Average Error: 7.4 → 0.2
Time: 11.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - x.im \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\]
\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(3 \cdot \left(x.im \cdot x.re\right)\right)
double f(double x_re, double x_im) {
        double r139552 = x_re;
        double r139553 = r139552 * r139552;
        double r139554 = x_im;
        double r139555 = r139554 * r139554;
        double r139556 = r139553 - r139555;
        double r139557 = r139556 * r139552;
        double r139558 = r139552 * r139554;
        double r139559 = r139554 * r139552;
        double r139560 = r139558 + r139559;
        double r139561 = r139560 * r139554;
        double r139562 = r139557 - r139561;
        return r139562;
}
↓
double f(double x_re, double x_im) {
        double r139563 = x_re;
        double r139564 = 3.0;
        double r139565 = pow(r139563, r139564);
        double r139566 = x_im;
        double r139567 = r139566 * r139563;
        double r139568 = r139564 * r139567;
        double r139569 = r139566 * r139568;
        double r139570 = r139565 - r139569;
        return r139570;
}

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
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.4

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

  2. Simplified7.4

    \[\leadsto \color{blue}{{x.re}^{3} - \left(x.im \cdot x.im\right) \cdot \left(3 \cdot x.re\right)}\]
  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. Taylor expanded around 0 0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019212 
(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)))