math.cube on complex, imaginary part

Average Error: 6.9 → 0.2

Time: 22.3s

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 r181964 = x_re;
        double r181965 = r181964 * r181964;
        double r181966 = x_im;
        double r181967 = r181966 * r181966;
        double r181968 = r181965 - r181967;
        double r181969 = r181968 * r181966;
        double r181970 = r181964 * r181966;
        double r181971 = r181966 * r181964;
        double r181972 = r181970 + r181971;
        double r181973 = r181972 * r181964;
        double r181974 = r181969 + r181973;
        return r181974;
}

↓

double f(double x_re, double x_im) {
        double r181975 = x_re;
        double r181976 = x_im;
        double r181977 = r181975 + r181976;
        double r181978 = r181975 - r181976;
        double r181979 = r181978 * r181976;
        double r181980 = r181977 * r181979;
        double r181981 = r181975 * r181976;
        double r181982 = r181976 * r181975;
        double r181983 = r181981 + r181982;
        double r181984 = r181983 * r181975;
        double r181985 = r181980 + r181984;
        return r181985;
}

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)))