math.cube on complex, imaginary part

Average Error: 7.1 → 0.3

Time: 26.1s

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 r8273463 = x_re;
        double r8273464 = r8273463 * r8273463;
        double r8273465 = x_im;
        double r8273466 = r8273465 * r8273465;
        double r8273467 = r8273464 - r8273466;
        double r8273468 = r8273467 * r8273465;
        double r8273469 = r8273463 * r8273465;
        double r8273470 = r8273465 * r8273463;
        double r8273471 = r8273469 + r8273470;
        double r8273472 = r8273471 * r8273463;
        double r8273473 = r8273468 + r8273472;
        return r8273473;
}

↓

double f(double x_re, double x_im) {
        double r8273474 = x_re;
        double r8273475 = x_im;
        double r8273476 = r8273474 - r8273475;
        double r8273477 = r8273476 * r8273475;
        double r8273478 = r8273475 + r8273474;
        double r8273479 = r8273477 * r8273478;
        double r8273480 = r8273474 * r8273475;
        double r8273481 = r8273480 + r8273480;
        double r8273482 = r8273474 * r8273481;
        double r8273483 = r8273479 + r8273482;
        return r8273483;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 7.1

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

3. Applied difference-of-squares 7.1

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

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 2019163 +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)))