Average Error: 7.5 → 0.2
Time: 16.4s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.im\]
\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
↓
{x.re}^{3} - \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.im
double f(double x_re, double x_im) {
        double r253876 = x_re;
        double r253877 = r253876 * r253876;
        double r253878 = x_im;
        double r253879 = r253878 * r253878;
        double r253880 = r253877 - r253879;
        double r253881 = r253880 * r253876;
        double r253882 = r253876 * r253878;
        double r253883 = r253878 * r253876;
        double r253884 = r253882 + r253883;
        double r253885 = r253884 * r253878;
        double r253886 = r253881 - r253885;
        return r253886;
}
↓
double f(double x_re, double x_im) {
        double r253887 = x_re;
        double r253888 = 3.0;
        double r253889 = pow(r253887, r253888);
        double r253890 = r253888 * r253887;
        double r253891 = x_im;
        double r253892 = r253890 * r253891;
        double r253893 = r253892 * r253891;
        double r253894 = r253889 - r253893;
        return r253894;
}

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

Original7.5
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 7.5

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

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

  6. Applied associate-*r*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020043 
(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)))