. Initial program 49.9

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

    2. Initial simplification49.9

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

    4. Applied add-sqr-sqrt49.9

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

    5. Applied *-un-lft-identity49.9

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

    6. Applied times-frac49.9

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

    8. Applied div-inv49.9

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

    9. Applied associate-*r*49.9

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

    11. Applied add-sqr-sqrt49.9

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

    12. Applied associate-*r*49.9

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

    13. Taylor expanded around inf 8.6

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

    14. Simplified8.6

      \[\leadsto \color{blue}{\frac{\log re}{\log 10}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.739157930024531 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -2.6401746271055864 \cdot 10^{-185}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^{3} \cdot \left(\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\right)}\\ \mathbf{elif}\;re \le 1.6510598051339403 \cdot 10^{-137}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log im\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 8.578878492887041 \cdot 10^{+100}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^{3} \cdot \left(\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Runtime

Time bar (total: 29.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.217.96.924.354.8%
herbie shell --seed 2018353 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))