math.cube on complex, imaginary part

Average Error: 6.9 → 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 r8409744 = x_re;
        double r8409745 = r8409744 * r8409744;
        double r8409746 = x_im;
        double r8409747 = r8409746 * r8409746;
        double r8409748 = r8409745 - r8409747;
        double r8409749 = r8409748 * r8409746;
        double r8409750 = r8409744 * r8409746;
        double r8409751 = r8409746 * r8409744;
        double r8409752 = r8409750 + r8409751;
        double r8409753 = r8409752 * r8409744;
        double r8409754 = r8409749 + r8409753;
        return r8409754;
}

↓

double f(double x_re, double x_im) {
        double r8409755 = x_re;
        double r8409756 = x_im;
        double r8409757 = r8409755 - r8409756;
        double r8409758 = r8409757 * r8409756;
        double r8409759 = r8409756 + r8409755;
        double r8409760 = r8409758 * r8409759;
        double r8409761 = r8409755 * r8409756;
        double r8409762 = r8409761 + r8409761;
        double r8409763 = r8409755 * r8409762;
        double r8409764 = r8409760 + r8409763;
        return r8409764;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.9
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 6.9

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

3. Applied difference-of-squares 6.9

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

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 2019137 
(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)))