Average Error: 31.6 → 0.5
Time: 23.7s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(re, im\right)\right), \log base, \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right)} \cdot \frac{1}{\mathsf{hypot}\left(\log base, 0.0\right)}\]
\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{\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 \frac{1}{\mathsf{hypot}\left(\log base, 0.0\right)}
double f(double re, double im, double base) {
        double r48849 = re;
        double r48850 = r48849 * r48849;
        double r48851 = im;
        double r48852 = r48851 * r48851;
        double r48853 = r48850 + r48852;
        double r48854 = sqrt(r48853);
        double r48855 = log(r48854);
        double r48856 = base;
        double r48857 = log(r48856);
        double r48858 = r48855 * r48857;
        double r48859 = atan2(r48851, r48849);
        double r48860 = 0.0;
        double r48861 = r48859 * r48860;
        double r48862 = r48858 + r48861;
        double r48863 = r48857 * r48857;
        double r48864 = r48860 * r48860;
        double r48865 = r48863 + r48864;
        double r48866 = r48862 / r48865;
        return r48866;
}
↓
double f(double re, double im, double base) {
        double r48867 = re;
        double r48868 = im;
        double r48869 = hypot(r48867, r48868);
        double r48870 = log(r48869);
        double r48871 = base;
        double r48872 = log(r48871);
        double r48873 = atan2(r48868, r48867);
        double r48874 = 0.0;
        double r48875 = r48873 * r48874;
        double r48876 = fma(r48870, r48872, r48875);
        double r48877 = hypot(r48872, r48874);
        double r48878 = r48876 / r48877;
        double r48879 = 1.0;
        double r48880 = r48879 / r48877;
        double r48881 = r48878 * r48880;
        return r48881;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 31.6

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

  2. Simplified0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-/r*0.4

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

  6. Simplified0.4

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

  8. Applied div-inv0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019325 +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))))