Average Error: 31.9 → 0.5
Time: 7.2s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}} \cdot \frac{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(re, im\right)\right), \log base, \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}\]
\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{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}} \cdot \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 1}
double f(double re, double im, double base) {
        double r43138 = re;
        double r43139 = r43138 * r43138;
        double r43140 = im;
        double r43141 = r43140 * r43140;
        double r43142 = r43139 + r43141;
        double r43143 = sqrt(r43142);
        double r43144 = log(r43143);
        double r43145 = base;
        double r43146 = log(r43145);
        double r43147 = r43144 * r43146;
        double r43148 = atan2(r43140, r43138);
        double r43149 = 0.0;
        double r43150 = r43148 * r43149;
        double r43151 = r43147 + r43150;
        double r43152 = r43146 * r43146;
        double r43153 = r43149 * r43149;
        double r43154 = r43152 + r43153;
        double r43155 = r43151 / r43154;
        return r43155;
}
↓
double f(double re, double im, double base) {
        double r43156 = 1.0;
        double r43157 = base;
        double r43158 = log(r43157);
        double r43159 = 0.0;
        double r43160 = hypot(r43158, r43159);
        double r43161 = r43160 / r43156;
        double r43162 = r43156 / r43161;
        double r43163 = re;
        double r43164 = im;
        double r43165 = hypot(r43163, r43164);
        double r43166 = log(r43165);
        double r43167 = atan2(r43164, r43163);
        double r43168 = r43167 * r43159;
        double r43169 = fma(r43166, r43158, r43168);
        double r43170 = r43160 * r43156;
        double r43171 = r43169 / r43170;
        double r43172 = r43162 * r43171;
        return r43172;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 31.9

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

  3. Applied hypot-def0.5

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

  5. Applied add-sqr-sqrt0.5

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

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

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

  7. Applied times-frac0.5

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

  8. Simplified0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2020018 +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))))