Average Error: 7.3 → 0.2
Time: 23.1s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(x.im, \left(-3 \cdot x.im\right) \cdot x.re, {x.re}^{3}\right)\]
\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(x.im, \left(-3 \cdot x.im\right) \cdot x.re, {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r120467 = x_re;
        double r120468 = r120467 * r120467;
        double r120469 = x_im;
        double r120470 = r120469 * r120469;
        double r120471 = r120468 - r120470;
        double r120472 = r120471 * r120467;
        double r120473 = r120467 * r120469;
        double r120474 = r120469 * r120467;
        double r120475 = r120473 + r120474;
        double r120476 = r120475 * r120469;
        double r120477 = r120472 - r120476;
        return r120477;
}
↓
double f(double x_re, double x_im) {
        double r120478 = x_im;
        double r120479 = -3.0;
        double r120480 = r120479 * r120478;
        double r120481 = x_re;
        double r120482 = r120480 * r120481;
        double r120483 = 3.0;
        double r120484 = pow(r120481, r120483);
        double r120485 = fma(r120478, r120482, r120484);
        return r120485;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.3
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 7.3

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

  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 associate-*r*0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +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)))