Average Error: 6.9 → 0.2
Time: 18.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im
↓
\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im
double f(double x_re, double x_im) {
        double r2913907 = x_re;
        double r2913908 = r2913907 * r2913907;
        double r2913909 = x_im;
        double r2913910 = r2913909 * r2913909;
        double r2913911 = r2913908 - r2913910;
        double r2913912 = r2913911 * r2913907;
        double r2913913 = r2913907 * r2913909;
        double r2913914 = r2913909 * r2913907;
        double r2913915 = r2913913 + r2913914;
        double r2913916 = r2913915 * r2913909;
        double r2913917 = r2913912 - r2913916;
        return r2913917;
}
↓
double f(double x_re, double x_im) {
        double r2913918 = x_re;
        double r2913919 = x_im;
        double r2913920 = r2913918 - r2913919;
        double r2913921 = r2913920 * r2913918;
        double r2913922 = r2913919 + r2913918;
        double r2913923 = r2913921 * r2913922;
        double r2913924 = r2913918 * r2913919;
        double r2913925 = r2913924 + r2913924;
        double r2913926 = r2913925 * r2913919;
        double r2913927 = r2913923 - r2913926;
        return r2913927;
}

Error

Bits error versus x.re

Bits error versus x.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.9
Target0.2
Herbie0.2
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 6.9

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]

  2. Taylor expanded around inf 6.8

    \[\leadsto \color{blue}{\left({x.re}^{3} - {x.im}^{2} \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]

  4. Final simplification0.2

    \[\leadsto \left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))