math.cube on complex, imaginary part

Average Error: 7.4 → 0.2

Time: 21.0s

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 r10215791 = x_re;
        double r10215792 = r10215791 * r10215791;
        double r10215793 = x_im;
        double r10215794 = r10215793 * r10215793;
        double r10215795 = r10215792 - r10215794;
        double r10215796 = r10215795 * r10215793;
        double r10215797 = r10215791 * r10215793;
        double r10215798 = r10215793 * r10215791;
        double r10215799 = r10215797 + r10215798;
        double r10215800 = r10215799 * r10215791;
        double r10215801 = r10215796 + r10215800;
        return r10215801;
}

↓

double f(double x_re, double x_im) {
        double r10215802 = x_re;
        double r10215803 = x_im;
        double r10215804 = r10215802 - r10215803;
        double r10215805 = r10215804 * r10215803;
        double r10215806 = r10215803 + r10215802;
        double r10215807 = r10215805 * r10215806;
        double r10215808 = r10215802 * r10215803;
        double r10215809 = r10215808 + r10215808;
        double r10215810 = r10215802 * r10215809;
        double r10215811 = r10215807 + r10215810;
        return r10215811;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.4
Target	0.2
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 7.4

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

3. Applied difference-of-squares 7.4

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

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. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019170 
(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)))