Average Error: 0.8 → 0.7
Time: 7.0s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\\ \frac{1}{\left(t_0 \cdot t_0\right) \cdot \sqrt[3]{t_0 \cdot {t_0}^{2}}} \end{array} \]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\\
\frac{1}{\left(t_0 \cdot t_0\right) \cdot \sqrt[3]{t_0 \cdot {t_0}^{2}}}
\end{array}

(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (cbrt (/ (log 10.0) (atan2 im re)))))
   (/ 1.0 (* (* t_0 t_0) (cbrt (* t_0 (pow t_0 2.0)))))))
double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}
↓
double code(double re, double im) {
	double t_0 = cbrt(log(10.0) / atan2(im, re));
	return 1.0 / ((t_0 * t_0) * cbrt(t_0 * pow(t_0, 2.0)));
}

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. Initial program 0.8

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]

  2. Applied clear-num_binary641.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}} \]

  3. Applied add-cube-cbrt_binary640.8

    \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}} \]

  4. Applied add-cbrt-cube_binary640.7

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}}} \]

  5. Applied pow1_binary640.7

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \color{blue}{{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right)}^{1}}\right) \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}} \]

  6. Applied pow1_binary640.7

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\left(\color{blue}{{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right)}^{1}} \cdot {\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right)}^{1}\right) \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}} \]

  7. Applied pow-sqr_binary640.7

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

  8. Final simplification0.7

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot {\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right)}^{2}}} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))