math.cube on complex, imaginary part

Average Error: 7.3 → 0.3

Time: 24.6s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r223632 = x_re;
        double r223633 = r223632 * r223632;
        double r223634 = x_im;
        double r223635 = r223634 * r223634;
        double r223636 = r223633 - r223635;
        double r223637 = r223636 * r223634;
        double r223638 = r223632 * r223634;
        double r223639 = r223634 * r223632;
        double r223640 = r223638 + r223639;
        double r223641 = r223640 * r223632;
        double r223642 = r223637 + r223641;
        return r223642;
}

↓

double f(double x_re, double x_im) {
        double r223643 = x_re;
        double r223644 = x_im;
        double r223645 = r223643 + r223644;
        double r223646 = r223643 - r223644;
        double r223647 = r223646 * r223644;
        double r223648 = r223645 * r223647;
        double r223649 = r223643 * r223644;
        double r223650 = r223644 * r223643;
        double r223651 = r223649 + r223650;
        double r223652 = r223651 * r223643;
        double r223653 = r223648 + r223652;
        return r223653;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.3
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 7.3

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

3. Applied difference-of-squares 7.3

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

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

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :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)))