Average Error: 30.9 → 17.4
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-1}{re} \le -6.144636439206768 \cdot 10^{+250}:\\ \;\;\;\;\left(2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log im\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{if}\;\frac{-1}{re} \le -1.401615215011707 \cdot 10^{-107}:\\ \;\;\;\;\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{if}\;\frac{-1}{re} \le 8.846763691197842 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(-2 \cdot \log re\right) \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{-\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{if}\;\frac{-1}{re} \le 1.580234197536095 \cdot 10^{-82}:\\ \;\;\;\;\left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{if}\;\frac{-1}{re} \le 9.435904736789276 \cdot 10^{+243} \lor \neg \left(\frac{-1}{re} \le 1.0906249298466369 \cdot 10^{+296}\right):\\ \;\;\;\;\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log im\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}\]

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 4 regimes

  2. if (/ -1 re) < -6.144636439206768e+250 or 9.435904736789276e+243 < (/ -1 re) < 1.0906249298466369e+296

    1. Initial program 31.4

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

    3. Applied add-sqr-sqrt31.4

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

    4. Applied pow1/231.4

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

    5. Applied log-pow31.4

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

    6. Applied times-frac31.4

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

    8. Applied add-sqr-sqrt31.5

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

    9. Applied associate-*l*31.4

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

    10. Taylor expanded around 0 34.6

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log im\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right)}\]

    if -6.144636439206768e+250 < (/ -1 re) < -1.401615215011707e-107 or 1.580234197536095e-82 < (/ -1 re) < 9.435904736789276e+243 or 1.0906249298466369e+296 < (/ -1 re)

    1. Initial program 19.5

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

    3. Applied add-sqr-sqrt19.5

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

    4. Applied pow1/219.5

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

    5. Applied log-pow19.5

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

    6. Applied times-frac19.5

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

    8. Applied add-sqr-sqrt19.5

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

    9. Applied associate-*l*19.4

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

    if -1.401615215011707e-107 < (/ -1 re) < 8.846763691197842e-306

    1. Initial program 52.0

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

    3. Applied add-sqr-sqrt52.0

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

    4. Applied pow1/252.0

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

    5. Applied log-pow52.0

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

    6. Applied times-frac52.0

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

    7. Taylor expanded around inf 9.5

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

    8. Applied simplify9.5

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

    if 8.846763691197842e-306 < (/ -1 re) < 1.580234197536095e-82

    1. Initial program 48.0

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

    3. Applied add-sqr-sqrt48.0

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

    4. Applied pow1/248.0

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

    5. Applied log-pow48.0

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

    6. Applied times-frac48.0

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

    7. Taylor expanded around -inf 10.0

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify17.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{-1}{re} \le -6.144636439206768 \cdot 10^{+250}:\\ \;\;\;\;\left(2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log im\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{if}\;\frac{-1}{re} \le -1.401615215011707 \cdot 10^{-107}:\\ \;\;\;\;\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{if}\;\frac{-1}{re} \le 8.846763691197842 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(-2 \cdot \log re\right) \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{-\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{if}\;\frac{-1}{re} \le 1.580234197536095 \cdot 10^{-82}:\\ \;\;\;\;\left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{if}\;\frac{-1}{re} \le 9.435904736789276 \cdot 10^{+243} \lor \neg \left(\frac{-1}{re} \le 1.0906249298466369 \cdot 10^{+296}\right):\\ \;\;\;\;\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log im\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018207 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))