Average Error: 31.0 → 17.4
Time: 51.1s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.075911959668439 \cdot 10^{+117}:\\ \;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -1.9812566793834287 \cdot 10^{-186}:\\ \;\;\;\;\frac{\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 3.4378810267307503 \cdot 10^{-261}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log im\right) \cdot 2\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{elif}\;re \le 1.3755683907547103 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(im \cdot im + re \cdot re\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(2 \cdot \left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if re < -5.075911959668439e+117

    1. Initial program 52.7

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

    2. Initial simplification52.7

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

    4. Applied add-sqr-sqrt52.7

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

    5. Applied pow1/252.7

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

    6. Applied log-pow52.7

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

    7. Applied times-frac52.7

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

    9. Applied div-inv52.7

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

    10. Applied associate-*r*52.7

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

    11. Taylor expanded around -inf 8.9

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

    12. Simplified8.9

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

    if -5.075911959668439e+117 < re < -1.9812566793834287e-186

    1. Initial program 17.5

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

    2. Initial simplification17.5

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

    4. Applied add-sqr-sqrt17.5

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

    5. Applied pow1/217.5

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

    6. Applied log-pow17.5

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

    7. Applied times-frac17.5

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

    9. Applied div-inv17.4

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

    10. Applied associate-*r*17.4

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

    12. Applied associate-*l/17.5

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

    13. Applied associate-*l/17.5

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

    14. Simplified17.5

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

    if -1.9812566793834287e-186 < re < 3.4378810267307503e-261

    1. Initial program 31.2

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

    2. Initial simplification31.2

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

    4. Applied add-sqr-sqrt31.2

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

    5. Applied pow1/231.2

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

    6. Applied log-pow31.2

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

    7. Applied times-frac31.2

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

    9. Applied add-sqr-sqrt31.2

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

    10. Applied associate-*l*31.1

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

    11. Taylor expanded around 0 31.7

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

    if 3.4378810267307503e-261 < re < 1.3755683907547103e+136

    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 add-sqr-sqrt19.5

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

    10. 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 1.3755683907547103e+136 < re

    1. Initial program 57.4

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

    2. Initial simplification57.4

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

    4. Applied add-sqr-sqrt57.4

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

    5. Applied pow1/257.4

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

    6. Applied log-pow57.4

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

    7. Applied times-frac57.4

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

    9. Applied add-sqr-sqrt57.4

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

    10. Applied associate-*l*57.3

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

    11. Taylor expanded around inf 8.0

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

    12. Simplified8.0

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.075911959668439 \cdot 10^{+117}:\\ \;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -1.9812566793834287 \cdot 10^{-186}:\\ \;\;\;\;\frac{\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 3.4378810267307503 \cdot 10^{-261}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log im\right) \cdot 2\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{elif}\;re \le 1.3755683907547103 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(im \cdot im + re \cdot re\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(2 \cdot \left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}\]

Runtime

Time bar (total: 51.1s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))