Average Error: 31.6 → 0.4
Time: 5.7s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{2 \cdot \log \left({\left(\frac{1}{base}\right)}^{\frac{1}{3}}\right) + \log \left(\sqrt[3]{\frac{1}{base}}\right)}\]
\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}}{2 \cdot \log \left({\left(\frac{1}{base}\right)}^{\frac{1}{3}}\right) + \log \left(\sqrt[3]{\frac{1}{base}}\right)}
double code(double re, double im, double base) {
	return ((double) (((double) (((double) (((double) atan2(im, re)) * ((double) log(base)))) - ((double) (((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) * 0.0)))) / ((double) (((double) (((double) log(base)) * ((double) log(base)))) + ((double) (0.0 * 0.0))))));
}
↓
double code(double re, double im, double base) {
	return ((double) (-1.0 * ((double) (((double) atan2(im, re)) / ((double) (((double) (2.0 * ((double) log(((double) pow(((double) (1.0 / base)), 0.3333333333333333)))))) + ((double) log(((double) cbrt(((double) (1.0 / base))))))))))));
}

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 31.6

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

  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. Using strategy rm

  8. Applied pow1/30.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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))))