Average Error: 7.2 → 0.2
Time: 26.1s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(-3, \left(x.im \cdot x.re\right) \cdot x.im, {x.re}^{3}\right)\]
\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.im \cdot x.re\right) \cdot x.im, {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r96631 = x_re;
        double r96632 = r96631 * r96631;
        double r96633 = x_im;
        double r96634 = r96633 * r96633;
        double r96635 = r96632 - r96634;
        double r96636 = r96635 * r96631;
        double r96637 = r96631 * r96633;
        double r96638 = r96633 * r96631;
        double r96639 = r96637 + r96638;
        double r96640 = r96639 * r96633;
        double r96641 = r96636 - r96640;
        return r96641;
}
↓
double f(double x_re, double x_im) {
        double r96642 = -3.0;
        double r96643 = x_im;
        double r96644 = x_re;
        double r96645 = r96643 * r96644;
        double r96646 = r96645 * r96643;
        double r96647 = 3.0;
        double r96648 = pow(r96644, r96647);
        double r96649 = fma(r96642, r96646, r96648);
        return r96649;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.2
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 7.2

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

  2. Simplified0.2

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied associate-*l*0.2

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

  6. Taylor expanded around 0 7.1

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +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)))