math.cube on complex, imaginary part

Average Error: 6.7 → 0.2

Time: 22.2s

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 r9999757 = x_re;
        double r9999758 = r9999757 * r9999757;
        double r9999759 = x_im;
        double r9999760 = r9999759 * r9999759;
        double r9999761 = r9999758 - r9999760;
        double r9999762 = r9999761 * r9999759;
        double r9999763 = r9999757 * r9999759;
        double r9999764 = r9999759 * r9999757;
        double r9999765 = r9999763 + r9999764;
        double r9999766 = r9999765 * r9999757;
        double r9999767 = r9999762 + r9999766;
        return r9999767;
}

↓

double f(double x_re, double x_im) {
        double r9999768 = x_re;
        double r9999769 = x_im;
        double r9999770 = r9999768 - r9999769;
        double r9999771 = r9999770 * r9999769;
        double r9999772 = r9999769 + r9999768;
        double r9999773 = r9999771 * r9999772;
        double r9999774 = r9999768 * r9999769;
        double r9999775 = r9999774 + r9999774;
        double r9999776 = r9999768 * r9999775;
        double r9999777 = r9999773 + r9999776;
        return r9999777;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.7
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 6.7

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

3. Applied difference-of-squares 6.7

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

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 2019158 
(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)))