Average Error: 7.4 → 0.2
Time: 26.4s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(x.re, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)\]
\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
↓
\mathsf{fma}\left(x.re, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)
double f(double x_re, double x_im) {
        double r207712 = x_re;
        double r207713 = r207712 * r207712;
        double r207714 = x_im;
        double r207715 = r207714 * r207714;
        double r207716 = r207713 - r207715;
        double r207717 = r207716 * r207714;
        double r207718 = r207712 * r207714;
        double r207719 = r207714 * r207712;
        double r207720 = r207718 + r207719;
        double r207721 = r207720 * r207712;
        double r207722 = r207717 + r207721;
        return r207722;
}
↓
double f(double x_re, double x_im) {
        double r207723 = x_re;
        double r207724 = 3.0;
        double r207725 = x_im;
        double r207726 = r207724 * r207725;
        double r207727 = r207726 * r207723;
        double r207728 = pow(r207725, r207724);
        double r207729 = -r207728;
        double r207730 = fma(r207723, r207727, r207729);
        return r207730;
}

Error

Bits error versus x.re

Bits error versus x.im

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. Simplified0.2

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

  4. Applied associate-*r*0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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)))