Average Error: 7.6 → 0.2
Time: 2.1s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.im\]
\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
↓
{x.re}^{3} - \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.im
double f(double x_re, double x_im) {
        double r224883 = x_re;
        double r224884 = r224883 * r224883;
        double r224885 = x_im;
        double r224886 = r224885 * r224885;
        double r224887 = r224884 - r224886;
        double r224888 = r224887 * r224883;
        double r224889 = r224883 * r224885;
        double r224890 = r224885 * r224883;
        double r224891 = r224889 + r224890;
        double r224892 = r224891 * r224885;
        double r224893 = r224888 - r224892;
        return r224893;
}
↓
double f(double x_re, double x_im) {
        double r224894 = x_re;
        double r224895 = 3.0;
        double r224896 = pow(r224894, r224895);
        double r224897 = x_im;
        double r224898 = r224895 * r224897;
        double r224899 = r224898 * r224894;
        double r224900 = r224899 * r224897;
        double r224901 = r224896 - r224900;
        return r224901;
}

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

Original7.6
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 7.6

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

    \[\leadsto {x.re}^{3} - \color{blue}{\left(3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}\]
  5. Using strategy rm

  6. Applied associate-*r*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :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)))