Average Error: 31.1 → 17.7
Time: 55.7s
Precision: 64
Internal Precision: 320
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-1}{re} \le -4.469079448017009 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(\sqrt[3]{im \cdot im + re \cdot re} \cdot \sqrt[3]{im \cdot im + re \cdot re}\right) + \log \left(\sqrt[3]{im \cdot im + re \cdot re}\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;\frac{-1}{re} \le -2.076505029864966 \cdot 10^{+163}:\\ \;\;\;\;\frac{-\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\log re \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;\frac{-1}{re} \le -5.71089831282543 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(\sqrt[3]{im \cdot im + re \cdot re} \cdot \sqrt[3]{im \cdot im + re \cdot re}\right) + \log \left(\sqrt[3]{im \cdot im + re \cdot re}\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;\frac{-1}{re} \le 1.1454248398791993 \cdot 10^{-308}:\\ \;\;\;\;\frac{-\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\log re \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;\frac{-1}{re} \le 5.973837020518981 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(\sqrt[3]{im \cdot im + re \cdot re} \cdot \sqrt[3]{im \cdot im + re \cdot re}\right) + \log \left(\sqrt[3]{im \cdot im + re \cdot re}\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \end{array}\]

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 3 regimes

  2. if (/ -1 re) < -4.469079448017009e+208 or -2.076505029864966e+163 < (/ -1 re) < -5.71089831282543e-113 or 5.973837020518981e-57 < (/ -1 re)

    1. Initial program 21.3

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

    2. Initial simplification21.3

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

    4. Applied add-sqr-sqrt21.3

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

    5. Applied pow1/221.3

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

    6. Applied log-pow21.3

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

    7. Applied times-frac21.2

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

    9. Applied add-sqr-sqrt21.3

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

    10. Applied associate-*l*21.2

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

    12. Applied add-cube-cbrt21.2

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

    13. Applied log-prod21.2

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

    if -4.469079448017009e+208 < (/ -1 re) < -2.076505029864966e+163 or -5.71089831282543e-113 < (/ -1 re) < 1.1454248398791993e-308

    1. Initial program 46.9

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

    2. Initial simplification46.9

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

    4. Applied add-sqr-sqrt46.9

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

    5. Applied pow1/246.9

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

    6. Applied log-pow46.9

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

    7. Applied times-frac46.8

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

    8. Taylor expanded around inf 15.1

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

    9. Simplified15.1

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

    if 1.1454248398791993e-308 < (/ -1 re) < 5.973837020518981e-57

    1. Initial program 44.6

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

    2. Initial simplification44.6

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

    4. Applied add-sqr-sqrt44.6

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

    5. Applied pow1/244.6

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

    6. Applied log-pow44.6

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

    7. Applied times-frac44.6

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

    8. Taylor expanded around -inf 10.2

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1}{re} \le -4.469079448017009 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(\sqrt[3]{im \cdot im + re \cdot re} \cdot \sqrt[3]{im \cdot im + re \cdot re}\right) + \log \left(\sqrt[3]{im \cdot im + re \cdot re}\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;\frac{-1}{re} \le -2.076505029864966 \cdot 10^{+163}:\\ \;\;\;\;\frac{-\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\log re \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;\frac{-1}{re} \le -5.71089831282543 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(\sqrt[3]{im \cdot im + re \cdot re} \cdot \sqrt[3]{im \cdot im + re \cdot re}\right) + \log \left(\sqrt[3]{im \cdot im + re \cdot re}\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;\frac{-1}{re} \le 1.1454248398791993 \cdot 10^{-308}:\\ \;\;\;\;\frac{-\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\log re \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;\frac{-1}{re} \le 5.973837020518981 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(\sqrt[3]{im \cdot im + re \cdot re} \cdot \sqrt[3]{im \cdot im + re \cdot re}\right) + \log \left(\sqrt[3]{im \cdot im + re \cdot re}\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \end{array}\]

Runtime

Time bar (total: 55.7s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))