Average Error: 32.2 → 0.3
Time: 5.1s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]
\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}}{\log base}
double f(double re, double im, double base) {
        double r35127 = im;
        double r35128 = re;
        double r35129 = atan2(r35127, r35128);
        double r35130 = base;
        double r35131 = log(r35130);
        double r35132 = r35129 * r35131;
        double r35133 = r35128 * r35128;
        double r35134 = r35127 * r35127;
        double r35135 = r35133 + r35134;
        double r35136 = sqrt(r35135);
        double r35137 = log(r35136);
        double r35138 = 0.0;
        double r35139 = r35137 * r35138;
        double r35140 = r35132 - r35139;
        double r35141 = r35131 * r35131;
        double r35142 = r35138 * r35138;
        double r35143 = r35141 + r35142;
        double r35144 = r35140 / r35143;
        return r35144;
}
↓
double f(double re, double im, double base) {
        double r35145 = im;
        double r35146 = re;
        double r35147 = atan2(r35145, r35146);
        double r35148 = base;
        double r35149 = log(r35148);
        double r35150 = r35147 / r35149;
        return r35150;
}

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 0 0.3

    \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}}\]

  3. Final simplification0.3

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]

Reproduce

herbie shell --seed 2020057 
(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))))