math.cube on complex, imaginary part

Average Error: 6.9 → 0.2

Time: 23.4s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r164291 = x_re;
        double r164292 = r164291 * r164291;
        double r164293 = x_im;
        double r164294 = r164293 * r164293;
        double r164295 = r164292 - r164294;
        double r164296 = r164295 * r164293;
        double r164297 = r164291 * r164293;
        double r164298 = r164293 * r164291;
        double r164299 = r164297 + r164298;
        double r164300 = r164299 * r164291;
        double r164301 = r164296 + r164300;
        return r164301;
}

↓

double f(double x_re, double x_im) {
        double r164302 = x_re;
        double r164303 = x_im;
        double r164304 = r164302 + r164303;
        double r164305 = r164302 - r164303;
        double r164306 = r164305 * r164303;
        double r164307 = r164304 * r164306;
        double r164308 = r164302 * r164303;
        double r164309 = r164303 * r164302;
        double r164310 = r164308 + r164309;
        double r164311 = r164310 * r164302;
        double r164312 = r164307 + r164311;
        return r164312;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.9
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 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* 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019303 
(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)))