math.cube on complex, real part

Average Error: 7.0 → 0.2

Time: 23.8s

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 r9983873 = x_re;
        double r9983874 = r9983873 * r9983873;
        double r9983875 = x_im;
        double r9983876 = r9983875 * r9983875;
        double r9983877 = r9983874 - r9983876;
        double r9983878 = r9983877 * r9983873;
        double r9983879 = r9983873 * r9983875;
        double r9983880 = r9983875 * r9983873;
        double r9983881 = r9983879 + r9983880;
        double r9983882 = r9983881 * r9983875;
        double r9983883 = r9983878 - r9983882;
        return r9983883;
}

↓

double f(double x_re, double x_im) {
        double r9983884 = x_re;
        double r9983885 = x_im;
        double r9983886 = r9983884 - r9983885;
        double r9983887 = r9983886 * r9983884;
        double r9983888 = r9983885 + r9983884;
        double r9983889 = r9983887 * r9983888;
        double r9983890 = r9983884 * r9983885;
        double r9983891 = r9983890 + r9983890;
        double r9983892 = r9983891 * r9983885;
        double r9983893 = r9983889 - r9983892;
        return r9983893;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
Target	0.3
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 7.0

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

3. Applied difference-of-squares 7.0

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

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 2019158 
(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)))