math.cube on complex, imaginary part

Average Error: 7.7 → 0.2

Time: 23.0s

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 r8188677 = x_re;
        double r8188678 = r8188677 * r8188677;
        double r8188679 = x_im;
        double r8188680 = r8188679 * r8188679;
        double r8188681 = r8188678 - r8188680;
        double r8188682 = r8188681 * r8188679;
        double r8188683 = r8188677 * r8188679;
        double r8188684 = r8188679 * r8188677;
        double r8188685 = r8188683 + r8188684;
        double r8188686 = r8188685 * r8188677;
        double r8188687 = r8188682 + r8188686;
        return r8188687;
}

↓

double f(double x_re, double x_im) {
        double r8188688 = x_re;
        double r8188689 = x_im;
        double r8188690 = r8188688 - r8188689;
        double r8188691 = r8188690 * r8188689;
        double r8188692 = r8188689 + r8188688;
        double r8188693 = r8188691 * r8188692;
        double r8188694 = r8188688 * r8188689;
        double r8188695 = r8188694 + r8188694;
        double r8188696 = r8188688 * r8188695;
        double r8188697 = r8188693 + r8188696;
        return r8188697;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.7
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 7.7

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

3. Applied difference-of-squares 7.7

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

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 2019171 
(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)))