Average Error: 31.2 → 0.4
Time: 4.7s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}\]
\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}
↓
\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}
double f(double re, double im, double base) {
        double r39243 = im;
        double r39244 = re;
        double r39245 = atan2(r39243, r39244);
        double r39246 = base;
        double r39247 = log(r39246);
        double r39248 = r39245 * r39247;
        double r39249 = r39244 * r39244;
        double r39250 = r39243 * r39243;
        double r39251 = r39249 + r39250;
        double r39252 = sqrt(r39251);
        double r39253 = log(r39252);
        double r39254 = 0.0;
        double r39255 = r39253 * r39254;
        double r39256 = r39248 - r39255;
        double r39257 = r39247 * r39247;
        double r39258 = r39254 * r39254;
        double r39259 = r39257 + r39258;
        double r39260 = r39256 / r39259;
        return r39260;
}
↓
double f(double re, double im, double base) {
        double r39261 = im;
        double r39262 = re;
        double r39263 = atan2(r39261, r39262);
        double r39264 = 1.0;
        double r39265 = base;
        double r39266 = log(r39265);
        double r39267 = r39264 / r39266;
        double r39268 = r39263 * r39267;
        return r39268;
}

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 31.2

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

  2. Taylor expanded around 0 0.3

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

  4. Applied div-inv0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(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))))