math.cube on complex, imaginary part

Average Error: 7.5 → 0.3

Time: 13.6s

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 r173584 = x_re;
        double r173585 = r173584 * r173584;
        double r173586 = x_im;
        double r173587 = r173586 * r173586;
        double r173588 = r173585 - r173587;
        double r173589 = r173588 * r173586;
        double r173590 = r173584 * r173586;
        double r173591 = r173586 * r173584;
        double r173592 = r173590 + r173591;
        double r173593 = r173592 * r173584;
        double r173594 = r173589 + r173593;
        return r173594;
}

↓

double f(double x_re, double x_im) {
        double r173595 = x_re;
        double r173596 = x_im;
        double r173597 = r173595 - r173596;
        double r173598 = r173597 * r173596;
        double r173599 = r173596 + r173595;
        double r173600 = r173598 * r173599;
        double r173601 = r173595 * r173596;
        double r173602 = r173601 + r173601;
        double r173603 = r173595 * r173602;
        double r173604 = r173600 + r173603;
        return r173604;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.5
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 7.5

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

3. Applied difference-of-squares 7.5

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

4. Applied associate-*l* 0.3

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

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 2019174 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))