Average Error: 31.4 → 19.5
Time: 55.0s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.8480056928576853 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -1.3605484803417976 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}}\\ \mathbf{elif}\;re \le 3.367065929769179 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \frac{\frac{1}{2}}{\log im}}\\ \mathbf{elif}\;re \le 2.0218161440293318 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

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 5 regimes

  2. if re < -1.8480056928576853e+75

    1. Initial program 45.6

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

    2. Initial simplification45.6

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

    4. Applied pow1/245.6

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

    5. Applied log-pow45.6

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

    6. Applied associate-/l*45.6

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

    7. Taylor expanded around -inf 9.7

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

    8. Simplified9.7

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

    if -1.8480056928576853e+75 < re < -1.3605484803417976e-159

    1. Initial program 16.7

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

    2. Initial simplification16.7

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

    4. Applied pow1/216.7

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

    5. Applied log-pow16.7

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

    6. Applied associate-/l*16.7

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

    8. Applied pow116.7

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

    9. Applied log-pow16.7

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

    10. Applied add-sqr-sqrt16.7

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

    11. Applied times-frac16.8

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

    12. Applied associate-/r*16.6

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

    14. Applied add-sqr-sqrt16.6

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

    15. Applied associate-/l*16.5

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

    16. Simplified16.5

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

    if -1.3605484803417976e-159 < re < 3.367065929769179e-95

    1. Initial program 28.6

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

    2. Initial simplification28.6

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

    4. Applied pow1/228.6

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

    5. Applied log-pow28.6

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

    6. Applied associate-/l*28.6

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

    8. Applied pow128.6

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

    9. Applied log-pow28.6

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

    10. Applied add-sqr-sqrt28.6

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

    11. Applied times-frac28.7

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

    12. Applied associate-/r*28.6

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

    13. Taylor expanded around 0 35.4

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

    14. Simplified35.4

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

    if 3.367065929769179e-95 < re < 2.0218161440293318e+124

    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 pow1/216.0

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

    5. Applied log-pow16.0

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

    6. Applied associate-/l*16.1

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

    8. Applied pow116.1

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

    9. Applied log-pow16.1

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

    10. Applied add-sqr-sqrt16.1

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

    11. Applied times-frac16.2

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

    12. Applied associate-/r*16.0

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

    14. Applied associate-/r/15.8

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

    if 2.0218161440293318e+124 < re

    1. Initial program 55.5

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

    2. Initial simplification55.5

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

    3. Taylor expanded around inf 8.5

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

  4. Final simplification19.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.8480056928576853 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -1.3605484803417976 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}}\\ \mathbf{elif}\;re \le 3.367065929769179 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \frac{\frac{1}{2}}{\log im}}\\ \mathbf{elif}\;re \le 2.0218161440293318 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Runtime

Time bar (total: 55.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.219.57.523.749.5%
herbie shell --seed 2018339 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))