Average Error: 32.5 → 17.8
Time: 7.7s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.859463483918923369547260927451545843966 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -1.167098834567443703136578031923709302405 \cdot 10^{-290}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}^{3}}\right)\\ \mathbf{elif}\;re \le 3.812412734570747037549119511317842097867 \cdot 10^{-267}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 1520535341655445535810607142928115832979000:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.859463483918923369547260927451545843966 \cdot 10^{84}:\\
\;\;\;\;\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 -1.167098834567443703136578031923709302405 \cdot 10^{-290}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}^{3}}\right)\\

\mathbf{elif}\;re \le 3.812412734570747037549119511317842097867 \cdot 10^{-267}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{elif}\;re \le 1520535341655445535810607142928115832979000:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}^{3}}\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 r106539 = re;
        double r106540 = r106539 * r106539;
        double r106541 = im;
        double r106542 = r106541 * r106541;
        double r106543 = r106540 + r106542;
        double r106544 = sqrt(r106543);
        double r106545 = log(r106544);
        double r106546 = 10.0;
        double r106547 = log(r106546);
        double r106548 = r106545 / r106547;
        return r106548;
}
↓
double f(double re, double im) {
        double r106549 = re;
        double r106550 = -3.8594634839189234e+84;
        bool r106551 = r106549 <= r106550;
        double r106552 = 1.0;
        double r106553 = 10.0;
        double r106554 = log(r106553);
        double r106555 = sqrt(r106554);
        double r106556 = r106552 / r106555;
        double r106557 = -1.0;
        double r106558 = r106557 / r106549;
        double r106559 = r106552 / r106554;
        double r106560 = sqrt(r106559);
        double r106561 = -r106560;
        double r106562 = pow(r106558, r106561);
        double r106563 = log(r106562);
        double r106564 = r106556 * r106563;
        double r106565 = -1.1670988345674437e-290;
        bool r106566 = r106549 <= r106565;
        double r106567 = r106549 * r106549;
        double r106568 = im;
        double r106569 = r106568 * r106568;
        double r106570 = r106567 + r106569;
        double r106571 = sqrt(r106570);
        double r106572 = pow(r106571, r106556);
        double r106573 = 3.0;
        double r106574 = pow(r106572, r106573);
        double r106575 = cbrt(r106574);
        double r106576 = log(r106575);
        double r106577 = r106556 * r106576;
        double r106578 = 3.812412734570747e-267;
        bool r106579 = r106549 <= r106578;
        double r106580 = pow(r106568, r106556);
        double r106581 = log(r106580);
        double r106582 = r106556 * r106581;
        double r106583 = 1.5205353416554455e+42;
        bool r106584 = r106549 <= r106583;
        double r106585 = pow(r106549, r106556);
        double r106586 = log(r106585);
        double r106587 = r106556 * r106586;
        double r106588 = r106584 ? r106577 : r106587;
        double r106589 = r106579 ? r106582 : r106588;
        double r106590 = r106566 ? r106577 : r106589;
        double r106591 = r106551 ? r106564 : r106590;
        return r106591;
}

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 4 regimes

  2. if re < -3.8594634839189234e+84

    1. Initial program 50.0

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

    3. Applied add-sqr-sqrt50.0

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

    4. Applied pow150.0

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

    5. Applied log-pow50.0

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

    6. Applied times-frac50.0

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

    8. Applied add-log-exp50.0

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

    9. Simplified49.9

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

    10. Taylor expanded around -inf 9.6

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

    11. Simplified9.5

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

    if -3.8594634839189234e+84 < re < -1.1670988345674437e-290 or 3.812412734570747e-267 < re < 1.5205353416554455e+42

    1. Initial program 21.9

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

    3. Applied add-sqr-sqrt21.9

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

    4. Applied pow121.9

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

    5. Applied log-pow21.9

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

    6. Applied times-frac21.8

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

    8. Applied add-log-exp21.8

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

    9. Simplified21.7

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

    11. Applied add-cbrt-cube21.7

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

    12. Simplified21.7

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

    if -1.1670988345674437e-290 < re < 3.812412734570747e-267

    1. Initial program 30.6

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

    3. Applied add-sqr-sqrt30.6

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

    4. Applied pow130.6

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

    5. Applied log-pow30.6

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

    6. Applied times-frac30.6

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

    8. Applied add-log-exp30.6

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

    9. Simplified30.4

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

    10. Taylor expanded around 0 30.6

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

    if 1.5205353416554455e+42 < re

    1. Initial program 45.8

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

    3. Applied add-sqr-sqrt45.8

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

    4. Applied pow145.8

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

    5. Applied log-pow45.8

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

    6. Applied times-frac45.8

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

    8. Applied add-log-exp45.8

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

    9. Simplified45.7

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

    10. Taylor expanded around inf 11.9

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\color{blue}{re}}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.859463483918923369547260927451545843966 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -1.167098834567443703136578031923709302405 \cdot 10^{-290}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}^{3}}\right)\\ \mathbf{elif}\;re \le 3.812412734570747037549119511317842097867 \cdot 10^{-267}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 1520535341655445535810607142928115832979000:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))