Average Error: 31.8 → 0.3
Time: 6.5s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left(e^{\log \left({base}^{\frac{1}{3}}\right)}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]
\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left(e^{\log \left({base}^{\frac{1}{3}}\right)}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)}
double f(double re, double im, double base) {
        double r46850 = im;
        double r46851 = re;
        double r46852 = atan2(r46850, r46851);
        double r46853 = base;
        double r46854 = log(r46853);
        double r46855 = r46852 * r46854;
        double r46856 = r46851 * r46851;
        double r46857 = r46850 * r46850;
        double r46858 = r46856 + r46857;
        double r46859 = sqrt(r46858);
        double r46860 = log(r46859);
        double r46861 = 0.0;
        double r46862 = r46860 * r46861;
        double r46863 = r46855 - r46862;
        double r46864 = r46854 * r46854;
        double r46865 = r46861 * r46861;
        double r46866 = r46864 + r46865;
        double r46867 = r46863 / r46866;
        return r46867;
}
↓
double f(double re, double im, double base) {
        double r46868 = -1.0;
        double r46869 = im;
        double r46870 = re;
        double r46871 = atan2(r46869, r46870);
        double r46872 = 2.0;
        double r46873 = -r46872;
        double r46874 = base;
        double r46875 = 0.3333333333333333;
        double r46876 = pow(r46874, r46875);
        double r46877 = log(r46876);
        double r46878 = exp(r46877);
        double r46879 = log(r46878);
        double r46880 = r46873 * r46879;
        double r46881 = 1.0;
        double r46882 = cbrt(r46874);
        double r46883 = r46881 / r46882;
        double r46884 = log(r46883);
        double r46885 = r46880 + r46884;
        double r46886 = r46871 / r46885;
        double r46887 = r46868 * r46886;
        return r46887;
}

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. Initial program 31.8

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

  2. Taylor expanded around inf 0.3

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied times-frac0.3

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

  7. Applied log-prod0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied add-exp-log0.4

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))