Average Error: 32.3 → 17.5
Time: 23.8s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.983262521343274363476799981820042586015 \cdot 10^{136}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \end{array}\]
\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(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\

\mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\sqrt{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(\sqrt{\frac{1}{\log 10}}\right)}\right)\\

\end{array}
double f(double re, double im) {
        double r62318 = re;
        double r62319 = r62318 * r62318;
        double r62320 = im;
        double r62321 = r62320 * r62320;
        double r62322 = r62319 + r62321;
        double r62323 = sqrt(r62322);
        double r62324 = log(r62323);
        double r62325 = 10.0;
        double r62326 = log(r62325);
        double r62327 = r62324 / r62326;
        return r62327;
}
↓
double f(double re, double im) {
        double r62328 = re;
        double r62329 = -9.983262521343274e+136;
        bool r62330 = r62328 <= r62329;
        double r62331 = 1.0;
        double r62332 = 10.0;
        double r62333 = log(r62332);
        double r62334 = sqrt(r62333);
        double r62335 = r62331 / r62334;
        double r62336 = -1.0;
        double r62337 = r62336 / r62328;
        double r62338 = r62331 / r62333;
        double r62339 = sqrt(r62338);
        double r62340 = -r62339;
        double r62341 = pow(r62337, r62340);
        double r62342 = log(r62341);
        double r62343 = r62335 * r62342;
        double r62344 = 2.7153468834491098e+73;
        bool r62345 = r62328 <= r62344;
        double r62346 = r62328 * r62328;
        double r62347 = im;
        double r62348 = r62347 * r62347;
        double r62349 = r62346 + r62348;
        double r62350 = sqrt(r62349);
        double r62351 = pow(r62350, r62335);
        double r62352 = log(r62351);
        double r62353 = r62335 * r62352;
        double r62354 = pow(r62328, r62339);
        double r62355 = log(r62354);
        double r62356 = r62335 * r62355;
        double r62357 = r62345 ? r62353 : r62356;
        double r62358 = r62330 ? r62343 : r62357;
        return r62358;
}

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 div-inv59.8

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

    10. Applied add-log-exp59.8

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

    11. Simplified59.8

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

    12. Taylor expanded around -inf 8.1

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

    13. Simplified8.1

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

    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 div-inv22.0

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

    10. Applied add-log-exp22.0

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

    11. Simplified22.0

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

    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 div-inv47.0

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

    10. Applied add-log-exp47.0

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

    11. Simplified47.0

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

    12. Taylor expanded around inf 9.2

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

    13. Simplified9.2

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

  4. Final simplification17.5

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

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)))