Average Error: 31.9 → 0.5
Time: 8.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(re, im\right)\right), \log base, \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
\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{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(re, im\right)\right), \log base, \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}
double f(double re, double im, double base) {
        double r46885 = re;
        double r46886 = r46885 * r46885;
        double r46887 = im;
        double r46888 = r46887 * r46887;
        double r46889 = r46886 + r46888;
        double r46890 = sqrt(r46889);
        double r46891 = log(r46890);
        double r46892 = base;
        double r46893 = log(r46892);
        double r46894 = r46891 * r46893;
        double r46895 = atan2(r46887, r46885);
        double r46896 = 0.0;
        double r46897 = r46895 * r46896;
        double r46898 = r46894 + r46897;
        double r46899 = r46893 * r46893;
        double r46900 = r46896 * r46896;
        double r46901 = r46899 + r46900;
        double r46902 = r46898 / r46901;
        return r46902;
}
↓
double f(double re, double im, double base) {
        double r46903 = 1.0;
        double r46904 = base;
        double r46905 = log(r46904);
        double r46906 = 0.0;
        double r46907 = hypot(r46905, r46906);
        double r46908 = re;
        double r46909 = im;
        double r46910 = hypot(r46908, r46909);
        double r46911 = log(r46910);
        double r46912 = atan2(r46909, r46908);
        double r46913 = r46912 * r46906;
        double r46914 = fma(r46911, r46905, r46913);
        double r46915 = r46907 / r46914;
        double r46916 = r46903 / r46915;
        double r46917 = r46905 * r46905;
        double r46918 = r46906 * r46906;
        double r46919 = r46917 + r46918;
        double r46920 = sqrt(r46919);
        double r46921 = r46916 / r46920;
        return r46921;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 31.9

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

  3. Applied hypot-def0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied associate-/r*0.4

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

  7. Simplified0.4

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

  9. Applied clear-num0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2020060 +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))))