math.cube on complex, imaginary part

Average Error: 7.0 → 0.2

Time: 17.6s

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 r12001687 = x_re;
        double r12001688 = r12001687 * r12001687;
        double r12001689 = x_im;
        double r12001690 = r12001689 * r12001689;
        double r12001691 = r12001688 - r12001690;
        double r12001692 = r12001691 * r12001689;
        double r12001693 = r12001687 * r12001689;
        double r12001694 = r12001689 * r12001687;
        double r12001695 = r12001693 + r12001694;
        double r12001696 = r12001695 * r12001687;
        double r12001697 = r12001692 + r12001696;
        return r12001697;
}

↓

double f(double x_re, double x_im) {
        double r12001698 = x_re;
        double r12001699 = x_im;
        double r12001700 = r12001698 - r12001699;
        double r12001701 = r12001700 * r12001699;
        double r12001702 = r12001699 + r12001698;
        double r12001703 = r12001701 * r12001702;
        double r12001704 = r12001698 * r12001699;
        double r12001705 = r12001704 + r12001704;
        double r12001706 = r12001698 * r12001705;
        double r12001707 = r12001703 + r12001706;
        return r12001707;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
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 7.0

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

3. Applied difference-of-squares 7.0

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

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 2019192 
(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)))