math.cube on complex, imaginary part

Average Error: 6.9 → 0.3

Time: 25.8s

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 r6721298 = x_re;
        double r6721299 = r6721298 * r6721298;
        double r6721300 = x_im;
        double r6721301 = r6721300 * r6721300;
        double r6721302 = r6721299 - r6721301;
        double r6721303 = r6721302 * r6721300;
        double r6721304 = r6721298 * r6721300;
        double r6721305 = r6721300 * r6721298;
        double r6721306 = r6721304 + r6721305;
        double r6721307 = r6721306 * r6721298;
        double r6721308 = r6721303 + r6721307;
        return r6721308;
}

↓

double f(double x_re, double x_im) {
        double r6721309 = x_re;
        double r6721310 = x_im;
        double r6721311 = r6721309 - r6721310;
        double r6721312 = r6721311 * r6721310;
        double r6721313 = r6721310 + r6721309;
        double r6721314 = r6721312 * r6721313;
        double r6721315 = r6721309 * r6721310;
        double r6721316 = r6721315 + r6721315;
        double r6721317 = r6721309 * r6721316;
        double r6721318 = r6721314 + r6721317;
        return r6721318;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.9
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 6.9

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

3. Applied difference-of-squares 6.9

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

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 2019165 +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)))