Average Error: 31.3 → 18.4
Time: 50.9s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.9126520588893617 \cdot 10^{+89}:\\ \;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -1.0485843961505591 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 1.6776212883736178 \cdot 10^{-164}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 3384984853102974.0:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(im \cdot im + re \cdot re\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.9126520588893617 \cdot 10^{+89}:\\
\;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\

\mathbf{elif}\;re \le -1.0485843961505591 \cdot 10^{-243}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right)\\

\mathbf{elif}\;re \le 1.6776212883736178 \cdot 10^{-164}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\\

\mathbf{elif}\;re \le 3384984853102974.0:\\
\;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(im \cdot im + re \cdot re\right)}{\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 r2208134 = re;
        double r2208135 = r2208134 * r2208134;
        double r2208136 = im;
        double r2208137 = r2208136 * r2208136;
        double r2208138 = r2208135 + r2208137;
        double r2208139 = sqrt(r2208138);
        double r2208140 = log(r2208139);
        double r2208141 = 10.0;
        double r2208142 = log(r2208141);
        double r2208143 = r2208140 / r2208142;
        return r2208143;
}
↓
double f(double re, double im) {
        double r2208144 = re;
        double r2208145 = -1.9126520588893617e+89;
        bool r2208146 = r2208144 <= r2208145;
        double r2208147 = -1.0;
        double r2208148 = r2208147 / r2208144;
        double r2208149 = log(r2208148);
        double r2208150 = -2.0;
        double r2208151 = r2208149 * r2208150;
        double r2208152 = 0.5;
        double r2208153 = 10.0;
        double r2208154 = log(r2208153);
        double r2208155 = sqrt(r2208154);
        double r2208156 = r2208152 / r2208155;
        double r2208157 = r2208151 * r2208156;
        double r2208158 = 1.0;
        double r2208159 = r2208158 / r2208155;
        double r2208160 = r2208157 * r2208159;
        double r2208161 = -1.0485843961505591e-243;
        bool r2208162 = r2208144 <= r2208161;
        double r2208163 = im;
        double r2208164 = r2208163 * r2208163;
        double r2208165 = r2208144 * r2208144;
        double r2208166 = r2208164 + r2208165;
        double r2208167 = log(r2208166);
        double r2208168 = r2208167 * r2208156;
        double r2208169 = r2208159 * r2208168;
        double r2208170 = 1.6776212883736178e-164;
        bool r2208171 = r2208144 <= r2208170;
        double r2208172 = log(r2208163);
        double r2208173 = r2208158 / r2208154;
        double r2208174 = sqrt(r2208173);
        double r2208175 = r2208172 * r2208174;
        double r2208176 = r2208159 * r2208175;
        double r2208177 = 3384984853102974.0;
        bool r2208178 = r2208144 <= r2208177;
        double r2208179 = sqrt(r2208156);
        double r2208180 = r2208167 / r2208155;
        double r2208181 = r2208179 * r2208180;
        double r2208182 = r2208179 * r2208181;
        double r2208183 = log(r2208144);
        double r2208184 = r2208183 * r2208174;
        double r2208185 = r2208159 * r2208184;
        double r2208186 = r2208178 ? r2208182 : r2208185;
        double r2208187 = r2208171 ? r2208176 : r2208186;
        double r2208188 = r2208162 ? r2208169 : r2208187;
        double r2208189 = r2208146 ? r2208160 : r2208188;
        return r2208189;
}

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 5 regimes

  2. if re < -1.9126520588893617e+89

    1. Initial program 48.3

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

    3. Applied add-sqr-sqrt48.3

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

    4. Applied pow148.3

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

    5. Applied sqrt-pow148.3

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

    6. Applied log-pow48.3

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

    7. Applied times-frac48.3

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

    8. Simplified48.3

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

    10. Applied div-inv48.2

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

    11. Applied associate-*r*48.2

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

    12. Taylor expanded around -inf 9.6

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

    13. Simplified9.6

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

    if -1.9126520588893617e+89 < re < -1.0485843961505591e-243

    1. Initial program 20.5

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

    3. Applied add-sqr-sqrt20.5

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

    4. Applied pow120.5

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

    5. Applied sqrt-pow120.5

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

    6. Applied log-pow20.5

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

    7. Applied times-frac20.5

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

    8. Simplified20.5

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

    10. Applied div-inv20.3

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

    11. Applied associate-*r*20.3

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

    if -1.0485843961505591e-243 < re < 1.6776212883736178e-164

    1. Initial program 30.5

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

    3. Applied add-sqr-sqrt30.5

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

    4. Applied pow130.5

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

    5. Applied sqrt-pow130.5

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

    6. Applied log-pow30.5

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

    7. Applied times-frac30.5

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

    8. Simplified30.5

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

    10. Applied div-inv30.4

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

    11. Applied associate-*r*30.4

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

    12. Taylor expanded around 0 35.4

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

    if 1.6776212883736178e-164 < re < 3384984853102974.0

    1. Initial program 15.9

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

    3. Applied add-sqr-sqrt15.9

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

    4. Applied pow115.9

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

    5. Applied sqrt-pow115.9

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

    6. Applied log-pow15.9

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

    7. Applied times-frac15.9

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

    8. Simplified15.9

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

    10. Applied add-sqr-sqrt15.9

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

    11. Applied associate-*l*15.8

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

    if 3384984853102974.0 < re

    1. Initial program 40.3

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

    3. Applied add-sqr-sqrt40.3

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

    4. Applied pow140.3

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

    5. Applied sqrt-pow140.3

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

    6. Applied log-pow40.3

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

    7. Applied times-frac40.3

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

    8. Simplified40.3

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

    10. Applied div-inv40.2

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

    11. Applied associate-*r*40.2

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

    12. Taylor expanded around inf 12.0

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

    13. Simplified12.0

      \[\leadsto \color{blue}{\left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)} \cdot \frac{1}{\sqrt{\log 10}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.9126520588893617 \cdot 10^{+89}:\\ \;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -1.0485843961505591 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 1.6776212883736178 \cdot 10^{-164}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 3384984853102974.0:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(im \cdot im + re \cdot re\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))