Average Error: 31.9 → 17.7
Time: 20.1s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.366950917668637798933900247544484583658 \cdot 10^{69}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.366950917668637798933900247544484583658 \cdot 10^{69}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)\right)\\

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

\end{array}
double f(double re, double im) {
        double r41102 = re;
        double r41103 = r41102 * r41102;
        double r41104 = im;
        double r41105 = r41104 * r41104;
        double r41106 = r41103 + r41105;
        double r41107 = sqrt(r41106);
        double r41108 = log(r41107);
        double r41109 = 10.0;
        double r41110 = log(r41109);
        double r41111 = r41108 / r41110;
        return r41111;
}
↓
double f(double re, double im) {
        double r41112 = re;
        double r41113 = -2.3669509176686378e+69;
        bool r41114 = r41112 <= r41113;
        double r41115 = 1.0;
        double r41116 = 10.0;
        double r41117 = log(r41116);
        double r41118 = sqrt(r41117);
        double r41119 = r41115 / r41118;
        double r41120 = -r41112;
        double r41121 = pow(r41120, r41119);
        double r41122 = log(r41121);
        double r41123 = r41119 * r41122;
        double r41124 = 7.747777771049568e+94;
        bool r41125 = r41112 <= r41124;
        double r41126 = r41119 / r41118;
        double r41127 = im;
        double r41128 = r41127 * r41127;
        double r41129 = r41112 * r41112;
        double r41130 = r41128 + r41129;
        double r41131 = sqrt(r41130);
        double r41132 = cbrt(r41131);
        double r41133 = r41132 * r41132;
        double r41134 = r41132 * r41133;
        double r41135 = log(r41134);
        double r41136 = r41126 * r41135;
        double r41137 = pow(r41112, r41119);
        double r41138 = log(r41137);
        double r41139 = r41138 * r41119;
        double r41140 = r41125 ? r41136 : r41139;
        double r41141 = r41114 ? r41123 : r41140;
        return r41141;
}

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 < -2.3669509176686378e+69

    1. Initial program 46.4

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

    3. Applied add-sqr-sqrt46.4

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

    4. Applied pow146.4

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

    5. Applied log-pow46.4

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

    6. Applied times-frac46.4

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

    8. Applied add-log-exp46.4

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

    9. Simplified46.4

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

    10. Taylor expanded around -inf 10.5

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

    11. Simplified10.5

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

    if -2.3669509176686378e+69 < re < 7.747777771049568e+94

    1. Initial program 22.5

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

    3. Applied add-sqr-sqrt22.5

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

    4. Applied pow122.5

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

    5. Applied log-pow22.5

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

    6. Applied times-frac22.4

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

    8. Applied add-log-exp22.4

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

    9. Simplified22.3

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

    11. Applied log-pow22.3

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

    12. Applied associate-*r*22.3

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

    13. Simplified22.3

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

    15. Applied add-cube-cbrt22.3

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

    if 7.747777771049568e+94 < re

    1. Initial program 49.1

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

    3. Applied add-sqr-sqrt49.1

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

    4. Applied pow149.1

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

    5. Applied log-pow49.1

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

    6. Applied times-frac49.1

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

    8. Applied add-log-exp49.1

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

    9. Simplified49.0

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

    10. Taylor expanded around inf 9.8

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.366950917668637798933900247544484583658 \cdot 10^{69}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))