math.log10 on complex, imaginary part

Average Error: 0.8 → 0.8

Time: 3.3s

Precision: 64

Math

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

↓

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

C

double code(double re, double im) {
	return ((double) (((double) atan2(im, re)) / ((double) log(10.0))));
}

↓

double code(double re, double im) {
	return ((double) (((double) (((double) sqrt(((double) (1.0 / ((double) log(10.0)))))) / ((double) sqrt(((double) log(10.0)))))) * ((double) atan2(im, re))));
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.8

   \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.8

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

4. Applied *-un-lft-identity 0.8

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

5. Applied times-frac 0.8

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

6. Taylor expanded around 0 0.8

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

8. Applied *-commutative 0.8

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

9. Applied associate-*r* 0.8

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

10. Simplified 0.8

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

11. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020113 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))