math.log10 on complex, imaginary part

Average Error: 0.9 → 0.1

Time: 23.9s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r22019 = im;
        double r22020 = re;
        double r22021 = atan2(r22019, r22020);
        double r22022 = 10.0;
        double r22023 = log(r22022);
        double r22024 = r22021 / r22023;
        return r22024;
}

↓

double f(double re, double im) {
        double r22025 = 1.0;
        double r22026 = 10.0;
        double r22027 = log(r22026);
        double r22028 = sqrt(r22027);
        double r22029 = r22025 / r22028;
        double r22030 = sqrt(r22029);
        double r22031 = cbrt(r22028);
        double r22032 = r22025 / r22031;
        double r22033 = sqrt(r22032);
        double r22034 = im;
        double r22035 = re;
        double r22036 = atan2(r22034, r22035);
        double r22037 = r22036 / r22028;
        double r22038 = r22033 * r22037;
        double r22039 = r22031 * r22031;
        double r22040 = r22025 / r22039;
        double r22041 = sqrt(r22040);
        double r22042 = r22038 * r22041;
        double r22043 = r22030 * r22042;
        return r22043;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.9

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

3. Applied add-sqr-sqrt 0.9

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

4. Applied *-un-lft-identity 0.9

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

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. Using strategy rm

7. Applied add-sqr-sqrt 0.8

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

8. Applied associate-*l* 0.9

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

10. Applied add-cube-cbrt 0.1

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

11. Applied add-cube-cbrt 0.1

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

12. Applied times-frac 0.1

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

13. Applied sqrt-prod 0.1

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

14. Applied associate-*l* 0.1

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

15. Simplified 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019315 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))