Average Error: 32.0 → 0.4
Time: 4.1s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}\]
\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 r30875 = im;
        double r30876 = re;
        double r30877 = atan2(r30875, r30876);
        double r30878 = base;
        double r30879 = log(r30878);
        double r30880 = r30877 * r30879;
        double r30881 = r30876 * r30876;
        double r30882 = r30875 * r30875;
        double r30883 = r30881 + r30882;
        double r30884 = sqrt(r30883);
        double r30885 = log(r30884);
        double r30886 = 0.0;
        double r30887 = r30885 * r30886;
        double r30888 = r30880 - r30887;
        double r30889 = r30879 * r30879;
        double r30890 = r30886 * r30886;
        double r30891 = r30889 + r30890;
        double r30892 = r30888 / r30891;
        return r30892;
}
↓
double f(double re, double im, double base) {
        double r30893 = im;
        double r30894 = re;
        double r30895 = atan2(r30893, r30894);
        double r30896 = 1.0;
        double r30897 = base;
        double r30898 = log(r30897);
        double r30899 = r30896 / r30898;
        double r30900 = r30895 * r30899;
        return r30900;
}

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 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 2020003 +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))))