math.cube on complex, imaginary part

Average Error: 7.5 → 0.2

Time: 14.4s

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 r116301 = x_re;
        double r116302 = r116301 * r116301;
        double r116303 = x_im;
        double r116304 = r116303 * r116303;
        double r116305 = r116302 - r116304;
        double r116306 = r116305 * r116303;
        double r116307 = r116301 * r116303;
        double r116308 = r116303 * r116301;
        double r116309 = r116307 + r116308;
        double r116310 = r116309 * r116301;
        double r116311 = r116306 + r116310;
        return r116311;
}

↓

double f(double x_re, double x_im) {
        double r116312 = x_re;
        double r116313 = x_im;
        double r116314 = r116312 - r116313;
        double r116315 = r116314 * r116313;
        double r116316 = r116313 + r116312;
        double r116317 = r116315 * r116316;
        double r116318 = r116312 * r116313;
        double r116319 = r116318 + r116318;
        double r116320 = r116312 * r116319;
        double r116321 = r116317 + r116320;
        return r116321;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.5
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 7.5

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
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* 0.2

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

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 2019196 
(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)))