Average Error: 7.2 → 0.2
Time: 2.8s
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 r298278 = x_re;
        double r298279 = r298278 * r298278;
        double r298280 = x_im;
        double r298281 = r298280 * r298280;
        double r298282 = r298279 - r298281;
        double r298283 = r298282 * r298278;
        double r298284 = r298278 * r298280;
        double r298285 = r298280 * r298278;
        double r298286 = r298284 + r298285;
        double r298287 = r298286 * r298280;
        double r298288 = r298283 - r298287;
        return r298288;
}
↓
double f(double x_re, double x_im) {
        double r298289 = 3.0;
        double r298290 = x_re;
        double r298291 = x_im;
        double r298292 = r298290 * r298291;
        double r298293 = -r298291;
        double r298294 = r298292 * r298293;
        double r298295 = pow(r298290, r298289);
        double r298296 = fma(r298289, r298294, r298295);
        return r298296;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.2
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 7.2

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

  2. Simplified7.1

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

  4. Applied distribute-rgt-neg-in7.1

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

  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 2020049 +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)))