Average Error: 7.3 → 0.2
Time: 28.5s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(x.re, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)\]
\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, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)
double f(double x_re, double x_im) {
        double r165920 = x_re;
        double r165921 = r165920 * r165920;
        double r165922 = x_im;
        double r165923 = r165922 * r165922;
        double r165924 = r165921 - r165923;
        double r165925 = r165924 * r165922;
        double r165926 = r165920 * r165922;
        double r165927 = r165922 * r165920;
        double r165928 = r165926 + r165927;
        double r165929 = r165928 * r165920;
        double r165930 = r165925 + r165929;
        return r165930;
}
↓
double f(double x_re, double x_im) {
        double r165931 = x_re;
        double r165932 = 3.0;
        double r165933 = x_im;
        double r165934 = r165932 * r165933;
        double r165935 = r165934 * r165931;
        double r165936 = pow(r165933, r165932);
        double r165937 = -r165936;
        double r165938 = fma(r165931, r165935, r165937);
        return r165938;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.3
Target0.3
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 7.3

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, 3 \cdot \left(x.im \cdot x.re\right), -{x.im}^{3}\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*0.2

    \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\left(3 \cdot x.im\right) \cdot x.re}, -{x.im}^{3}\right)\]

  5. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(x.re, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :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)))