Average Error: 31.8 → 0.3
Time: 16.9s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\left(\sqrt[3]{\frac{1}{base}} \cdot {base}^{\frac{-1}{3}}\right) \cdot {base}^{\frac{-1}{3}}\right)}\]
\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}}{\log \left(\left(\sqrt[3]{\frac{1}{base}} \cdot {base}^{\frac{-1}{3}}\right) \cdot {base}^{\frac{-1}{3}}\right)}
double f(double re, double im, double base) {
        double r105164 = im;
        double r105165 = re;
        double r105166 = atan2(r105164, r105165);
        double r105167 = base;
        double r105168 = log(r105167);
        double r105169 = r105166 * r105168;
        double r105170 = r105165 * r105165;
        double r105171 = r105164 * r105164;
        double r105172 = r105170 + r105171;
        double r105173 = sqrt(r105172);
        double r105174 = log(r105173);
        double r105175 = 0.0;
        double r105176 = r105174 * r105175;
        double r105177 = r105169 - r105176;
        double r105178 = r105168 * r105168;
        double r105179 = r105175 * r105175;
        double r105180 = r105178 + r105179;
        double r105181 = r105177 / r105180;
        return r105181;
}
↓
double f(double re, double im, double base) {
        double r105182 = -1.0;
        double r105183 = im;
        double r105184 = re;
        double r105185 = atan2(r105183, r105184);
        double r105186 = 1.0;
        double r105187 = base;
        double r105188 = r105186 / r105187;
        double r105189 = cbrt(r105188);
        double r105190 = -0.3333333333333333;
        double r105191 = pow(r105187, r105190);
        double r105192 = r105189 * r105191;
        double r105193 = r105192 * r105191;
        double r105194 = log(r105193);
        double r105195 = r105185 / r105194;
        double r105196 = r105182 * r105195;
        return r105196;
}

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

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

  5. Applied log-prod0.4

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

  6. Simplified0.4

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

  7. Taylor expanded around 0 0.3

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

  9. Applied add-log-exp0.3

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

  10. Applied sum-log0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019294 
(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))))