math.cube on complex, imaginary part

Average Error: 7.3 → 0.3

Time: 25.0s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r178321 = x_re;
        double r178322 = r178321 * r178321;
        double r178323 = x_im;
        double r178324 = r178323 * r178323;
        double r178325 = r178322 - r178324;
        double r178326 = r178325 * r178323;
        double r178327 = r178321 * r178323;
        double r178328 = r178323 * r178321;
        double r178329 = r178327 + r178328;
        double r178330 = r178329 * r178321;
        double r178331 = r178326 + r178330;
        return r178331;
}

↓

double f(double x_re, double x_im) {
        double r178332 = x_re;
        double r178333 = x_im;
        double r178334 = r178332 + r178333;
        double r178335 = r178332 - r178333;
        double r178336 = r178335 * r178333;
        double r178337 = r178334 * r178336;
        double r178338 = r178332 * r178333;
        double r178339 = r178333 * r178332;
        double r178340 = r178338 + r178339;
        double r178341 = r178340 * r178332;
        double r178342 = r178337 + r178341;
        return r178342;
}

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)))