Average Error: 31.1 → 0.4
Time: 1.2m
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log base}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}
↓
\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log base}
double f(double re, double im, double base) {
        double r1817890 = re;
        double r1817891 = r1817890 * r1817890;
        double r1817892 = im;
        double r1817893 = r1817892 * r1817892;
        double r1817894 = r1817891 + r1817893;
        double r1817895 = sqrt(r1817894);
        double r1817896 = log(r1817895);
        double r1817897 = base;
        double r1817898 = log(r1817897);
        double r1817899 = r1817896 * r1817898;
        double r1817900 = atan2(r1817892, r1817890);
        double r1817901 = 0.0;
        double r1817902 = r1817900 * r1817901;
        double r1817903 = r1817899 + r1817902;
        double r1817904 = r1817898 * r1817898;
        double r1817905 = r1817901 * r1817901;
        double r1817906 = r1817904 + r1817905;
        double r1817907 = r1817903 / r1817906;
        return r1817907;
}
↓
double f(double re, double im, double base) {
        double r1817908 = re;
        double r1817909 = im;
        double r1817910 = hypot(r1817908, r1817909);
        double r1817911 = log(r1817910);
        double r1817912 = 1.0;
        double r1817913 = base;
        double r1817914 = log(r1817913);
        double r1817915 = r1817912 / r1817914;
        double r1817916 = r1817911 * r1817915;
        return r1817916;
}

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 31.1

    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}}\]
  3. Using strategy rm

  4. Applied div-inv0.4

    \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log base}}\]

  5. Final simplification0.4

    \[\leadsto \log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log base}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))