math.log/2 on complex, imaginary part

Average Error: 31.8 → 0.4

Time: 7.3s

Precision: 64

Math

\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

↓

\[-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right) + \left(\log \left(\frac{1}{\sqrt[3]{base}}\right) - \frac{-2}{3} \cdot \log \left(\frac{\sqrt[3]{1}}{\sqrt{base}}\right)\right)}\]

C

double f(double re, double im, double base) {
        double r54882 = im;
        double r54883 = re;
        double r54884 = atan2(r54882, r54883);
        double r54885 = base;
        double r54886 = log(r54885);
        double r54887 = r54884 * r54886;
        double r54888 = r54883 * r54883;
        double r54889 = r54882 * r54882;
        double r54890 = r54888 + r54889;
        double r54891 = sqrt(r54890);
        double r54892 = log(r54891);
        double r54893 = 0.0;
        double r54894 = r54892 * r54893;
        double r54895 = r54887 - r54894;
        double r54896 = r54886 * r54886;
        double r54897 = r54893 * r54893;
        double r54898 = r54896 + r54897;
        double r54899 = r54895 / r54898;
        return r54899;
}

↓

double f(double re, double im, double base) {
        double r54900 = -1.0;
        double r54901 = im;
        double r54902 = re;
        double r54903 = atan2(r54901, r54902);
        double r54904 = 2.0;
        double r54905 = -r54904;
        double r54906 = 1.0;
        double r54907 = cbrt(r54906);
        double r54908 = r54907 * r54907;
        double r54909 = base;
        double r54910 = sqrt(r54909);
        double r54911 = r54908 / r54910;
        double r54912 = -0.3333333333333333;
        double r54913 = pow(r54911, r54912);
        double r54914 = log(r54913);
        double r54915 = r54905 * r54914;
        double r54916 = cbrt(r54909);
        double r54917 = r54906 / r54916;
        double r54918 = log(r54917);
        double r54919 = -0.6666666666666666;
        double r54920 = r54907 / r54910;
        double r54921 = log(r54920);
        double r54922 = r54919 * r54921;
        double r54923 = r54918 - r54922;
        double r54924 = r54915 + r54923;
        double r54925 = r54903 / r54924;
        double r54926 = r54900 * r54925;
        return r54926;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 31.8

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

2. Taylor expanded around inf 0.3

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

4. Applied add-cube-cbrt 0.3

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

5. Applied add-cube-cbrt 0.3

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

6. Applied times-frac 0.3

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

7. Applied log-prod 0.4

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

8. Simplified 0.4

   \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\left(-2\right) \cdot \log \left(\sqrt[3]{base}\right)} + \log \left(\frac{\sqrt[3]{1}}{\sqrt[3]{base}}\right)}\]

9. Simplified 0.4

   \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left(\sqrt[3]{base}\right) + \color{blue}{\log \left(\frac{1}{\sqrt[3]{base}}\right)}}\]

10. Taylor expanded around inf 0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \color{blue}{\left({\left(\frac{1}{base}\right)}^{\frac{-1}{3}}\right)} + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]
11. Using strategy rm

12. Applied add-sqr-sqrt 0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left({\left(\frac{1}{\color{blue}{\sqrt{base} \cdot \sqrt{base}}}\right)}^{\frac{-1}{3}}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]

13. Applied add-cube-cbrt 0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left({\left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{base} \cdot \sqrt{base}}\right)}^{\frac{-1}{3}}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]

14. Applied times-frac 0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left({\color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{base}} \cdot \frac{\sqrt[3]{1}}{\sqrt{base}}\right)}}^{\frac{-1}{3}}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]

15. Applied unpow-prod-down 0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \color{blue}{\left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}} \cdot {\left(\frac{\sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right)} + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]

16. Applied log-prod 0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \color{blue}{\left(\log \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right) + \log \left({\left(\frac{\sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right)\right)} + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]

17. Applied distribute-lft-in 0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\left(\left(-2\right) \cdot \log \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right) + \left(-2\right) \cdot \log \left({\left(\frac{\sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right)\right)} + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]

18. Applied associate-+l+ 0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\left(-2\right) \cdot \log \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right) + \left(\left(-2\right) \cdot \log \left({\left(\frac{\sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)\right)}}\]

19. Simplified 0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right) + \color{blue}{\left(\log \left(\frac{1}{\sqrt[3]{base}}\right) - \frac{-2}{3} \cdot \log \left(\frac{\sqrt[3]{1}}{\sqrt{base}}\right)\right)}}\]

20. Final simplification 0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{base}}\right)}^{\frac{-1}{3}}\right) + \left(\log \left(\frac{1}{\sqrt[3]{base}}\right) - \frac{-2}{3} \cdot \log \left(\frac{\sqrt[3]{1}}{\sqrt{base}}\right)\right)}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))