math.log10 on complex, imaginary part

Average Error: 0.9 → 0.1

Time: 21.6s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r22243 = im;
        double r22244 = re;
        double r22245 = atan2(r22243, r22244);
        double r22246 = 10.0;
        double r22247 = log(r22246);
        double r22248 = r22245 / r22247;
        return r22248;
}

↓

double f(double re, double im) {
        double r22249 = 1.0;
        double r22250 = 10.0;
        double r22251 = log(r22250);
        double r22252 = sqrt(r22251);
        double r22253 = r22249 / r22252;
        double r22254 = sqrt(r22253);
        double r22255 = im;
        double r22256 = re;
        double r22257 = atan2(r22255, r22256);
        double r22258 = r22257 / r22252;
        double r22259 = cbrt(r22252);
        double r22260 = r22249 / r22259;
        double r22261 = sqrt(r22260);
        double r22262 = r22258 * r22261;
        double r22263 = r22259 * r22259;
        double r22264 = r22249 / r22263;
        double r22265 = sqrt(r22264);
        double r22266 = r22262 * r22265;
        double r22267 = r22254 * r22266;
        return r22267;
}

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

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

16. Final simplification 0.1

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

Reproduce

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