math.log10 on complex, imaginary part

Average Error: 0.8 → 0.1

Time: 19.9s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r1735220 = im;
        double r1735221 = re;
        double r1735222 = atan2(r1735220, r1735221);
        double r1735223 = 10.0;
        double r1735224 = log(r1735223);
        double r1735225 = r1735222 / r1735224;
        return r1735225;
}

↓

double f(double re, double im) {
        double r1735226 = 1.0;
        double r1735227 = 10.0;
        double r1735228 = log(r1735227);
        double r1735229 = sqrt(r1735228);
        double r1735230 = r1735226 / r1735229;
        double r1735231 = sqrt(r1735230);
        double r1735232 = im;
        double r1735233 = re;
        double r1735234 = atan2(r1735232, r1735233);
        double r1735235 = r1735230 * r1735234;
        double r1735236 = r1735231 * r1735235;
        double r1735237 = sqrt(r1735231);
        double r1735238 = r1735236 * r1735237;
        double r1735239 = r1735238 * r1735237;
        return r1735239;
}

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. Using strategy rm

7. Applied div-inv 0.8

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

9. Applied add-sqr-sqrt 0.8

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

10. Applied associate-*l* 0.8

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

12. Applied add-sqr-sqrt 0.8

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

13. Applied sqrt-prod 0.1

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

14. Applied associate-*l* 0.1

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

15. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019192 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10.0)))