Average Error: 7.4 → 0.2
Time: 57.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
↓
\mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r4669659 = x_re;
        double r4669660 = r4669659 * r4669659;
        double r4669661 = x_im;
        double r4669662 = r4669661 * r4669661;
        double r4669663 = r4669660 - r4669662;
        double r4669664 = r4669663 * r4669661;
        double r4669665 = r4669659 * r4669661;
        double r4669666 = r4669661 * r4669659;
        double r4669667 = r4669665 + r4669666;
        double r4669668 = r4669667 * r4669659;
        double r4669669 = r4669664 + r4669668;
        return r4669669;
}
↓
double f(double x_re, double x_im) {
        double r4669670 = x_im;
        double r4669671 = x_re;
        double r4669672 = r4669670 + r4669671;
        double r4669673 = r4669671 - r4669670;
        double r4669674 = r4669673 * r4669670;
        double r4669675 = r4669671 * r4669670;
        double r4669676 = r4669675 + r4669675;
        double r4669677 = r4669676 * r4669671;
        double r4669678 = fma(r4669672, r4669674, r4669677);
        return r4669678;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.4
Target0.3
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2.0 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 7.4

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

  3. Applied difference-of-squares7.4

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

  4. Applied associate-*l*0.3

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

  6. Applied fma-def0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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