Average Error: 7.3 → 0.2
Time: 2.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.im\]
\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 r209666 = x_re;
        double r209667 = r209666 * r209666;
        double r209668 = x_im;
        double r209669 = r209668 * r209668;
        double r209670 = r209667 - r209669;
        double r209671 = r209670 * r209666;
        double r209672 = r209666 * r209668;
        double r209673 = r209668 * r209666;
        double r209674 = r209672 + r209673;
        double r209675 = r209674 * r209668;
        double r209676 = r209671 - r209675;
        return r209676;
}
↓
double f(double x_re, double x_im) {
        double r209677 = x_re;
        double r209678 = 3.0;
        double r209679 = pow(r209677, r209678);
        double r209680 = x_im;
        double r209681 = r209678 * r209680;
        double r209682 = r209681 * r209677;
        double r209683 = r209682 * r209680;
        double r209684 = r209679 - r209683;
        return r209684;
}

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

    \[\leadsto \color{blue}{{x.re}^{3} - 3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)}\]
  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 2019356 
(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)))