Average Error: 32.4 → 0.4
Time: 10.6s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot -1}{\left(\log \left(\frac{\sqrt{1}}{{\left({base}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{base}\right)}^{\frac{1}{3}}}\right) + \log \left(\sqrt{1}\right)\right) - \frac{2}{3} \cdot \log base}\]
\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\frac{\tan^{-1}_* \frac{im}{re} \cdot -1}{\left(\log \left(\frac{\sqrt{1}}{{\left({base}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{base}\right)}^{\frac{1}{3}}}\right) + \log \left(\sqrt{1}\right)\right) - \frac{2}{3} \cdot \log base}
double f(double re, double im, double base) {
        double r97138 = im;
        double r97139 = re;
        double r97140 = atan2(r97138, r97139);
        double r97141 = base;
        double r97142 = log(r97141);
        double r97143 = r97140 * r97142;
        double r97144 = r97139 * r97139;
        double r97145 = r97138 * r97138;
        double r97146 = r97144 + r97145;
        double r97147 = sqrt(r97146);
        double r97148 = log(r97147);
        double r97149 = 0.0;
        double r97150 = r97148 * r97149;
        double r97151 = r97143 - r97150;
        double r97152 = r97142 * r97142;
        double r97153 = r97149 * r97149;
        double r97154 = r97152 + r97153;
        double r97155 = r97151 / r97154;
        return r97155;
}
↓
double f(double re, double im, double base) {
        double r97156 = im;
        double r97157 = re;
        double r97158 = atan2(r97156, r97157);
        double r97159 = -1.0;
        double r97160 = r97158 * r97159;
        double r97161 = 1.0;
        double r97162 = sqrt(r97161);
        double r97163 = base;
        double r97164 = 0.3333333333333333;
        double r97165 = pow(r97163, r97164);
        double r97166 = 0.6666666666666666;
        double r97167 = pow(r97165, r97166);
        double r97168 = cbrt(r97163);
        double r97169 = pow(r97168, r97164);
        double r97170 = r97167 * r97169;
        double r97171 = r97162 / r97170;
        double r97172 = log(r97171);
        double r97173 = log(r97162);
        double r97174 = r97172 + r97173;
        double r97175 = log(r97163);
        double r97176 = r97166 * r97175;
        double r97177 = r97174 - r97176;
        double r97178 = r97160 / r97177;
        return r97178;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.4

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

  2. Taylor expanded around inf 0.3

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

  4. Applied add-cube-cbrt0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{1}{\color{blue}{\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}}}\right)}\]

  5. Applied add-sqr-sqrt0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}}\right)}\]

  6. Applied times-frac0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left(\frac{\sqrt{1}}{\sqrt[3]{base} \cdot \sqrt[3]{base}} \cdot \frac{\sqrt{1}}{\sqrt[3]{base}}\right)}}\]

  7. Applied log-prod0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left(\frac{\sqrt{1}}{\sqrt[3]{base} \cdot \sqrt[3]{base}}\right) + \log \left(\frac{\sqrt{1}}{\sqrt[3]{base}}\right)}}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\sqrt{1}}{\sqrt[3]{base} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{base}} \cdot \sqrt{\sqrt[3]{base}}\right)}}\right) + \log \left(\frac{\sqrt{1}}{\sqrt[3]{base}}\right)}\]

  10. Applied add-sqr-sqrt0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\sqrt{1}}{\color{blue}{\left(\sqrt{\sqrt[3]{base}} \cdot \sqrt{\sqrt[3]{base}}\right)} \cdot \left(\sqrt{\sqrt[3]{base}} \cdot \sqrt{\sqrt[3]{base}}\right)}\right) + \log \left(\frac{\sqrt{1}}{\sqrt[3]{base}}\right)}\]

  11. Applied swap-sqr0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\sqrt{1}}{\color{blue}{\left(\sqrt{\sqrt[3]{base}} \cdot \sqrt{\sqrt[3]{base}}\right) \cdot \left(\sqrt{\sqrt[3]{base}} \cdot \sqrt{\sqrt[3]{base}}\right)}}\right) + \log \left(\frac{\sqrt{1}}{\sqrt[3]{base}}\right)}\]

  12. Simplified0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\sqrt{1}}{\color{blue}{{base}^{\frac{1}{3}}} \cdot \left(\sqrt{\sqrt[3]{base}} \cdot \sqrt{\sqrt[3]{base}}\right)}\right) + \log \left(\frac{\sqrt{1}}{\sqrt[3]{base}}\right)}\]

  13. Simplified0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\sqrt{1}}{{base}^{\frac{1}{3}} \cdot \color{blue}{{base}^{\frac{1}{3}}}}\right) + \log \left(\frac{\sqrt{1}}{\sqrt[3]{base}}\right)}\]
  14. Using strategy rm

  15. Applied add-cube-cbrt0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\sqrt{1}}{{base}^{\frac{1}{3}} \cdot {base}^{\frac{1}{3}}}\right) + \log \left(\frac{\sqrt{1}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{base}} \cdot \sqrt[3]{\sqrt[3]{base}}\right) \cdot \sqrt[3]{\sqrt[3]{base}}}}\right)}\]

  16. Simplified0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\sqrt{1}}{{base}^{\frac{1}{3}} \cdot {base}^{\frac{1}{3}}}\right) + \log \left(\frac{\sqrt{1}}{\color{blue}{{\left({base}^{\frac{1}{3}}\right)}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{base}}}\right)}\]

  17. Simplified0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\sqrt{1}}{{base}^{\frac{1}{3}} \cdot {base}^{\frac{1}{3}}}\right) + \log \left(\frac{\sqrt{1}}{{\left({base}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot \color{blue}{{\left(\sqrt[3]{base}\right)}^{\frac{1}{3}}}}\right)}\]

  18. Final simplification0.4

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot -1}{\left(\log \left(\frac{\sqrt{1}}{{\left({base}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{base}\right)}^{\frac{1}{3}}}\right) + \log \left(\sqrt{1}\right)\right) - \frac{2}{3} \cdot \log base}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))