math.log10 on complex, imaginary part

Average Error: 0.8 → 0.1

Time: 3.5s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r26273 = im;
        double r26274 = re;
        double r26275 = atan2(r26273, r26274);
        double r26276 = 10.0;
        double r26277 = log(r26276);
        double r26278 = r26275 / r26277;
        return r26278;
}

↓

double f(double re, double im) {
        double r26279 = 1.0;
        double r26280 = 10.0;
        double r26281 = log(r26280);
        double r26282 = sqrt(r26281);
        double r26283 = r26279 / r26282;
        double r26284 = im;
        double r26285 = re;
        double r26286 = atan2(r26284, r26285);
        double r26287 = sqrt(r26283);
        double r26288 = r26286 * r26287;
        double r26289 = sqrt(r26279);
        double r26290 = r26289 / r26282;
        double r26291 = sqrt(r26290);
        double r26292 = sqrt(r26291);
        double r26293 = r26288 * r26292;
        double r26294 = r26293 * r26292;
        double r26295 = r26283 * r26294;
        return r26295;
}

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

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

9. Applied add-sqr-sqrt 0.8

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

10. Applied times-frac 0.8

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

11. Applied sqrt-prod 0.8

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

12. Applied associate-*r* 0.8

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

13. Simplified 0.8

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

15. Applied add-sqr-sqrt 0.8

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

16. Applied sqrt-prod 0.1

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

17. Applied associate-*r* 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))