math.cube on complex, imaginary part

Average Error: 6.9 → 0.2

Time: 17.9s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r2256865 = x_re;
        double r2256866 = r2256865 * r2256865;
        double r2256867 = x_im;
        double r2256868 = r2256867 * r2256867;
        double r2256869 = r2256866 - r2256868;
        double r2256870 = r2256869 * r2256867;
        double r2256871 = r2256865 * r2256867;
        double r2256872 = r2256867 * r2256865;
        double r2256873 = r2256871 + r2256872;
        double r2256874 = r2256873 * r2256865;
        double r2256875 = r2256870 + r2256874;
        return r2256875;
}

↓

double f(double x_re, double x_im) {
        double r2256876 = x_im;
        double r2256877 = x_re;
        double r2256878 = r2256876 + r2256877;
        double r2256879 = r2256877 - r2256876;
        double r2256880 = r2256879 * r2256876;
        double r2256881 = r2256877 * r2256876;
        double r2256882 = r2256881 + r2256881;
        double r2256883 = r2256882 * r2256877;
        double r2256884 = fma(r2256878, r2256880, r2256883);
        return r2256884;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.9
Target	0.3
Herbie	0.2

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

Derivation

1. Initial program 6.9

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

3. Applied difference-of-squares 6.9

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

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-def 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019129 +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)))