Average Error: 0.8 → 0.1
Time: 4.5s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]
↓
\[\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot {\left(\sqrt{\frac{1}{\log 10}}\right)}^{1.5} \]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot {\left(\sqrt{\frac{1}{\log 10}}\right)}^{1.5}

(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (*
  (* (sqrt (/ 1.0 (sqrt (log 10.0)))) (atan2 im re))
  (pow (sqrt (/ 1.0 (log 10.0))) 1.5)))
double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}
↓
double code(double re, double im) {
	return (sqrt(1.0 / sqrt(log(10.0))) * atan2(im, re)) * pow(sqrt(1.0 / log(10.0)), 1.5);
}

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 add-sqr-sqrt_binary640.8

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

  3. Applied *-un-lft-identity_binary640.8

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

  4. Applied times-frac_binary640.8

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

  5. Taylor expanded in im around 0 0.8

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

  6. Applied add-sqr-sqrt_binary640.8

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

  7. Applied associate-*l*_binary640.8

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

  8. Simplified0.1

    \[\leadsto \sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\sqrt{\frac{1}{\log 10}}\right)}^{1.5}\right)} \]

  9. Applied associate-*r*_binary640.1

    \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot {\left(\sqrt{\frac{1}{\log 10}}\right)}^{1.5}} \]

  10. Final simplification0.1

    \[\leadsto \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot {\left(\sqrt{\frac{1}{\log 10}}\right)}^{1.5} \]

Reproduce

herbie shell --seed 2021357 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))