math.log/2 on complex, real part

Average Error: 32.1 → 0.4

Time: 8.5s

Precision: 64

Math

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

↓

\[\frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}\]

C

double f(double re, double im, double base) {
        double r48913 = re;
        double r48914 = r48913 * r48913;
        double r48915 = im;
        double r48916 = r48915 * r48915;
        double r48917 = r48914 + r48916;
        double r48918 = sqrt(r48917);
        double r48919 = log(r48918);
        double r48920 = base;
        double r48921 = log(r48920);
        double r48922 = r48919 * r48921;
        double r48923 = atan2(r48915, r48913);
        double r48924 = 0.0;
        double r48925 = r48923 * r48924;
        double r48926 = r48922 + r48925;
        double r48927 = r48921 * r48921;
        double r48928 = r48924 * r48924;
        double r48929 = r48927 + r48928;
        double r48930 = r48926 / r48929;
        return r48930;
}

↓

double f(double re, double im, double base) {
        double r48931 = base;
        double r48932 = log(r48931);
        double r48933 = re;
        double r48934 = im;
        double r48935 = hypot(r48933, r48934);
        double r48936 = log(r48935);
        double r48937 = atan2(r48934, r48933);
        double r48938 = 0.0;
        double r48939 = r48937 * r48938;
        double r48940 = fma(r48932, r48936, r48939);
        double r48941 = hypot(r48932, r48938);
        double r48942 = 1.0;
        double r48943 = r48941 * r48942;
        double r48944 = r48940 / r48943;
        double r48945 = 2.0;
        double r48946 = cbrt(r48931);
        double r48947 = log(r48946);
        double r48948 = r48945 * r48947;
        double r48949 = r48932 * r48948;
        double r48950 = r48932 * r48947;
        double r48951 = r48949 + r48950;
        double r48952 = r48938 * r48938;
        double r48953 = r48951 + r48952;
        double r48954 = sqrt(r48953);
        double r48955 = r48944 / r48954;
        return r48955;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 32.1

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

3. Applied *-un-lft-identity 32.1

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

4. Applied sqrt-prod 32.1

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

5. Simplified 32.1

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

6. Simplified 0.5

   \[\leadsto \frac{\log \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
7. Using strategy rm

8. Applied add-sqr-sqrt 0.5

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

9. Applied associate-/r* 0.4

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

10. Simplified 0.4

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}{\sqrt{\log base \cdot \log \color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)} + 0.0 \cdot 0.0}}\]

13. Applied log-prod 0.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}{\sqrt{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}}\]

14. Applied distribute-lft-in 0.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}{\sqrt{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}}\]

15. Simplified 0.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}{\sqrt{\left(\color{blue}{\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right)} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}\]

16. Final simplification 0.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))