Average Error: 30.9 → 17.4
Time: 21.1s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.5084829120832133 \cdot 10^{+136}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right)}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}\\ \mathbf{elif}\;re \le 441347509.722609:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\log re \cdot 2}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.5084829120832133 \cdot 10^{+136}:\\
\;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right)}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}\\

\mathbf{elif}\;re \le 441347509.722609:\\
\;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\log re \cdot 2}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}\\

\end{array}
double f(double re, double im) {
        double r885960 = re;
        double r885961 = r885960 * r885960;
        double r885962 = im;
        double r885963 = r885962 * r885962;
        double r885964 = r885961 + r885963;
        double r885965 = sqrt(r885964);
        double r885966 = log(r885965);
        double r885967 = 10.0;
        double r885968 = log(r885967);
        double r885969 = r885966 / r885968;
        return r885969;
}
↓
double f(double re, double im) {
        double r885970 = re;
        double r885971 = -1.5084829120832133e+136;
        bool r885972 = r885970 <= r885971;
        double r885973 = 0.5;
        double r885974 = cbrt(r885973);
        double r885975 = 10.0;
        double r885976 = log(r885975);
        double r885977 = sqrt(r885976);
        double r885978 = sqrt(r885977);
        double r885979 = r885974 / r885978;
        double r885980 = -2.0;
        double r885981 = -1.0;
        double r885982 = r885981 / r885970;
        double r885983 = log(r885982);
        double r885984 = r885980 * r885983;
        double r885985 = r885984 / r885977;
        double r885986 = r885979 * r885985;
        double r885987 = r885974 * r885974;
        double r885988 = r885987 / r885978;
        double r885989 = r885986 * r885988;
        double r885990 = 441347509.722609;
        bool r885991 = r885970 <= r885990;
        double r885992 = r885973 / r885977;
        double r885993 = 1.0;
        double r885994 = r885993 / r885977;
        double r885995 = r885970 * r885970;
        double r885996 = im;
        double r885997 = r885996 * r885996;
        double r885998 = r885995 + r885997;
        double r885999 = log(r885998);
        double r886000 = r885994 * r885999;
        double r886001 = r885992 * r886000;
        double r886002 = log(r885970);
        double r886003 = 2.0;
        double r886004 = r886002 * r886003;
        double r886005 = r886004 / r885977;
        double r886006 = r885979 * r886005;
        double r886007 = r886006 * r885988;
        double r886008 = r885991 ? r886001 : r886007;
        double r886009 = r885972 ? r885989 : r886008;
        return r886009;
}

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 < -1.5084829120832133e+136

    1. Initial program 57.0

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

    3. Applied add-sqr-sqrt57.0

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

    4. Applied pow157.0

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

    5. Applied sqrt-pow157.0

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

    6. Applied log-pow57.0

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

    7. Applied times-frac56.9

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

    8. Simplified56.9

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

    10. Applied add-sqr-sqrt57.0

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

    11. Applied add-cube-cbrt56.9

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

    12. Applied times-frac56.9

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

    13. Applied associate-*l*56.9

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

    14. Taylor expanded around -inf 6.5

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

    15. Simplified6.5

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

    if -1.5084829120832133e+136 < re < 441347509.722609

    1. Initial program 21.6

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

    3. Applied add-sqr-sqrt21.6

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

    4. Applied pow121.6

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

    5. Applied sqrt-pow121.6

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

    6. Applied log-pow21.6

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

    7. Applied times-frac21.6

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

    8. Simplified21.6

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

    10. Applied div-inv21.5

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

    if 441347509.722609 < re

    1. Initial program 39.5

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

    3. Applied add-sqr-sqrt39.5

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

    4. Applied pow139.5

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

    5. Applied sqrt-pow139.5

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

    6. Applied log-pow39.5

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

    7. Applied times-frac39.4

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

    8. Simplified39.4

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

    10. Applied add-sqr-sqrt39.7

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

    11. Applied add-cube-cbrt39.4

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

    12. Applied times-frac39.4

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

    13. Applied associate-*l*39.4

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

    14. Taylor expanded around inf 13.3

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

    15. Simplified13.3

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.5084829120832133 \cdot 10^{+136}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right)}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}\\ \mathbf{elif}\;re \le 441347509.722609:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\log re \cdot 2}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))