math.cube on complex, imaginary part

Average Error: 7.1 → 0.3

Time: 24.8s

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 r7951602 = x_re;
        double r7951603 = r7951602 * r7951602;
        double r7951604 = x_im;
        double r7951605 = r7951604 * r7951604;
        double r7951606 = r7951603 - r7951605;
        double r7951607 = r7951606 * r7951604;
        double r7951608 = r7951602 * r7951604;
        double r7951609 = r7951604 * r7951602;
        double r7951610 = r7951608 + r7951609;
        double r7951611 = r7951610 * r7951602;
        double r7951612 = r7951607 + r7951611;
        return r7951612;
}

↓

double f(double x_re, double x_im) {
        double r7951613 = x_re;
        double r7951614 = x_im;
        double r7951615 = r7951613 - r7951614;
        double r7951616 = r7951615 * r7951614;
        double r7951617 = r7951614 + r7951613;
        double r7951618 = r7951616 * r7951617;
        double r7951619 = r7951613 * r7951614;
        double r7951620 = r7951619 + r7951619;
        double r7951621 = r7951613 * r7951620;
        double r7951622 = r7951618 + r7951621;
        return r7951622;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
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 7.1

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

3. Applied difference-of-squares 7.1

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

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 2019163 +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)))