Average Error: 7.5 → 0.2
Time: 3.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(3, \left(x.re \cdot x.im\right) \cdot \left(-x.im\right), {x.re}^{3}\right)\]
\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
↓
\mathsf{fma}\left(3, \left(x.re \cdot x.im\right) \cdot \left(-x.im\right), {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r299632 = x_re;
        double r299633 = r299632 * r299632;
        double r299634 = x_im;
        double r299635 = r299634 * r299634;
        double r299636 = r299633 - r299635;
        double r299637 = r299636 * r299632;
        double r299638 = r299632 * r299634;
        double r299639 = r299634 * r299632;
        double r299640 = r299638 + r299639;
        double r299641 = r299640 * r299634;
        double r299642 = r299637 - r299641;
        return r299642;
}
↓
double f(double x_re, double x_im) {
        double r299643 = 3.0;
        double r299644 = x_re;
        double r299645 = x_im;
        double r299646 = r299644 * r299645;
        double r299647 = -r299645;
        double r299648 = r299646 * r299647;
        double r299649 = pow(r299644, r299643);
        double r299650 = fma(r299643, r299648, r299649);
        return r299650;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.5
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 7.5

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

  2. Simplified7.5

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

  4. Applied distribute-rgt-neg-in7.5

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

  5. Applied associate-*r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))