math.cube on complex, imaginary part

Average Error: 7.4 → 0.3

Time: 23.9s

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 r304299 = x_re;
        double r304300 = r304299 * r304299;
        double r304301 = x_im;
        double r304302 = r304301 * r304301;
        double r304303 = r304300 - r304302;
        double r304304 = r304303 * r304301;
        double r304305 = r304299 * r304301;
        double r304306 = r304301 * r304299;
        double r304307 = r304305 + r304306;
        double r304308 = r304307 * r304299;
        double r304309 = r304304 + r304308;
        return r304309;
}

↓

double f(double x_re, double x_im) {
        double r304310 = x_re;
        double r304311 = x_im;
        double r304312 = r304310 + r304311;
        double r304313 = r304310 - r304311;
        double r304314 = r304313 * r304311;
        double r304315 = r304312 * r304314;
        double r304316 = r304310 * r304311;
        double r304317 = r304311 * r304310;
        double r304318 = r304316 + r304317;
        double r304319 = r304318 * r304310;
        double r304320 = r304315 + r304319;
        return r304320;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.4
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 7.4

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

3. Applied difference-of-squares 7.4

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

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 2019322 
(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)))