Average Error: 31.4 → 17.6
Time: 14.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.8015950926867568 \cdot 10^{144}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -2.603232334857776 \cdot 10^{-212}:\\ \;\;\;\;\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -6.08456462896641645 \cdot 10^{-228}:\\ \;\;\;\;-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le 4.4853367152010175 \cdot 10^{105}:\\ \;\;\;\;\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.8015950926867568 \cdot 10^{144}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{elif}\;re \le -2.603232334857776 \cdot 10^{-212}:\\
\;\;\;\;\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\

\mathbf{elif}\;re \le -6.08456462896641645 \cdot 10^{-228}:\\
\;\;\;\;-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}\\

\mathbf{elif}\;re \le 4.4853367152010175 \cdot 10^{105}:\\
\;\;\;\;\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\

\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 r47648 = re;
        double r47649 = r47648 * r47648;
        double r47650 = im;
        double r47651 = r47650 * r47650;
        double r47652 = r47649 + r47651;
        double r47653 = sqrt(r47652);
        double r47654 = log(r47653);
        double r47655 = 10.0;
        double r47656 = log(r47655);
        double r47657 = r47654 / r47656;
        return r47657;
}
↓
double f(double re, double im) {
        double r47658 = re;
        double r47659 = -2.8015950926867568e+144;
        bool r47660 = r47658 <= r47659;
        double r47661 = 1.0;
        double r47662 = 10.0;
        double r47663 = log(r47662);
        double r47664 = sqrt(r47663);
        double r47665 = r47661 / r47664;
        double r47666 = -r47658;
        double r47667 = pow(r47666, r47665);
        double r47668 = log(r47667);
        double r47669 = r47665 * r47668;
        double r47670 = -2.6032323348577763e-212;
        bool r47671 = r47658 <= r47670;
        double r47672 = r47658 * r47658;
        double r47673 = im;
        double r47674 = r47673 * r47673;
        double r47675 = r47672 + r47674;
        double r47676 = sqrt(r47675);
        double r47677 = pow(r47676, r47665);
        double r47678 = log(r47677);
        double r47679 = r47678 * r47665;
        double r47680 = -6.084564628966416e-228;
        bool r47681 = r47658 <= r47680;
        double r47682 = -1.0;
        double r47683 = r47682 / r47658;
        double r47684 = log(r47683);
        double r47685 = r47684 / r47663;
        double r47686 = -r47685;
        double r47687 = 4.4853367152010175e+105;
        bool r47688 = r47658 <= r47687;
        double r47689 = pow(r47658, r47665);
        double r47690 = log(r47689);
        double r47691 = r47665 * r47690;
        double r47692 = r47688 ? r47679 : r47691;
        double r47693 = r47681 ? r47686 : r47692;
        double r47694 = r47671 ? r47679 : r47693;
        double r47695 = r47660 ? r47669 : r47694;
        return r47695;
}

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 < -2.8015950926867568e+144

    1. Initial program 60.9

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

    3. Applied add-sqr-sqrt60.9

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

    4. Applied pow160.9

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

    5. Applied log-pow60.9

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

    6. Applied times-frac60.9

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

    8. Applied div-inv60.9

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

    10. Applied add-log-exp60.9

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

    11. Simplified60.9

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

    12. Taylor expanded around -inf 6.9

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

    13. Simplified6.9

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

    if -2.8015950926867568e+144 < re < -2.6032323348577763e-212 or -6.084564628966416e-228 < re < 4.4853367152010175e+105

    1. Initial program 21.3

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

    3. Applied add-sqr-sqrt21.3

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

    4. Applied pow121.3

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

    5. Applied log-pow21.3

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

    6. Applied times-frac21.3

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

    8. Applied div-inv21.1

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

    10. Applied add-log-exp21.1

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

    11. Simplified21.1

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

    13. Applied *-un-lft-identity21.1

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

    if -2.6032323348577763e-212 < re < -6.084564628966416e-228

    1. Initial program 31.4

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

    2. Taylor expanded around -inf 49.3

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

    3. Simplified49.3

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

    if 4.4853367152010175e+105 < re

    1. Initial program 51.3

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

    3. Applied add-sqr-sqrt51.3

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

    4. Applied pow151.3

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

    5. Applied log-pow51.3

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

    6. Applied times-frac51.3

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

    8. Applied div-inv51.3

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

    10. Applied add-log-exp51.3

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

    11. Simplified51.3

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

    12. Taylor expanded around inf 8.2

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.8015950926867568 \cdot 10^{144}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -2.603232334857776 \cdot 10^{-212}:\\ \;\;\;\;\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -6.08456462896641645 \cdot 10^{-228}:\\ \;\;\;\;-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le 4.4853367152010175 \cdot 10^{105}:\\ \;\;\;\;\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))