Average Error: 7.4 → 0.2
Time: 18.2s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(x.re + x.im, x.re \cdot \left(x.re - x.im\right), -x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\]
\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.re + x.im, x.re \cdot \left(x.re - x.im\right), -x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)
double f(double x_re, double x_im) {
        double r13115497 = x_re;
        double r13115498 = r13115497 * r13115497;
        double r13115499 = x_im;
        double r13115500 = r13115499 * r13115499;
        double r13115501 = r13115498 - r13115500;
        double r13115502 = r13115501 * r13115497;
        double r13115503 = r13115497 * r13115499;
        double r13115504 = r13115499 * r13115497;
        double r13115505 = r13115503 + r13115504;
        double r13115506 = r13115505 * r13115499;
        double r13115507 = r13115502 - r13115506;
        return r13115507;
}
↓
double f(double x_re, double x_im) {
        double r13115508 = x_re;
        double r13115509 = x_im;
        double r13115510 = r13115508 + r13115509;
        double r13115511 = r13115508 - r13115509;
        double r13115512 = r13115508 * r13115511;
        double r13115513 = r13115509 * r13115508;
        double r13115514 = r13115513 + r13115513;
        double r13115515 = r13115509 * r13115514;
        double r13115516 = -r13115515;
        double r13115517 = fma(r13115510, r13115512, r13115516);
        return r13115517;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.4
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 7.4

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
  2. Using strategy rm

  3. Applied difference-of-squares7.4

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

  4. Applied associate-*l*0.3

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

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

  9. Applied *-commutative0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 +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 (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))