math.cube on complex, imaginary part

Average Error: 6.5 → 0.2

Time: 46.1s

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 r33633659 = x_re;
        double r33633660 = r33633659 * r33633659;
        double r33633661 = x_im;
        double r33633662 = r33633661 * r33633661;
        double r33633663 = r33633660 - r33633662;
        double r33633664 = r33633663 * r33633661;
        double r33633665 = r33633659 * r33633661;
        double r33633666 = r33633661 * r33633659;
        double r33633667 = r33633665 + r33633666;
        double r33633668 = r33633667 * r33633659;
        double r33633669 = r33633664 + r33633668;
        return r33633669;
}

↓

double f(double x_re, double x_im) {
        double r33633670 = x_re;
        double r33633671 = x_im;
        double r33633672 = r33633670 - r33633671;
        double r33633673 = r33633672 * r33633671;
        double r33633674 = r33633671 + r33633670;
        double r33633675 = r33633673 * r33633674;
        double r33633676 = r33633670 * r33633671;
        double r33633677 = r33633676 + r33633676;
        double r33633678 = r33633670 * r33633677;
        double r33633679 = r33633675 + r33633678;
        return r33633679;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.5
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 6.5

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

3. Applied difference-of-squares 6.5

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

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 2019112 +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)))