Average Error: 31.4 → 0.5
Time: 23.7s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{1}{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re} - \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot 0.0\right)\]
\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}
↓
\frac{1}{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re} - \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot 0.0\right)
double f(double re, double im, double base) {
        double r2116850 = im;
        double r2116851 = re;
        double r2116852 = atan2(r2116850, r2116851);
        double r2116853 = base;
        double r2116854 = log(r2116853);
        double r2116855 = r2116852 * r2116854;
        double r2116856 = r2116851 * r2116851;
        double r2116857 = r2116850 * r2116850;
        double r2116858 = r2116856 + r2116857;
        double r2116859 = sqrt(r2116858);
        double r2116860 = log(r2116859);
        double r2116861 = 0.0;
        double r2116862 = r2116860 * r2116861;
        double r2116863 = r2116855 - r2116862;
        double r2116864 = r2116854 * r2116854;
        double r2116865 = r2116861 * r2116861;
        double r2116866 = r2116864 + r2116865;
        double r2116867 = r2116863 / r2116866;
        return r2116867;
}
↓
double f(double re, double im, double base) {
        double r2116868 = 1.0;
        double r2116869 = base;
        double r2116870 = log(r2116869);
        double r2116871 = 0.0;
        double r2116872 = r2116871 * r2116871;
        double r2116873 = fma(r2116870, r2116870, r2116872);
        double r2116874 = r2116868 / r2116873;
        double r2116875 = im;
        double r2116876 = re;
        double r2116877 = atan2(r2116875, r2116876);
        double r2116878 = r2116870 * r2116877;
        double r2116879 = hypot(r2116875, r2116876);
        double r2116880 = log(r2116879);
        double r2116881 = r2116880 * r2116871;
        double r2116882 = r2116878 - r2116881;
        double r2116883 = r2116874 * r2116882;
        return r2116883;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 31.4

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

  2. Simplified0.4

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

  4. Applied div-inv0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))