Average Error: 31.0 → 17.5
Time: 35.9s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.747857693765517 \cdot 10^{+140}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{-re}\right)}}\\ \mathbf{elif}\;re \le -9.94316899238669 \cdot 10^{-208}:\\ \;\;\;\;\frac{3}{\frac{e^{\log \left(\frac{\log 10}{\frac{1}{3}}\right)}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;re \le 4.822589790424056 \cdot 10^{-238}:\\ \;\;\;\;\frac{3}{\frac{\frac{\log 10}{\frac{1}{3}}}{\log im}}\\ \mathbf{elif}\;re \le 6.438474756079632 \cdot 10^{+84}:\\ \;\;\;\;\frac{3}{\frac{e^{\log \left(\frac{\log 10}{\frac{1}{3}}\right)}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{re}\right)}}\\ \end{array}\]

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 4 regimes

  2. if re < -5.747857693765517e+140

    1. Initial program 58.6

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

    3. Applied add-cube-cbrt58.6

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

    5. Applied pow158.6

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

    6. Applied pow158.6

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

    7. Applied pow-plus58.6

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

    8. Applied pow-prod-up58.6

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

    9. Applied log-pow58.6

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

    10. Applied associate-/l*58.6

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

    11. Simplified58.6

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

    12. Taylor expanded around -inf 7.9

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

    13. Simplified7.9

      \[\leadsto \frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\color{blue}{-re}}\right)}}\]

    if -5.747857693765517e+140 < re < -9.94316899238669e-208 or 4.822589790424056e-238 < re < 6.438474756079632e+84

    1. Initial program 18.8

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

    3. Applied add-cube-cbrt18.8

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

    5. Applied pow118.8

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

    6. Applied pow118.8

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

    7. Applied pow-plus18.8

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

    8. Applied pow-prod-up18.8

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

    9. Applied log-pow18.8

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

    10. Applied associate-/l*18.8

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

    11. Simplified18.8

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

    13. Applied pow1/319.0

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

    14. Applied log-pow19.0

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

    15. Applied associate-/r*18.8

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

    17. Applied add-exp-log18.6

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

    if -9.94316899238669e-208 < re < 4.822589790424056e-238

    1. Initial program 30.9

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

    3. Applied add-cube-cbrt30.9

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

    5. Applied pow130.9

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

    6. Applied pow130.9

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

    7. Applied pow-plus30.9

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

    8. Applied pow-prod-up30.9

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

    9. Applied log-pow30.9

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

    10. Applied associate-/l*30.9

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

    11. Simplified30.9

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

    13. Applied pow1/331.0

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

    14. Applied log-pow31.0

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

    15. Applied associate-/r*30.9

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

    16. Taylor expanded around 0 32.5

      \[\leadsto \frac{3}{\frac{\frac{\log 10}{\frac{1}{3}}}{\log \color{blue}{im}}}\]

    if 6.438474756079632e+84 < re

    1. Initial program 47.0

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

    3. Applied add-cube-cbrt47.0

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

    5. Applied pow147.0

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

    6. Applied pow147.0

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

    7. Applied pow-plus47.0

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

    8. Applied pow-prod-up47.0

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

    9. Applied log-pow47.0

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

    10. Applied associate-/l*47.0

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

    11. Simplified47.0

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

    12. Taylor expanded around inf 10.4

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.747857693765517 \cdot 10^{+140}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{-re}\right)}}\\ \mathbf{elif}\;re \le -9.94316899238669 \cdot 10^{-208}:\\ \;\;\;\;\frac{3}{\frac{e^{\log \left(\frac{\log 10}{\frac{1}{3}}\right)}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;re \le 4.822589790424056 \cdot 10^{-238}:\\ \;\;\;\;\frac{3}{\frac{\frac{\log 10}{\frac{1}{3}}}{\log im}}\\ \mathbf{elif}\;re \le 6.438474756079632 \cdot 10^{+84}:\\ \;\;\;\;\frac{3}{\frac{e^{\log \left(\frac{\log 10}{\frac{1}{3}}\right)}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{re}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 35.9s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))