Average Error: 7.4 → 0.2
Time: 2.1s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - {x.im}^{3}\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
↓
\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - {x.im}^{3}
double f(double x_re, double x_im) {
        double r226611 = x_re;
        double r226612 = r226611 * r226611;
        double r226613 = x_im;
        double r226614 = r226613 * r226613;
        double r226615 = r226612 - r226614;
        double r226616 = r226615 * r226613;
        double r226617 = r226611 * r226613;
        double r226618 = r226613 * r226611;
        double r226619 = r226617 + r226618;
        double r226620 = r226619 * r226611;
        double r226621 = r226616 + r226620;
        return r226621;
}
↓
double f(double x_re, double x_im) {
        double r226622 = 3.0;
        double r226623 = x_im;
        double r226624 = r226622 * r226623;
        double r226625 = x_re;
        double r226626 = r226624 * r226625;
        double r226627 = r226626 * r226625;
        double r226628 = pow(r226623, r226622);
        double r226629 = r226627 - r226628;
        return r226629;
}

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

Original7.4
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 7.4

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

  2. Simplified7.3

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

  4. Applied associate-*r*0.2

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

  6. Applied associate-*r*0.2

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

  8. Applied associate-*r*0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020024 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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