Average Error: 32.0 → 0.4
Time: 6.8s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(\log \left({\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right)}^{\left(-\frac{1}{3}\right)}\right) - \frac{2}{3} \cdot \log base\right) + \log \left({\left(\sqrt[3]{base}\right)}^{\left(-\frac{1}{3}\right)}\right)}\]
\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}
↓
-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(\log \left({\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right)}^{\left(-\frac{1}{3}\right)}\right) - \frac{2}{3} \cdot \log base\right) + \log \left({\left(\sqrt[3]{base}\right)}^{\left(-\frac{1}{3}\right)}\right)}
double f(double re, double im, double base) {
        double r101102 = im;
        double r101103 = re;
        double r101104 = atan2(r101102, r101103);
        double r101105 = base;
        double r101106 = log(r101105);
        double r101107 = r101104 * r101106;
        double r101108 = r101103 * r101103;
        double r101109 = r101102 * r101102;
        double r101110 = r101108 + r101109;
        double r101111 = sqrt(r101110);
        double r101112 = log(r101111);
        double r101113 = 0.0;
        double r101114 = r101112 * r101113;
        double r101115 = r101107 - r101114;
        double r101116 = r101106 * r101106;
        double r101117 = r101113 * r101113;
        double r101118 = r101116 + r101117;
        double r101119 = r101115 / r101118;
        return r101119;
}
↓
double f(double re, double im, double base) {
        double r101120 = -1.0;
        double r101121 = im;
        double r101122 = re;
        double r101123 = atan2(r101121, r101122);
        double r101124 = base;
        double r101125 = cbrt(r101124);
        double r101126 = r101125 * r101125;
        double r101127 = 0.3333333333333333;
        double r101128 = -r101127;
        double r101129 = pow(r101126, r101128);
        double r101130 = log(r101129);
        double r101131 = 0.6666666666666666;
        double r101132 = log(r101124);
        double r101133 = r101131 * r101132;
        double r101134 = r101130 - r101133;
        double r101135 = pow(r101125, r101128);
        double r101136 = log(r101135);
        double r101137 = r101134 + r101136;
        double r101138 = r101123 / r101137;
        double r101139 = r101120 * r101138;
        return r101139;
}

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 32.0

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

  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-cube-cbrt0.3

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

  6. Applied times-frac0.3

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

  7. Applied log-prod0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied pow1/30.3

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

  12. Applied pow-flip0.3

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

  14. Applied add-cube-cbrt0.3

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

  15. Applied unpow-prod-down0.3

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

  16. Applied log-prod0.4

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

  17. Applied associate-+r+0.4

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

  18. Simplified0.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2020062 
(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))))