math.cube on complex, real part

Average Error: 7.2 → 0.2

Time: 14.3s

Precision: 64

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

Math

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

↓

\[{x.re}^{3} - \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.im\]

C

double f(double x_re, double x_im) {
        double r310929 = x_re;
        double r310930 = r310929 * r310929;
        double r310931 = x_im;
        double r310932 = r310931 * r310931;
        double r310933 = r310930 - r310932;
        double r310934 = r310933 * r310929;
        double r310935 = r310929 * r310931;
        double r310936 = r310931 * r310929;
        double r310937 = r310935 + r310936;
        double r310938 = r310937 * r310931;
        double r310939 = r310934 - r310938;
        return r310939;
}

↓

double f(double x_re, double x_im) {
        double r310940 = x_re;
        double r310941 = 3.0;
        double r310942 = pow(r310940, r310941);
        double r310943 = r310941 * r310940;
        double r310944 = x_im;
        double r310945 = r310943 * r310944;
        double r310946 = r310945 * r310944;
        double r310947 = r310942 - r310946;
        return r310947;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.2
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 7.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{{x.re}^{3} - 3 \cdot \left(\left(x.re \cdot x.im\right) \cdot x.im\right)}\]
3. Using strategy rm

4. Applied associate-*r* 0.2

   \[\leadsto {x.re}^{3} - \color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im}\]
5. Using strategy rm

6. Applied associate-*r* 0.2

   \[\leadsto {x.re}^{3} - \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.im\right)} \cdot x.im\]

7. Final simplification 0.2

   \[\leadsto {x.re}^{3} - \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.im\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :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)))