math.cube on complex, imaginary part

Average Error: 6.8 → 0.3

Time: 41.8s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r54694853 = x_re;
        double r54694854 = r54694853 * r54694853;
        double r54694855 = x_im;
        double r54694856 = r54694855 * r54694855;
        double r54694857 = r54694854 - r54694856;
        double r54694858 = r54694857 * r54694855;
        double r54694859 = r54694853 * r54694855;
        double r54694860 = r54694855 * r54694853;
        double r54694861 = r54694859 + r54694860;
        double r54694862 = r54694861 * r54694853;
        double r54694863 = r54694858 + r54694862;
        return r54694863;
}

↓

double f(double x_re, double x_im) {
        double r54694864 = x_re;
        double r54694865 = x_im;
        double r54694866 = r54694864 - r54694865;
        double r54694867 = r54694866 * r54694865;
        double r54694868 = r54694865 + r54694864;
        double r54694869 = r54694867 * r54694868;
        double r54694870 = r54694864 * r54694865;
        double r54694871 = r54694870 + r54694870;
        double r54694872 = r54694864 * r54694871;
        double r54694873 = r54694869 + r54694872;
        return r54694873;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.8
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 6.8

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

3. Applied difference-of-squares 6.8

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

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. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019104 
(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)))