math.cube on complex, real part

Average Error: 7.1 → 0.3

Time: 22.5s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r8202748 = x_re;
        double r8202749 = r8202748 * r8202748;
        double r8202750 = x_im;
        double r8202751 = r8202750 * r8202750;
        double r8202752 = r8202749 - r8202751;
        double r8202753 = r8202752 * r8202748;
        double r8202754 = r8202748 * r8202750;
        double r8202755 = r8202750 * r8202748;
        double r8202756 = r8202754 + r8202755;
        double r8202757 = r8202756 * r8202750;
        double r8202758 = r8202753 - r8202757;
        return r8202758;
}

↓

double f(double x_re, double x_im) {
        double r8202759 = x_re;
        double r8202760 = x_im;
        double r8202761 = r8202759 - r8202760;
        double r8202762 = r8202761 * r8202759;
        double r8202763 = r8202760 + r8202759;
        double r8202764 = r8202762 * r8202763;
        double r8202765 = r8202759 * r8202760;
        double r8202766 = r8202765 + r8202765;
        double r8202767 = r8202766 * r8202760;
        double r8202768 = r8202764 - r8202767;
        return r8202768;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
Target	0.3
Herbie	0.3

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

1. Initial program 7.1

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

3. Applied difference-of-squares 7.1

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

4. Applied associate-*l* 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))