Average Error: 31.7 → 0.5
Time: 19.8s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), 0.0 \cdot \tan^{-1}_* \frac{im}{re}\right)}{\sqrt{\mathsf{fma}\left(0.0, 0.0, \log base \cdot \log base\right)}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(0.0, 0.0, \log base \cdot \log base\right)}}\]
\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 base, \log \left(\mathsf{hypot}\left(re, im\right)\right), 0.0 \cdot \tan^{-1}_* \frac{im}{re}\right)}{\sqrt{\mathsf{fma}\left(0.0, 0.0, \log base \cdot \log base\right)}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(0.0, 0.0, \log base \cdot \log base\right)}}
double f(double re, double im, double base) {
        double r1562818 = re;
        double r1562819 = r1562818 * r1562818;
        double r1562820 = im;
        double r1562821 = r1562820 * r1562820;
        double r1562822 = r1562819 + r1562821;
        double r1562823 = sqrt(r1562822);
        double r1562824 = log(r1562823);
        double r1562825 = base;
        double r1562826 = log(r1562825);
        double r1562827 = r1562824 * r1562826;
        double r1562828 = atan2(r1562820, r1562818);
        double r1562829 = 0.0;
        double r1562830 = r1562828 * r1562829;
        double r1562831 = r1562827 + r1562830;
        double r1562832 = r1562826 * r1562826;
        double r1562833 = r1562829 * r1562829;
        double r1562834 = r1562832 + r1562833;
        double r1562835 = r1562831 / r1562834;
        return r1562835;
}
↓
double f(double re, double im, double base) {
        double r1562836 = base;
        double r1562837 = log(r1562836);
        double r1562838 = re;
        double r1562839 = im;
        double r1562840 = hypot(r1562838, r1562839);
        double r1562841 = log(r1562840);
        double r1562842 = 0.0;
        double r1562843 = atan2(r1562839, r1562838);
        double r1562844 = r1562842 * r1562843;
        double r1562845 = fma(r1562837, r1562841, r1562844);
        double r1562846 = r1562837 * r1562837;
        double r1562847 = fma(r1562842, r1562842, r1562846);
        double r1562848 = sqrt(r1562847);
        double r1562849 = r1562845 / r1562848;
        double r1562850 = 1.0;
        double r1562851 = r1562850 / r1562848;
        double r1562852 = r1562849 * r1562851;
        return r1562852;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 31.7

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

  2. Simplified0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied *-un-lft-identity0.5

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

  6. Applied times-frac0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))