Average Error: 31.8 → 17.7
Time: 23.7s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.968701183945851689901531641860092880161 \cdot 10^{116}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(-2 \cdot \left({\left(\frac{1}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}\right)}^{\frac{1}{4}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\\ \mathbf{elif}\;re \le 85920091.530095160007476806640625:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.968701183945851689901531641860092880161 \cdot 10^{116}:\\
\;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(-2 \cdot \left({\left(\frac{1}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}\right)}^{\frac{1}{4}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\\

\mathbf{elif}\;re \le 85920091.530095160007476806640625:\\
\;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\\

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

\end{array}
double f(double re, double im) {
        double r3721507 = re;
        double r3721508 = r3721507 * r3721507;
        double r3721509 = im;
        double r3721510 = r3721509 * r3721509;
        double r3721511 = r3721508 + r3721510;
        double r3721512 = sqrt(r3721511);
        double r3721513 = log(r3721512);
        double r3721514 = 10.0;
        double r3721515 = log(r3721514);
        double r3721516 = r3721513 / r3721515;
        return r3721516;
}
↓
double f(double re, double im) {
        double r3721517 = re;
        double r3721518 = -1.9687011839458517e+116;
        bool r3721519 = r3721517 <= r3721518;
        double r3721520 = 0.5;
        double r3721521 = 10.0;
        double r3721522 = log(r3721521);
        double r3721523 = sqrt(r3721522);
        double r3721524 = r3721520 / r3721523;
        double r3721525 = sqrt(r3721524);
        double r3721526 = -2.0;
        double r3721527 = 1.0;
        double r3721528 = r3721522 * r3721522;
        double r3721529 = r3721528 * r3721522;
        double r3721530 = r3721527 / r3721529;
        double r3721531 = 0.25;
        double r3721532 = pow(r3721530, r3721531);
        double r3721533 = -1.0;
        double r3721534 = r3721533 / r3721517;
        double r3721535 = log(r3721534);
        double r3721536 = sqrt(r3721520);
        double r3721537 = r3721535 * r3721536;
        double r3721538 = r3721532 * r3721537;
        double r3721539 = r3721526 * r3721538;
        double r3721540 = r3721525 * r3721539;
        double r3721541 = 85920091.53009516;
        bool r3721542 = r3721517 <= r3721541;
        double r3721543 = cbrt(r3721520);
        double r3721544 = r3721543 * r3721543;
        double r3721545 = sqrt(r3721523);
        double r3721546 = r3721544 / r3721545;
        double r3721547 = r3721543 / r3721545;
        double r3721548 = r3721517 * r3721517;
        double r3721549 = im;
        double r3721550 = r3721549 * r3721549;
        double r3721551 = r3721548 + r3721550;
        double r3721552 = log(r3721551);
        double r3721553 = r3721547 * r3721552;
        double r3721554 = r3721546 * r3721553;
        double r3721555 = r3721527 / r3721523;
        double r3721556 = r3721554 * r3721555;
        double r3721557 = r3721527 / r3721522;
        double r3721558 = sqrt(r3721557);
        double r3721559 = log(r3721517);
        double r3721560 = r3721558 * r3721559;
        double r3721561 = r3721560 * r3721555;
        double r3721562 = r3721542 ? r3721556 : r3721561;
        double r3721563 = r3721519 ? r3721540 : r3721562;
        return r3721563;
}

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 < -1.9687011839458517e+116

    1. Initial program 54.5

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

    3. Applied add-sqr-sqrt54.5

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

    4. Applied pow1/254.5

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

    5. Applied log-pow54.5

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

    6. Applied times-frac54.5

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

    8. Applied add-sqr-sqrt54.5

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

    9. Applied associate-*l*54.5

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

    10. Taylor expanded around -inf 7.7

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(-2 \cdot \left(\left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right)}\]

    11. Simplified7.7

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(-2 \cdot \left({\left(\frac{1}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}\right)}^{\frac{1}{4}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right)\right)\right)}\]

    if -1.9687011839458517e+116 < re < 85920091.53009516

    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 pow1/222.2

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

    5. Applied log-pow22.2

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

    6. Applied times-frac22.2

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

    8. Applied div-inv22.1

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

    9. Applied associate-*r*22.1

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

    11. Applied add-sqr-sqrt22.1

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

    12. Applied sqrt-prod22.5

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

    13. Applied add-cube-cbrt22.1

      \[\leadsto \left(\frac{\color{blue}{\left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{1}{2}}}}{\sqrt{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\log 10}}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]

    14. Applied times-frac22.1

      \[\leadsto \left(\color{blue}{\left(\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}\right)} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]

    15. Applied associate-*l*22.1

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\right)} \cdot \frac{1}{\sqrt{\log 10}}\]

    if 85920091.53009516 < re

    1. Initial program 40.1

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

    3. Applied add-sqr-sqrt40.1

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

    4. Applied pow1/240.1

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

    5. Applied log-pow40.1

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

    6. Applied times-frac40.1

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

    8. Applied div-inv40.0

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

    9. Applied associate-*r*40.0

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

    10. Taylor expanded around inf 13.4

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

    11. Simplified13.4

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.968701183945851689901531641860092880161 \cdot 10^{116}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(-2 \cdot \left({\left(\frac{1}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}\right)}^{\frac{1}{4}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\\ \mathbf{elif}\;re \le 85920091.530095160007476806640625:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))