Average Error: 31.9 → 17.4
Time: 13.8s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.6872226106627852 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(-\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 6.23122095692666968 \cdot 10^{141}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.6872226106627852 \cdot 10^{117}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(-\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\

\mathbf{elif}\;re \le 6.23122095692666968 \cdot 10^{141}:\\
\;\;\;\;\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)\\

\end{array}
double f(double re, double im) {
        double r49997 = re;
        double r49998 = r49997 * r49997;
        double r49999 = im;
        double r50000 = r49999 * r49999;
        double r50001 = r49998 + r50000;
        double r50002 = sqrt(r50001);
        double r50003 = log(r50002);
        double r50004 = 10.0;
        double r50005 = log(r50004);
        double r50006 = r50003 / r50005;
        return r50006;
}
↓
double f(double re, double im) {
        double r50007 = re;
        double r50008 = -4.687222610662785e+117;
        bool r50009 = r50007 <= r50008;
        double r50010 = 1.0;
        double r50011 = 10.0;
        double r50012 = log(r50011);
        double r50013 = sqrt(r50012);
        double r50014 = r50010 / r50013;
        double r50015 = -1.0;
        double r50016 = r50015 / r50007;
        double r50017 = log(r50016);
        double r50018 = r50010 / r50012;
        double r50019 = sqrt(r50018);
        double r50020 = r50017 * r50019;
        double r50021 = -r50020;
        double r50022 = r50014 * r50021;
        double r50023 = 6.23122095692667e+141;
        bool r50024 = r50007 <= r50023;
        double r50025 = cbrt(r50013);
        double r50026 = r50025 * r50025;
        double r50027 = r50010 / r50026;
        double r50028 = sqrt(r50027);
        double r50029 = sqrt(r50014);
        double r50030 = 3.0;
        double r50031 = pow(r50029, r50030);
        double r50032 = r50007 * r50007;
        double r50033 = im;
        double r50034 = r50033 * r50033;
        double r50035 = r50032 + r50034;
        double r50036 = sqrt(r50035);
        double r50037 = log(r50036);
        double r50038 = r50031 * r50037;
        double r50039 = r50010 / r50025;
        double r50040 = sqrt(r50039);
        double r50041 = r50038 * r50040;
        double r50042 = r50028 * r50041;
        double r50043 = log(r50007);
        double r50044 = r50043 * r50019;
        double r50045 = r50014 * r50044;
        double r50046 = r50024 ? r50042 : r50045;
        double r50047 = r50009 ? r50022 : r50046;
        return r50047;
}

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 re < -4.687222610662785e+117

    1. Initial program 54.9

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

    3. Applied add-sqr-sqrt54.9

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

    4. Applied pow154.9

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

    5. Applied log-pow54.9

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

    6. Applied times-frac54.9

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

    7. Taylor expanded around -inf 8.7

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

    8. Simplified8.7

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

    if -4.687222610662785e+117 < re < 6.23122095692667e+141

    1. Initial program 21.4

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

    3. Applied add-sqr-sqrt21.4

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

    4. Applied pow121.4

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

    5. Applied log-pow21.4

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

    6. Applied times-frac21.3

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

    8. Applied add-sqr-sqrt21.3

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

    9. Applied associate-*l*21.4

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

    11. Applied add-cube-cbrt21.3

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

    12. Applied *-un-lft-identity21.3

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

    13. Applied times-frac21.3

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

    14. Applied sqrt-prod21.3

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

    15. Applied associate-*l*21.3

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

    16. Simplified21.2

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

    if 6.23122095692667e+141 < re

    1. Initial program 60.0

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

    3. Applied add-sqr-sqrt60.0

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

    4. Applied pow160.0

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

    5. Applied log-pow60.0

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

    6. Applied times-frac60.0

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

    7. Taylor expanded around inf 7.8

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

    8. Simplified7.8

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.6872226106627852 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(-\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 6.23122095692666968 \cdot 10^{141}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))