Average Error: 31.0 → 17.3
Time: 20.9s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.9173717249563345 \cdot 10^{+124}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le 3.425958633542213 \cdot 10^{+105}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.9173717249563345 \cdot 10^{+124}:\\
\;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\

\mathbf{elif}\;re \le 3.425958633542213 \cdot 10^{+105}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\log re}{\log 10}\\

\end{array}
double f(double re, double im) {
        double r856359 = re;
        double r856360 = r856359 * r856359;
        double r856361 = im;
        double r856362 = r856361 * r856361;
        double r856363 = r856360 + r856362;
        double r856364 = sqrt(r856363);
        double r856365 = log(r856364);
        double r856366 = 10.0;
        double r856367 = log(r856366);
        double r856368 = r856365 / r856367;
        return r856368;
}
↓
double f(double re, double im) {
        double r856369 = re;
        double r856370 = -1.9173717249563345e+124;
        bool r856371 = r856369 <= r856370;
        double r856372 = -r856369;
        double r856373 = log(r856372);
        double r856374 = 10.0;
        double r856375 = log(r856374);
        double r856376 = r856373 / r856375;
        double r856377 = 3.425958633542213e+105;
        bool r856378 = r856369 <= r856377;
        double r856379 = 1.0;
        double r856380 = sqrt(r856375);
        double r856381 = r856379 / r856380;
        double r856382 = im;
        double r856383 = r856382 * r856382;
        double r856384 = r856369 * r856369;
        double r856385 = r856383 + r856384;
        double r856386 = cbrt(r856385);
        double r856387 = fabs(r856386);
        double r856388 = sqrt(r856386);
        double r856389 = r856387 * r856388;
        double r856390 = log(r856389);
        double r856391 = r856381 * r856390;
        double r856392 = r856381 * r856391;
        double r856393 = log(r856369);
        double r856394 = r856393 / r856375;
        double r856395 = r856378 ? r856392 : r856394;
        double r856396 = r856371 ? r856376 : r856395;
        return r856396;
}

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 < -1.9173717249563345e+124

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 8.1

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

    3. Simplified8.1

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

    if -1.9173717249563345e+124 < re < 3.425958633542213e+105

    1. Initial program 21.2

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

    3. Applied add-cube-cbrt21.2

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

    4. Applied sqrt-prod21.2

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

    5. Simplified21.2

      \[\leadsto \frac{\log \left(\color{blue}{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log 10}\]
    6. Using strategy rm

    7. Applied add-exp-log21.2

      \[\leadsto \frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\color{blue}{e^{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}}\right)}{\log 10}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt21.2

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

    10. Applied pow121.2

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

    11. Applied pow121.2

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

    12. Applied pow-prod-down21.2

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

    13. Applied log-pow21.2

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

    14. Applied times-frac21.2

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

    15. Simplified21.2

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\sqrt{\log 10}}}\]
    16. Using strategy rm

    17. Applied div-inv21.1

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

    if 3.425958633542213e+105 < re

    1. Initial program 50.9

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.3

      \[\leadsto \color{blue}{\frac{\log re}{\log 10}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.9173717249563345 \cdot 10^{+124}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le 3.425958633542213 \cdot 10^{+105}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))