Average Error: 32.3 → 17.9
Time: 21.0s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.981250171867664465389722520730842139436 \cdot 10^{79}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le 9.926501183114802962177232372696119953466 \cdot 10^{132}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{1}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}\right)}^{\frac{1}{4}} \cdot \left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot \log re\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.981250171867664465389722520730842139436 \cdot 10^{79}:\\
\;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\

\mathbf{elif}\;re \le 9.926501183114802962177232372696119953466 \cdot 10^{132}:\\
\;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left({\left(\frac{1}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}\right)}^{\frac{1}{4}} \cdot \left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot \log re\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\

\end{array}
double f(double re, double im) {
        double r1444638 = re;
        double r1444639 = r1444638 * r1444638;
        double r1444640 = im;
        double r1444641 = r1444640 * r1444640;
        double r1444642 = r1444639 + r1444641;
        double r1444643 = sqrt(r1444642);
        double r1444644 = log(r1444643);
        double r1444645 = 10.0;
        double r1444646 = log(r1444645);
        double r1444647 = r1444644 / r1444646;
        return r1444647;
}
↓
double f(double re, double im) {
        double r1444648 = re;
        double r1444649 = -3.9812501718676645e+79;
        bool r1444650 = r1444648 <= r1444649;
        double r1444651 = 0.5;
        double r1444652 = 10.0;
        double r1444653 = log(r1444652);
        double r1444654 = sqrt(r1444653);
        double r1444655 = r1444651 / r1444654;
        double r1444656 = 1.0;
        double r1444657 = r1444656 / r1444653;
        double r1444658 = sqrt(r1444657);
        double r1444659 = -1.0;
        double r1444660 = r1444659 / r1444648;
        double r1444661 = log(r1444660);
        double r1444662 = r1444658 * r1444661;
        double r1444663 = -2.0;
        double r1444664 = r1444662 * r1444663;
        double r1444665 = r1444655 * r1444664;
        double r1444666 = 9.926501183114803e+132;
        bool r1444667 = r1444648 <= r1444666;
        double r1444668 = sqrt(r1444655);
        double r1444669 = r1444648 * r1444648;
        double r1444670 = im;
        double r1444671 = r1444670 * r1444670;
        double r1444672 = r1444669 + r1444671;
        double r1444673 = log(r1444672);
        double r1444674 = r1444673 / r1444654;
        double r1444675 = r1444674 * r1444668;
        double r1444676 = r1444668 * r1444675;
        double r1444677 = r1444653 * r1444653;
        double r1444678 = r1444677 * r1444653;
        double r1444679 = r1444656 / r1444678;
        double r1444680 = 0.25;
        double r1444681 = pow(r1444679, r1444680);
        double r1444682 = 2.0;
        double r1444683 = sqrt(r1444651);
        double r1444684 = log(r1444648);
        double r1444685 = r1444683 * r1444684;
        double r1444686 = r1444682 * r1444685;
        double r1444687 = r1444681 * r1444686;
        double r1444688 = r1444687 * r1444668;
        double r1444689 = r1444667 ? r1444676 : r1444688;
        double r1444690 = r1444650 ? r1444665 : r1444689;
        return r1444690;
}

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 < -3.9812501718676645e+79

    1. Initial program 49.4

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

    3. Applied add-sqr-sqrt49.4

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

    4. Applied pow1/249.4

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

    5. Applied log-pow49.4

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

    6. Applied times-frac49.4

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

    7. Taylor expanded around -inf 10.4

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

    if -3.9812501718676645e+79 < re < 9.926501183114803e+132

    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-frac22.1

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

    8. Applied add-sqr-sqrt22.1

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

    9. Applied associate-*l*22.0

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

    if 9.926501183114803e+132 < re

    1. Initial program 57.8

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

    3. Applied add-sqr-sqrt57.8

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

    4. Applied pow1/257.8

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

    5. Applied log-pow57.8

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

    6. Applied times-frac57.8

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

    8. Applied add-sqr-sqrt57.8

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

    9. Applied associate-*l*57.8

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

    10. Taylor expanded around inf 8.0

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

    11. Simplified8.0

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.981250171867664465389722520730842139436 \cdot 10^{79}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le 9.926501183114802962177232372696119953466 \cdot 10^{132}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{1}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}\right)}^{\frac{1}{4}} \cdot \left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot \log re\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))