math.log10 on complex, imaginary part

Average Error: 0.8 → 0.8

Time: 9.8s

Precision: 64

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

Math

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

↓

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

C

double f(double re, double im) {
        double r34310 = im;
        double r34311 = re;
        double r34312 = atan2(r34310, r34311);
        double r34313 = 10.0;
        double r34314 = log(r34313);
        double r34315 = r34312 / r34314;
        return r34315;
}

↓

double f(double re, double im) {
        double r34316 = 1.0;
        double r34317 = 10.0;
        double r34318 = log(r34317);
        double r34319 = im;
        double r34320 = re;
        double r34321 = atan2(r34319, r34320);
        double r34322 = r34318 / r34321;
        double r34323 = cbrt(r34322);
        double r34324 = r34323 * r34323;
        double r34325 = cbrt(r34318);
        double r34326 = r34325 * r34325;
        double r34327 = cbrt(r34326);
        double r34328 = sqrt(r34318);
        double r34329 = cbrt(r34328);
        double r34330 = cbrt(r34329);
        double r34331 = r34329 / r34321;
        double r34332 = cbrt(r34331);
        double r34333 = r34330 * r34332;
        double r34334 = r34327 * r34333;
        double r34335 = r34324 * r34334;
        double r34336 = r34316 / r34335;
        return r34336;
}

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 clear-num 1.0

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

5. Applied add-cube-cbrt 0.8

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

7. Applied *-un-lft-identity 0.8

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

8. Applied add-cube-cbrt 0.9

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

9. Applied times-frac 0.9

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

10. Applied cbrt-prod 0.8

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

11. Simplified 0.8

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

13. Applied *-un-lft-identity 0.8

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

14. Applied add-sqr-sqrt 0.8

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

15. Applied cbrt-prod 0.8

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

16. Applied times-frac 0.8

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

17. Applied cbrt-prod 0.8

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

18. Simplified 0.8

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

19. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))