Average Error: 7.7 → 0.2
Time: 3.8s
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 r196470 = x_re;
        double r196471 = r196470 * r196470;
        double r196472 = x_im;
        double r196473 = r196472 * r196472;
        double r196474 = r196471 - r196473;
        double r196475 = r196474 * r196470;
        double r196476 = r196470 * r196472;
        double r196477 = r196472 * r196470;
        double r196478 = r196476 + r196477;
        double r196479 = r196478 * r196472;
        double r196480 = r196475 - r196479;
        return r196480;
}
↓
double f(double x_re, double x_im) {
        double r196481 = 3.0;
        double r196482 = x_re;
        double r196483 = x_im;
        double r196484 = r196482 * r196483;
        double r196485 = -r196483;
        double r196486 = r196484 * r196485;
        double r196487 = pow(r196482, r196481);
        double r196488 = fma(r196481, r196486, r196487);
        return r196488;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.7
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 7.7

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

  2. Simplified7.7

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

  4. Applied distribute-rgt-neg-in7.7

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

  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 2020060 +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)))