Average Error: 7.5 → 0.2
Time: 29.1s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\]
\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.re + x.im, \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 r17052310 = x_re;
        double r17052311 = r17052310 * r17052310;
        double r17052312 = x_im;
        double r17052313 = r17052312 * r17052312;
        double r17052314 = r17052311 - r17052313;
        double r17052315 = r17052314 * r17052312;
        double r17052316 = r17052310 * r17052312;
        double r17052317 = r17052312 * r17052310;
        double r17052318 = r17052316 + r17052317;
        double r17052319 = r17052318 * r17052310;
        double r17052320 = r17052315 + r17052319;
        return r17052320;
}
↓
double f(double x_re, double x_im) {
        double r17052321 = x_re;
        double r17052322 = x_im;
        double r17052323 = r17052321 + r17052322;
        double r17052324 = r17052321 - r17052322;
        double r17052325 = r17052324 * r17052322;
        double r17052326 = r17052321 * r17052322;
        double r17052327 = r17052326 + r17052326;
        double r17052328 = r17052327 * r17052321;
        double r17052329 = fma(r17052323, r17052325, r17052328);
        return r17052329;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.5
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 7.5

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

  3. Applied difference-of-squares7.5

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

  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

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

Reproduce

herbie shell --seed 2019173 +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)))