math.cube on complex, real part

Average Error: 7.4 → 0.2

Time: 16.2s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r199315 = x_re;
        double r199316 = r199315 * r199315;
        double r199317 = x_im;
        double r199318 = r199317 * r199317;
        double r199319 = r199316 - r199318;
        double r199320 = r199319 * r199315;
        double r199321 = r199315 * r199317;
        double r199322 = r199317 * r199315;
        double r199323 = r199321 + r199322;
        double r199324 = r199323 * r199317;
        double r199325 = r199320 - r199324;
        return r199325;
}

↓

double f(double x_re, double x_im) {
        double r199326 = x_re;
        double r199327 = x_im;
        double r199328 = r199326 - r199327;
        double r199329 = r199328 * r199326;
        double r199330 = r199327 + r199326;
        double r199331 = r199329 * r199330;
        double r199332 = r199326 * r199327;
        double r199333 = r199332 + r199332;
        double r199334 = r199333 * r199327;
        double r199335 = r199331 - r199334;
        return r199335;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.4
Target	0.2
Herbie	0.2

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

1. Initial program 7.4

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

3. Applied difference-of-squares 7.4

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

4. Applied associate-*l* 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))