math.log10 on complex, imaginary part

Average Error: 0.8 → 0.1

Time: 3.5s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r81551 = im;
        double r81552 = re;
        double r81553 = atan2(r81551, r81552);
        double r81554 = 10.0;
        double r81555 = log(r81554);
        double r81556 = r81553 / r81555;
        return r81556;
}

↓

double f(double re, double im) {
        double r81557 = 1.0;
        double r81558 = 10.0;
        double r81559 = log(r81558);
        double r81560 = sqrt(r81559);
        double r81561 = r81557 / r81560;
        double r81562 = sqrt(r81561);
        double r81563 = sqrt(r81562);
        double r81564 = im;
        double r81565 = re;
        double r81566 = atan2(r81564, r81565);
        double r81567 = r81557 / r81559;
        double r81568 = sqrt(r81567);
        double r81569 = r81566 * r81568;
        double r81570 = r81562 * r81569;
        double r81571 = r81563 * r81570;
        double r81572 = r81563 * r81571;
        return r81572;
}

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

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

9. Applied associate-*l* 0.8

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

11. Applied add-sqr-sqrt 0.8

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

12. Applied sqrt-prod 0.1

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

13. Applied associate-*l* 0.1

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

14. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019344 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))