Average Error: 32.3 → 17.5
Time: 10.7s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.983262521343274363476799981820042586015 \cdot 10^{136}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.983262521343274363476799981820042586015 \cdot 10^{136}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

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

\end{array}
double f(double re, double im) {
        double r40140 = re;
        double r40141 = r40140 * r40140;
        double r40142 = im;
        double r40143 = r40142 * r40142;
        double r40144 = r40141 + r40143;
        double r40145 = sqrt(r40144);
        double r40146 = log(r40145);
        double r40147 = 10.0;
        double r40148 = log(r40147);
        double r40149 = r40146 / r40148;
        return r40149;
}
↓
double f(double re, double im) {
        double r40150 = re;
        double r40151 = -9.983262521343274e+136;
        bool r40152 = r40150 <= r40151;
        double r40153 = 1.0;
        double r40154 = 10.0;
        double r40155 = log(r40154);
        double r40156 = sqrt(r40155);
        double r40157 = r40153 / r40156;
        double r40158 = -1.0;
        double r40159 = r40158 * r40150;
        double r40160 = pow(r40159, r40157);
        double r40161 = log(r40160);
        double r40162 = r40157 * r40161;
        double r40163 = 2.7153468834491098e+73;
        bool r40164 = r40150 <= r40163;
        double r40165 = r40150 * r40150;
        double r40166 = im;
        double r40167 = r40166 * r40166;
        double r40168 = r40165 + r40167;
        double r40169 = cbrt(r40168);
        double r40170 = fabs(r40169);
        double r40171 = sqrt(r40169);
        double r40172 = r40170 * r40171;
        double r40173 = pow(r40172, r40157);
        double r40174 = log(r40173);
        double r40175 = r40157 * r40174;
        double r40176 = pow(r40150, r40157);
        double r40177 = log(r40176);
        double r40178 = r40157 * r40177;
        double r40179 = r40164 ? r40175 : r40178;
        double r40180 = r40152 ? r40162 : r40179;
        return r40180;
}

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 < -9.983262521343274e+136

    1. Initial program 59.8

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

    3. Applied add-sqr-sqrt59.8

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

    4. Applied pow159.8

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

    5. Applied log-pow59.8

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

    6. Applied times-frac59.8

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

    8. Applied add-log-exp59.8

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

    9. Simplified59.8

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

    10. Taylor expanded around -inf 8.1

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

    if -9.983262521343274e+136 < re < 2.7153468834491098e+73

    1. Initial program 22.2

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

    3. Applied add-sqr-sqrt22.2

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

    4. Applied pow122.2

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

    5. Applied log-pow22.2

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

    6. Applied times-frac22.1

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

    8. Applied add-log-exp22.1

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

    9. Simplified22.0

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

    11. Applied add-cube-cbrt22.0

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

    12. Applied sqrt-prod22.0

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

    13. Simplified22.0

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

    if 2.7153468834491098e+73 < re

    1. Initial program 47.1

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

    3. Applied add-sqr-sqrt47.1

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

    4. Applied pow147.1

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

    5. Applied log-pow47.1

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

    6. Applied times-frac47.1

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

    8. Applied add-log-exp47.1

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

    9. Simplified47.0

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

    10. Taylor expanded around inf 9.2

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.983262521343274363476799981820042586015 \cdot 10^{136}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))