Average Error: 32.4 → 0.3
Time: 4.7s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]
\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 r80225 = im;
        double r80226 = re;
        double r80227 = atan2(r80225, r80226);
        double r80228 = base;
        double r80229 = log(r80228);
        double r80230 = r80227 * r80229;
        double r80231 = r80226 * r80226;
        double r80232 = r80225 * r80225;
        double r80233 = r80231 + r80232;
        double r80234 = sqrt(r80233);
        double r80235 = log(r80234);
        double r80236 = 0.0;
        double r80237 = r80235 * r80236;
        double r80238 = r80230 - r80237;
        double r80239 = r80229 * r80229;
        double r80240 = r80236 * r80236;
        double r80241 = r80239 + r80240;
        double r80242 = r80238 / r80241;
        return r80242;
}
↓
double f(double re, double im, double base) {
        double r80243 = im;
        double r80244 = re;
        double r80245 = atan2(r80243, r80244);
        double r80246 = base;
        double r80247 = log(r80246);
        double r80248 = r80245 / r80247;
        return r80248;
}

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 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 2020083 +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))))