Average Error: 31.7 → 0.4
Time: 8.1s
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 r45132 = re;
        double r45133 = r45132 * r45132;
        double r45134 = im;
        double r45135 = r45134 * r45134;
        double r45136 = r45133 + r45135;
        double r45137 = sqrt(r45136);
        double r45138 = log(r45137);
        double r45139 = base;
        double r45140 = log(r45139);
        double r45141 = r45138 * r45140;
        double r45142 = atan2(r45134, r45132);
        double r45143 = 0.0;
        double r45144 = r45142 * r45143;
        double r45145 = r45141 + r45144;
        double r45146 = r45140 * r45140;
        double r45147 = r45143 * r45143;
        double r45148 = r45146 + r45147;
        double r45149 = r45145 / r45148;
        return r45149;
}
↓
double f(double re, double im, double base) {
        double r45150 = base;
        double r45151 = log(r45150);
        double r45152 = re;
        double r45153 = im;
        double r45154 = hypot(r45152, r45153);
        double r45155 = log(r45154);
        double r45156 = atan2(r45153, r45152);
        double r45157 = 0.0;
        double r45158 = r45156 * r45157;
        double r45159 = fma(r45151, r45155, r45158);
        double r45160 = hypot(r45151, r45157);
        double r45161 = 1.0;
        double r45162 = r45160 * r45161;
        double r45163 = r45159 / r45162;
        double r45164 = r45151 * r45151;
        double r45165 = r45157 * r45157;
        double r45166 = r45164 + r45165;
        double r45167 = sqrt(r45166);
        double r45168 = r45163 / r45167;
        return r45168;
}

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. Using strategy rm

  3. Applied *-un-lft-identity31.7

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

  4. Applied sqrt-prod31.7

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

  5. Simplified31.7

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

  6. Simplified0.5

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied associate-/r*0.4

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

  10. Simplified0.4

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

  11. 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 2020100 +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))))