Average Error: 6.9 → 0.2
Time: 16.1s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im\]
\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
↓
\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im
double f(double x_re, double x_im) {
        double r2670740 = x_re;
        double r2670741 = r2670740 * r2670740;
        double r2670742 = x_im;
        double r2670743 = r2670742 * r2670742;
        double r2670744 = r2670741 - r2670743;
        double r2670745 = r2670744 * r2670740;
        double r2670746 = r2670740 * r2670742;
        double r2670747 = r2670742 * r2670740;
        double r2670748 = r2670746 + r2670747;
        double r2670749 = r2670748 * r2670742;
        double r2670750 = r2670745 - r2670749;
        return r2670750;
}
↓
double f(double x_re, double x_im) {
        double r2670751 = x_re;
        double r2670752 = x_im;
        double r2670753 = r2670751 - r2670752;
        double r2670754 = r2670753 * r2670751;
        double r2670755 = r2670752 + r2670751;
        double r2670756 = r2670754 * r2670755;
        double r2670757 = r2670751 * r2670752;
        double r2670758 = r2670757 + r2670757;
        double r2670759 = r2670758 * r2670752;
        double r2670760 = r2670756 - r2670759;
        return r2670760;
}

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

Original6.9
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 6.9

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

  2. Taylor expanded around 0 6.8

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :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)))