Average Error: 6.5 → 0.2
Time: 37.1s
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 r3418647 = x_re;
        double r3418648 = r3418647 * r3418647;
        double r3418649 = x_im;
        double r3418650 = r3418649 * r3418649;
        double r3418651 = r3418648 - r3418650;
        double r3418652 = r3418651 * r3418649;
        double r3418653 = r3418647 * r3418649;
        double r3418654 = r3418649 * r3418647;
        double r3418655 = r3418653 + r3418654;
        double r3418656 = r3418655 * r3418647;
        double r3418657 = r3418652 + r3418656;
        return r3418657;
}
↓
double f(double x_re, double x_im) {
        double r3418658 = x_im;
        double r3418659 = x_re;
        double r3418660 = r3418658 + r3418659;
        double r3418661 = r3418659 - r3418658;
        double r3418662 = r3418661 * r3418658;
        double r3418663 = r3418659 * r3418658;
        double r3418664 = r3418663 + r3418663;
        double r3418665 = r3418664 * r3418659;
        double r3418666 = fma(r3418660, r3418662, r3418665);
        return r3418666;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.5
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.5

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

  3. Applied difference-of-squares6.5

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

  4. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
  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 2019139 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 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)))