Average Error: 32.4 → 0.4
Time: 7.6s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
\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{\log base \cdot \log base + 0.0 \cdot 0.0}}
double f(double re, double im, double base) {
        double r47075 = re;
        double r47076 = r47075 * r47075;
        double r47077 = im;
        double r47078 = r47077 * r47077;
        double r47079 = r47076 + r47078;
        double r47080 = sqrt(r47079);
        double r47081 = log(r47080);
        double r47082 = base;
        double r47083 = log(r47082);
        double r47084 = r47081 * r47083;
        double r47085 = atan2(r47077, r47075);
        double r47086 = 0.0;
        double r47087 = r47085 * r47086;
        double r47088 = r47084 + r47087;
        double r47089 = r47083 * r47083;
        double r47090 = r47086 * r47086;
        double r47091 = r47089 + r47090;
        double r47092 = r47088 / r47091;
        return r47092;
}
↓
double f(double re, double im, double base) {
        double r47093 = base;
        double r47094 = log(r47093);
        double r47095 = re;
        double r47096 = im;
        double r47097 = hypot(r47095, r47096);
        double r47098 = log(r47097);
        double r47099 = atan2(r47096, r47095);
        double r47100 = 0.0;
        double r47101 = r47099 * r47100;
        double r47102 = fma(r47094, r47098, r47101);
        double r47103 = hypot(r47094, r47100);
        double r47104 = 1.0;
        double r47105 = r47103 * r47104;
        double r47106 = r47102 / r47105;
        double r47107 = r47094 * r47094;
        double r47108 = r47100 * r47100;
        double r47109 = r47107 + r47108;
        double r47110 = sqrt(r47109);
        double r47111 = r47106 / r47110;
        return r47111;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 32.4

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

  3. Applied add-exp-log32.4

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

  4. Simplified0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied associate-/r*0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020059 +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))))