math.cube on complex, imaginary part

Average Error: 6.8 → 0.3

Time: 24.2s

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 r10419924 = x_re;
        double r10419925 = r10419924 * r10419924;
        double r10419926 = x_im;
        double r10419927 = r10419926 * r10419926;
        double r10419928 = r10419925 - r10419927;
        double r10419929 = r10419928 * r10419926;
        double r10419930 = r10419924 * r10419926;
        double r10419931 = r10419926 * r10419924;
        double r10419932 = r10419930 + r10419931;
        double r10419933 = r10419932 * r10419924;
        double r10419934 = r10419929 + r10419933;
        return r10419934;
}

↓

double f(double x_re, double x_im) {
        double r10419935 = x_re;
        double r10419936 = x_im;
        double r10419937 = r10419935 - r10419936;
        double r10419938 = r10419937 * r10419936;
        double r10419939 = r10419936 + r10419935;
        double r10419940 = r10419938 * r10419939;
        double r10419941 = r10419935 * r10419936;
        double r10419942 = r10419941 + r10419941;
        double r10419943 = r10419935 * r10419942;
        double r10419944 = r10419940 + r10419943;
        return r10419944;
}

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 2019151 
(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)))