Average Error: 32.2 → 0.3
Time: 7.7s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left({\left(\frac{1}{base}\right)}^{\frac{-1}{3}}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]
\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(-2\right) \cdot \log \left({\left(\frac{1}{base}\right)}^{\frac{-1}{3}}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)}
double f(double re, double im, double base) {
        double r52632 = im;
        double r52633 = re;
        double r52634 = atan2(r52632, r52633);
        double r52635 = base;
        double r52636 = log(r52635);
        double r52637 = r52634 * r52636;
        double r52638 = r52633 * r52633;
        double r52639 = r52632 * r52632;
        double r52640 = r52638 + r52639;
        double r52641 = sqrt(r52640);
        double r52642 = log(r52641);
        double r52643 = 0.0;
        double r52644 = r52642 * r52643;
        double r52645 = r52637 - r52644;
        double r52646 = r52636 * r52636;
        double r52647 = r52643 * r52643;
        double r52648 = r52646 + r52647;
        double r52649 = r52645 / r52648;
        return r52649;
}
↓
double f(double re, double im, double base) {
        double r52650 = -1.0;
        double r52651 = im;
        double r52652 = re;
        double r52653 = atan2(r52651, r52652);
        double r52654 = 2.0;
        double r52655 = -r52654;
        double r52656 = 1.0;
        double r52657 = base;
        double r52658 = r52656 / r52657;
        double r52659 = -0.3333333333333333;
        double r52660 = pow(r52658, r52659);
        double r52661 = log(r52660);
        double r52662 = r52655 * r52661;
        double r52663 = cbrt(r52657);
        double r52664 = r52656 / r52663;
        double r52665 = log(r52664);
        double r52666 = r52662 + r52665;
        double r52667 = r52653 / r52666;
        double r52668 = r52650 * r52667;
        return r52668;
}

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 32.2

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

  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. Taylor expanded around inf 0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019318 
(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))))