Average Error: 7.3 → 0.2
Time: 22.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.im\]
\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(x.im \cdot 3\right) \cdot x.re\right) \cdot x.im
double f(double x_re, double x_im) {
        double r119913 = x_re;
        double r119914 = r119913 * r119913;
        double r119915 = x_im;
        double r119916 = r119915 * r119915;
        double r119917 = r119914 - r119916;
        double r119918 = r119917 * r119913;
        double r119919 = r119913 * r119915;
        double r119920 = r119915 * r119913;
        double r119921 = r119919 + r119920;
        double r119922 = r119921 * r119915;
        double r119923 = r119918 - r119922;
        return r119923;
}
↓
double f(double x_re, double x_im) {
        double r119924 = x_re;
        double r119925 = 3.0;
        double r119926 = pow(r119924, r119925);
        double r119927 = x_im;
        double r119928 = r119927 * r119925;
        double r119929 = r119928 * r119924;
        double r119930 = r119929 * r119927;
        double r119931 = r119926 - r119930;
        return r119931;
}

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

Original7.3
Target0.3
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 7.3

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

    \[\leadsto {x.re}^{3} - \color{blue}{\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im}\]
  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. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019306 
(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)))