Average Error: 30.9 → 17.5
Time: 51.8s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.115666643649848 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -7.2009850172815975 \cdot 10^{-180}:\\ \;\;\;\;\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.315376429877619 \cdot 10^{-295}:\\ \;\;\;\;\left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.1322433632218456 \cdot 10^{+124}:\\ \;\;\;\;\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

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 4 regimes

  2. if re < -6.115666643649848e+106

    1. Initial program 50.5

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

    2. Initial simplification50.5

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

    4. Applied add-sqr-sqrt50.5

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

    5. Applied pow1/250.5

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

    6. Applied log-pow50.5

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

    7. Applied times-frac50.5

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

    8. Taylor expanded around -inf 8.9

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

    if -6.115666643649848e+106 < re < -7.2009850172815975e-180 or 1.315376429877619e-295 < re < 1.1322433632218456e+124

    1. Initial program 19.5

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

    2. Initial simplification19.5

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

    4. Applied add-sqr-sqrt19.5

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

    5. Applied pow1/219.5

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

    6. Applied log-pow19.5

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

    7. Applied times-frac19.5

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

    9. Applied div-inv19.4

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

    10. Applied associate-*r*19.4

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

    12. Applied *-commutative19.4

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

    if -7.2009850172815975e-180 < re < 1.315376429877619e-295

    1. Initial program 29.1

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

    2. Initial simplification29.1

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

    4. Applied add-sqr-sqrt29.1

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

    5. Applied pow1/229.1

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

    6. Applied log-pow29.1

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

    7. Applied times-frac29.1

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

    9. Applied div-inv29.0

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

    10. Applied associate-*r*29.0

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

    12. Applied *-commutative29.0

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

    13. Taylor expanded around 0 32.9

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

    if 1.1322433632218456e+124 < re

    1. Initial program 53.8

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

    2. Initial simplification53.8

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

    4. Applied add-sqr-sqrt53.8

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

    5. Applied pow1/253.8

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

    6. Applied log-pow53.8

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

    7. Applied times-frac53.8

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

    9. Applied div-inv53.8

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

    10. Applied associate-*r*53.8

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

    12. Applied *-commutative53.8

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

    13. Taylor expanded around inf 8.6

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

    14. Simplified8.6

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.115666643649848 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -7.2009850172815975 \cdot 10^{-180}:\\ \;\;\;\;\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.315376429877619 \cdot 10^{-295}:\\ \;\;\;\;\left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.1322433632218456 \cdot 10^{+124}:\\ \;\;\;\;\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Runtime

Time bar (total: 51.8s)Debug logProfile

herbie shell --seed 2018242 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))