Average Error: 31.3 → 17.2
Time: 29.3s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.6984489629736752 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -2.357019787417465 \cdot 10^{-280}:\\ \;\;\;\;\frac{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.4633824567839186 \cdot 10^{-164}:\\ \;\;\;\;\left(2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log im\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.3966114874101971 \cdot 10^{+116}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\sqrt[3]{\log 10}}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log re \cdot 2\right) \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \end{array}\]

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 < -2.6984489629736752e+143

    1. Initial program 58.7

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

    2. Initial simplification58.7

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

    4. Applied add-sqr-sqrt58.7

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

    5. Applied pow1/258.7

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

    6. Applied log-pow58.7

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

    7. Applied times-frac58.7

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

    8. Taylor expanded around -inf 7.0

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

    if -2.6984489629736752e+143 < re < -2.357019787417465e-280

    1. Initial program 19.6

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

    2. Initial simplification19.6

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

    4. Applied add-sqr-sqrt19.6

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

    5. Applied pow1/219.6

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

    6. Applied log-pow19.6

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

    7. Applied times-frac19.6

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

    9. Applied add-sqr-sqrt19.6

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

    10. Applied associate-*l*19.5

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

    12. Applied associate-*r/19.5

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

    13. Applied associate-*r/19.5

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

    if -2.357019787417465e-280 < re < 1.4633824567839186e-164

    1. Initial program 31.6

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

    2. Initial simplification31.6

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

    4. Applied add-sqr-sqrt31.6

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

    5. Applied pow1/231.6

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

    6. Applied log-pow31.6

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

    7. Applied times-frac31.5

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

    8. Taylor expanded around 0 32.9

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

    if 1.4633824567839186e-164 < re < 1.3966114874101971e+116

    1. Initial program 16.0

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

    2. Initial simplification16.0

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

    4. Applied add-sqr-sqrt16.0

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

    5. Applied pow1/216.0

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

    6. Applied log-pow16.0

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

    7. Applied times-frac16.0

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

    9. Applied add-sqr-sqrt16.0

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

    10. Applied associate-*l*15.9

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

    12. Applied associate-*r/15.9

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

    13. Applied associate-*r/15.9

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

    15. Applied add-cube-cbrt16.1

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

    16. Applied sqrt-prod16.1

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

    17. Applied times-frac15.9

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

    18. Simplified15.9

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

    if 1.3966114874101971e+116 < re

    1. Initial program 53.6

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

    2. Initial simplification53.6

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

    4. Applied add-sqr-sqrt53.6

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

    5. Applied pow1/253.6

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

    6. Applied log-pow53.6

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

    7. Applied times-frac53.6

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

    8. Taylor expanded around inf 8.0

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

    9. Simplified8.0

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\left(\log re \cdot 2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.6984489629736752 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -2.357019787417465 \cdot 10^{-280}:\\ \;\;\;\;\frac{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.4633824567839186 \cdot 10^{-164}:\\ \;\;\;\;\left(2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log im\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.3966114874101971 \cdot 10^{+116}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\sqrt[3]{\log 10}}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log re \cdot 2\right) \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \end{array}\]

Runtime

Time bar (total: 29.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.217.27.124.158.2%
herbie shell --seed 2018297 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))