math.log10 on complex, imaginary part

Average Error: 0.8 → 0.9

Time: 3.6s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r26964 = im;
        double r26965 = re;
        double r26966 = atan2(r26964, r26965);
        double r26967 = 10.0;
        double r26968 = log(r26967);
        double r26969 = r26966 / r26968;
        return r26969;
}

↓

double f(double re, double im) {
        double r26970 = 1.0;
        double r26971 = 10.0;
        double r26972 = log(r26971);
        double r26973 = sqrt(r26972);
        double r26974 = r26970 / r26973;
        double r26975 = im;
        double r26976 = re;
        double r26977 = atan2(r26975, r26976);
        double r26978 = sqrt(r26974);
        double r26979 = cbrt(r26978);
        double r26980 = r26979 * r26979;
        double r26981 = r26977 * r26980;
        double r26982 = r26981 * r26979;
        double r26983 = sqrt(r26970);
        double r26984 = r26983 / r26973;
        double r26985 = sqrt(r26984);
        double r26986 = r26982 * r26985;
        double r26987 = r26974 * r26986;
        return r26987;
}

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-cube-cbrt 0.8

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

16. Applied associate-*r* 0.9

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

17. Final simplification 0.9

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

Reproduce

herbie shell --seed 2020036 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))