Average Error: 31.0 → 17.3
Time: 36.4s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2708662347081621 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 2.6681696550259567 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le 1.2173518665647989 \cdot 10^{-204}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 5.066783586607469 \cdot 10^{+57}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^{3} \cdot \left(\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log re \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \end{array}\]

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 < -1.2708662347081621e+124

    1. Initial program 54.1

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

    2. Initial simplification54.1

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

    4. Applied add-sqr-sqrt54.1

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

    5. Applied *-un-lft-identity54.1

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

    6. Applied times-frac54.1

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

    8. Applied div-inv54.0

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

    9. Taylor expanded around -inf 7.7

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

    10. Simplified7.7

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

    if -1.2708662347081621e+124 < re < 2.6681696550259567e-293

    1. Initial program 21.1

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

    2. Initial simplification21.1

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

    4. Applied add-sqr-sqrt21.1

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

    5. Applied *-un-lft-identity21.1

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

    6. Applied times-frac21.1

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

    8. Applied div-inv20.9

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

    10. Applied add-sqr-sqrt20.9

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

    11. Applied associate-*l*21.0

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

    if 2.6681696550259567e-293 < re < 1.2173518665647989e-204

    1. Initial program 31.3

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

    2. Initial simplification31.3

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

    4. Applied add-sqr-sqrt31.3

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

    5. Applied *-un-lft-identity31.3

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

    6. Applied times-frac31.3

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

    8. Applied div-inv31.2

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

    9. Taylor expanded around 0 35.8

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

    if 1.2173518665647989e-204 < re < 5.066783586607469e+57

    1. Initial program 17.9

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

    2. Initial simplification17.9

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

    4. Applied add-sqr-sqrt17.9

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

    5. Applied *-un-lft-identity17.9

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

    6. Applied times-frac17.8

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

    8. Applied div-inv17.7

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

    10. Applied add-cbrt-cube17.7

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

    11. Applied add-cbrt-cube17.9

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

    12. Applied cbrt-unprod17.9

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

    13. Applied add-cbrt-cube17.9

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

    14. Applied cbrt-unprod17.8

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

    15. Simplified17.8

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

    if 5.066783586607469e+57 < re

    1. Initial program 44.9

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

    2. Initial simplification44.9

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

    4. Applied add-sqr-sqrt44.9

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

    5. Applied *-un-lft-identity44.9

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

    6. Applied times-frac44.9

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

    8. Applied div-inv44.8

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

    9. Taylor expanded around inf 10.9

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2708662347081621 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 2.6681696550259567 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le 1.2173518665647989 \cdot 10^{-204}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 5.066783586607469 \cdot 10^{+57}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^{3} \cdot \left(\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log re \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \end{array}\]

Runtime

Time bar (total: 36.4s)Debug logProfile

herbie shell --seed 2018349 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))