Average Error: 0.8 → 0.2
Time: 5.0s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\log 10}\\ \frac{\frac{1}{t_0}}{\frac{\sqrt{{\log 10}^{0.6666666666666666}}}{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{t_0}}}} \end{array} \]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\begin{array}{l}
t_0 := \sqrt{\log 10}\\
\frac{\frac{1}{t_0}}{\frac{\sqrt{{\log 10}^{0.6666666666666666}}}{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{t_0}}}}
\end{array}

(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (log 10.0))))
   (/
    (/ 1.0 t_0)
    (/
     (sqrt (pow (log 10.0) 0.6666666666666666))
     (/ (atan2 im re) (cbrt t_0))))))
double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}
↓
double code(double re, double im) {
	double t_0 = sqrt(log(10.0));
	return (1.0 / t_0) / (sqrt(pow(log(10.0), 0.6666666666666666)) / (atan2(im, re) / cbrt(t_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 *-un-lft-identity_binary641.0

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

  4. Applied add-sqr-sqrt_binary641.0

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

  5. Applied times-frac_binary641.0

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

  6. Applied associate-/r*_binary641.0

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

  7. Simplified1.0

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

  8. Applied add-cube-cbrt_binary641.5

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

  9. Applied sqrt-prod_binary641.5

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

  10. Applied associate-/l*_binary641.5

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

  11. Simplified0.2

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

  12. Applied pow1/3_binary640.2

    \[\leadsto \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\sqrt[3]{\log 10} \cdot \color{blue}{{\log 10}^{0.3333333333333333}}}}{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}}} \]

  13. Applied pow1/3_binary640.2

    \[\leadsto \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\color{blue}{{\log 10}^{0.3333333333333333}} \cdot {\log 10}^{0.3333333333333333}}}{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}}} \]

  14. Applied pow-sqr_binary640.2

    \[\leadsto \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\color{blue}{{\log 10}^{\left(2 \cdot 0.3333333333333333\right)}}}}{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}}} \]

  15. Final simplification0.2

    \[\leadsto \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{{\log 10}^{0.6666666666666666}}}{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}}} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))