math.cube on complex, real part

Average Error: 6.9 → 0.2

Time: 15.3s

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 r3808290 = x_re;
        double r3808291 = r3808290 * r3808290;
        double r3808292 = x_im;
        double r3808293 = r3808292 * r3808292;
        double r3808294 = r3808291 - r3808293;
        double r3808295 = r3808294 * r3808290;
        double r3808296 = r3808290 * r3808292;
        double r3808297 = r3808292 * r3808290;
        double r3808298 = r3808296 + r3808297;
        double r3808299 = r3808298 * r3808292;
        double r3808300 = r3808295 - r3808299;
        return r3808300;
}

↓

double f(double x_re, double x_im) {
        double r3808301 = x_re;
        double r3808302 = x_im;
        double r3808303 = r3808301 - r3808302;
        double r3808304 = r3808303 * r3808301;
        double r3808305 = r3808302 + r3808301;
        double r3808306 = r3808304 * r3808305;
        double r3808307 = r3808301 * r3808302;
        double r3808308 = r3808307 + r3808307;
        double r3808309 = r3808308 * r3808302;
        double r3808310 = r3808306 - r3808309;
        return r3808310;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.9
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 6.9

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

3. Applied difference-of-squares 6.9

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

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 2019153 
(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 x.im))))

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