Average Error: 31.6 → 17.7
Time: 25.8s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.477980765554152890668180029680723006559 \cdot 10^{121}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \le 5.925540416458275309943000169806911367311 \cdot 10^{52}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.477980765554152890668180029680723006559 \cdot 10^{121}:\\
\;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\

\mathbf{elif}\;re \le 5.925540416458275309943000169806911367311 \cdot 10^{52}:\\
\;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-\log re}{-\log base}\\

\end{array}
double f(double re, double im, double base) {
        double r42789 = re;
        double r42790 = r42789 * r42789;
        double r42791 = im;
        double r42792 = r42791 * r42791;
        double r42793 = r42790 + r42792;
        double r42794 = sqrt(r42793);
        double r42795 = log(r42794);
        double r42796 = base;
        double r42797 = log(r42796);
        double r42798 = r42795 * r42797;
        double r42799 = atan2(r42791, r42789);
        double r42800 = 0.0;
        double r42801 = r42799 * r42800;
        double r42802 = r42798 + r42801;
        double r42803 = r42797 * r42797;
        double r42804 = r42800 * r42800;
        double r42805 = r42803 + r42804;
        double r42806 = r42802 / r42805;
        return r42806;
}
↓
double f(double re, double im, double base) {
        double r42807 = re;
        double r42808 = -1.477980765554153e+121;
        bool r42809 = r42807 <= r42808;
        double r42810 = -1.0;
        double r42811 = r42810 / r42807;
        double r42812 = log(r42811);
        double r42813 = -r42812;
        double r42814 = base;
        double r42815 = log(r42814);
        double r42816 = r42813 / r42815;
        double r42817 = 5.925540416458275e+52;
        bool r42818 = r42807 <= r42817;
        double r42819 = 1.0;
        double r42820 = 2.0;
        double r42821 = pow(r42815, r42820);
        double r42822 = 0.0;
        double r42823 = r42822 * r42822;
        double r42824 = r42821 + r42823;
        double r42825 = sqrt(r42824);
        double r42826 = r42819 / r42825;
        double r42827 = r42807 * r42807;
        double r42828 = im;
        double r42829 = r42828 * r42828;
        double r42830 = r42827 + r42829;
        double r42831 = sqrt(r42830);
        double r42832 = log(r42831);
        double r42833 = r42832 * r42815;
        double r42834 = atan2(r42828, r42807);
        double r42835 = r42834 * r42822;
        double r42836 = r42833 + r42835;
        double r42837 = 6.0;
        double r42838 = pow(r42815, r42837);
        double r42839 = cbrt(r42838);
        double r42840 = r42839 + r42823;
        double r42841 = sqrt(r42840);
        double r42842 = r42836 / r42841;
        double r42843 = r42826 * r42842;
        double r42844 = log(r42807);
        double r42845 = -r42844;
        double r42846 = -r42815;
        double r42847 = r42845 / r42846;
        double r42848 = r42818 ? r42843 : r42847;
        double r42849 = r42809 ? r42816 : r42848;
        return r42849;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -1.477980765554153e+121

    1. Initial program 55.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt55.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    4. Applied *-un-lft-identity55.4

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    5. Applied times-frac55.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    6. Simplified55.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    7. Simplified55.4

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}\]

    8. Taylor expanded around -inf 64.0

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

    9. Simplified7.9

      \[\leadsto \color{blue}{\frac{-\log \left(\frac{-1}{re}\right)}{0 + \log base}}\]

    if -1.477980765554153e+121 < re < 5.925540416458275e+52

    1. Initial program 21.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt21.9

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    4. Applied *-un-lft-identity21.9

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    5. Applied times-frac21.9

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    6. Simplified21.9

      \[\leadsto \color{blue}{\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    7. Simplified21.9

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube22.0

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\color{blue}{\sqrt[3]{\left({\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2}\right) \cdot {\left(\log base\right)}^{2}}} + 0.0 \cdot 0.0}}\]

    10. Simplified21.9

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\sqrt[3]{\color{blue}{{\left(\log base\right)}^{6}}} + 0.0 \cdot 0.0}}\]

    if 5.925540416458275e+52 < re

    1. Initial program 45.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    2. Taylor expanded around inf 11.3

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

    3. Simplified11.3

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.477980765554152890668180029680723006559 \cdot 10^{121}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \le 5.925540416458275309943000169806911367311 \cdot 10^{52}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))