Average Error: 7.3 → 0.2
Time: 33.3s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.re, \left(\left(-x.im\right) \cdot x.re + \left(-x.im\right) \cdot x.re\right) \cdot x.im\right)\]
\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 + x.re, \left(x.re - x.im\right) \cdot x.re, \left(\left(-x.im\right) \cdot x.re + \left(-x.im\right) \cdot x.re\right) \cdot x.im\right)
double f(double x_re, double x_im) {
        double r4060417 = x_re;
        double r4060418 = r4060417 * r4060417;
        double r4060419 = x_im;
        double r4060420 = r4060419 * r4060419;
        double r4060421 = r4060418 - r4060420;
        double r4060422 = r4060421 * r4060417;
        double r4060423 = r4060417 * r4060419;
        double r4060424 = r4060419 * r4060417;
        double r4060425 = r4060423 + r4060424;
        double r4060426 = r4060425 * r4060419;
        double r4060427 = r4060422 - r4060426;
        return r4060427;
}
↓
double f(double x_re, double x_im) {
        double r4060428 = x_im;
        double r4060429 = x_re;
        double r4060430 = r4060428 + r4060429;
        double r4060431 = r4060429 - r4060428;
        double r4060432 = r4060431 * r4060429;
        double r4060433 = -r4060428;
        double r4060434 = r4060433 * r4060429;
        double r4060435 = r4060434 + r4060434;
        double r4060436 = r4060435 * r4060428;
        double r4060437 = fma(r4060430, r4060432, r4060436);
        return r4060437;
}

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. Using strategy rm

  3. Applied difference-of-squares7.3

    \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]

  4. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
  5. Using strategy rm

  6. Applied fma-neg0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :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)))