Average Error: 7.5 → 0.2
Time: 4.0s
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 r206325 = x_re;
        double r206326 = r206325 * r206325;
        double r206327 = x_im;
        double r206328 = r206327 * r206327;
        double r206329 = r206326 - r206328;
        double r206330 = r206329 * r206325;
        double r206331 = r206325 * r206327;
        double r206332 = r206327 * r206325;
        double r206333 = r206331 + r206332;
        double r206334 = r206333 * r206327;
        double r206335 = r206330 - r206334;
        return r206335;
}
↓
double f(double x_re, double x_im) {
        double r206336 = 3.0;
        double r206337 = x_re;
        double r206338 = x_im;
        double r206339 = r206337 * r206338;
        double r206340 = -r206338;
        double r206341 = r206339 * r206340;
        double r206342 = pow(r206337, r206336);
        double r206343 = fma(r206336, r206341, r206342);
        return r206343;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.5
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 7.5

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

  2. Simplified7.4

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

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

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

  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 2020062 +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)))