Average Error: 7.8 → 0.2
Time: 14.1s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - x.im \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)\]
\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} - x.im \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)
double f(double x_re, double x_im) {
        double r116772 = x_re;
        double r116773 = r116772 * r116772;
        double r116774 = x_im;
        double r116775 = r116774 * r116774;
        double r116776 = r116773 - r116775;
        double r116777 = r116776 * r116772;
        double r116778 = r116772 * r116774;
        double r116779 = r116774 * r116772;
        double r116780 = r116778 + r116779;
        double r116781 = r116780 * r116774;
        double r116782 = r116777 - r116781;
        return r116782;
}
↓
double f(double x_re, double x_im) {
        double r116783 = x_re;
        double r116784 = 3.0;
        double r116785 = pow(r116783, r116784);
        double r116786 = x_im;
        double r116787 = r116784 * r116783;
        double r116788 = r116786 * r116787;
        double r116789 = r116786 * r116788;
        double r116790 = r116785 - r116789;
        return r116790;
}

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

Original7.8
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 7.8

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

  2. Simplified7.7

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

  4. Applied associate-*l*0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019208 
(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)))