Average Error: 31.1 → 17.4
Time: 36.1s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.991328405351682 \cdot 10^{+100}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le 2.5049477191180382 \cdot 10^{+125}:\\ \;\;\;\;\frac{3}{\frac{1}{\frac{\log \left(\sqrt[3]{\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{im \cdot im + re \cdot re}\right|}\right)}{\log 10}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re \cdot \left(-3\right)}{\frac{\log 10}{\frac{-1}{3}}}\\ \end{array}\]

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 3 regimes

  2. if re < -3.991328405351682e+100

    1. Initial program 51.0

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

    2. Taylor expanded around -inf 9.3

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

    3. Simplified9.3

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

    if -3.991328405351682e+100 < re < 2.5049477191180382e+125

    1. Initial program 21.2

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

    3. Applied add-cube-cbrt21.2

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

    5. Applied pow321.2

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

    6. Applied log-pow21.2

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

    7. Applied associate-/l*21.2

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

    9. Applied add-cube-cbrt21.2

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

    10. Applied sqrt-prod21.2

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

    11. Simplified21.2

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

    13. Applied *-un-lft-identity21.2

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

    14. Applied associate-/l*21.2

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

    if 2.5049477191180382e+125 < re

    1. Initial program 55.2

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

    3. Applied add-cube-cbrt55.2

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

    5. Applied pow355.2

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

    6. Applied log-pow55.2

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

    7. Applied associate-/l*55.1

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

    8. Taylor expanded around inf 8.9

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

    9. Simplified8.7

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.991328405351682 \cdot 10^{+100}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le 2.5049477191180382 \cdot 10^{+125}:\\ \;\;\;\;\frac{3}{\frac{1}{\frac{\log \left(\sqrt[3]{\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{im \cdot im + re \cdot re}\right|}\right)}{\log 10}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re \cdot \left(-3\right)}{\frac{\log 10}{\frac{-1}{3}}}\\ \end{array}\]

Runtime

Time bar (total: 36.1s)Debug logProfile

herbie shell --seed 2018230 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))