math.cube on complex, imaginary part

Average Error: 7.5 → 0.2

Time: 14.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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 r219237 = x_re;
        double r219238 = r219237 * r219237;
        double r219239 = x_im;
        double r219240 = r219239 * r219239;
        double r219241 = r219238 - r219240;
        double r219242 = r219241 * r219239;
        double r219243 = r219237 * r219239;
        double r219244 = r219239 * r219237;
        double r219245 = r219243 + r219244;
        double r219246 = r219245 * r219237;
        double r219247 = r219242 + r219246;
        return r219247;
}

↓

double f(double x_re, double x_im) {
        double r219248 = x_re;
        double r219249 = x_im;
        double r219250 = r219248 + r219249;
        double r219251 = r219248 - r219249;
        double r219252 = r219251 * r219249;
        double r219253 = r219250 * r219252;
        double r219254 = r219248 * r219249;
        double r219255 = r219249 * r219248;
        double r219256 = r219254 + r219255;
        double r219257 = r219256 * r219248;
        double r219258 = r219253 + r219257;
        return r219258;
}

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

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

Reproduce

herbie shell --seed 2019350 
(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)))