math.log10 on complex, imaginary part

Average Error: 0.8 → 0.8

Time: 14.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double re, double im) {
        double r30940 = im;
        double r30941 = re;
        double r30942 = atan2(r30940, r30941);
        double r30943 = 10.0;
        double r30944 = log(r30943);
        double r30945 = r30942 / r30944;
        return r30945;
}

↓

double f(double re, double im) {
        double r30946 = 1.0;
        double r30947 = 10.0;
        double r30948 = log(r30947);
        double r30949 = sqrt(r30948);
        double r30950 = r30946 / r30949;
        double r30951 = im;
        double r30952 = re;
        double r30953 = atan2(r30951, r30952);
        double r30954 = sqrt(r30950);
        double r30955 = r30953 * r30954;
        double r30956 = sqrt(r30946);
        double r30957 = r30956 / r30949;
        double r30958 = sqrt(r30957);
        double r30959 = r30955 * r30958;
        double r30960 = r30950 * r30959;
        return r30960;
}

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 add-sqr-sqrt 0.8

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

9. Applied add-sqr-sqrt 0.8

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

10. Applied times-frac 0.8

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

11. Applied sqrt-prod 0.8

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

12. Applied associate-*r* 0.8

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

13. Simplified 0.8

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

14. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019350 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))