Average Error: 7.3 → 0.2
Time: 25.4s
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 r103282 = x_re;
        double r103283 = r103282 * r103282;
        double r103284 = x_im;
        double r103285 = r103284 * r103284;
        double r103286 = r103283 - r103285;
        double r103287 = r103286 * r103282;
        double r103288 = r103282 * r103284;
        double r103289 = r103284 * r103282;
        double r103290 = r103288 + r103289;
        double r103291 = r103290 * r103284;
        double r103292 = r103287 - r103291;
        return r103292;
}
↓
double f(double x_re, double x_im) {
        double r103293 = x_im;
        double r103294 = -3.0;
        double r103295 = r103294 * r103293;
        double r103296 = x_re;
        double r103297 = r103295 * r103296;
        double r103298 = 3.0;
        double r103299 = pow(r103296, r103298);
        double r103300 = fma(r103293, r103297, r103299);
        return r103300;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.3
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 7.3

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

  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 2019306 +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)))