math.cube on complex, imaginary part

Average Error: 7.3 → 0.2

Time: 1.5m

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r7107926 = x_re;
        double r7107927 = r7107926 * r7107926;
        double r7107928 = x_im;
        double r7107929 = r7107928 * r7107928;
        double r7107930 = r7107927 - r7107929;
        double r7107931 = r7107930 * r7107928;
        double r7107932 = r7107926 * r7107928;
        double r7107933 = r7107928 * r7107926;
        double r7107934 = r7107932 + r7107933;
        double r7107935 = r7107934 * r7107926;
        double r7107936 = r7107931 + r7107935;
        return r7107936;
}

↓

double f(double x_re, double x_im) {
        double r7107937 = x_re;
        double r7107938 = 3.0;
        double r7107939 = r7107937 * r7107938;
        double r7107940 = x_im;
        double r7107941 = r7107939 * r7107940;
        double r7107942 = r7107941 * r7107937;
        double r7107943 = pow(r7107940, r7107938);
        double r7107944 = r7107942 - r7107943;
        return r7107944;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.3
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 7.3

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

3. Applied difference-of-squares 7.3

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

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. Taylor expanded around 0 7.3

   \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot {x.re}^{2}\right) - {x.im}^{3}}\]

6. Simplified 0.3

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

8. Applied pow1 0.3

   \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - x.im \cdot \left(x.im \cdot \color{blue}{{x.im}^{1}}\right)\]

9. Applied pow1 0.3

   \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - x.im \cdot \left(\color{blue}{{x.im}^{1}} \cdot {x.im}^{1}\right)\]

10. Applied pow-prod-up 0.3

    \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - x.im \cdot \color{blue}{{x.im}^{\left(1 + 1\right)}}\]

11. Applied pow1 0.3

    \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - \color{blue}{{x.im}^{1}} \cdot {x.im}^{\left(1 + 1\right)}\]

12. Applied pow-prod-up 0.2

    \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - \color{blue}{{x.im}^{\left(1 + \left(1 + 1\right)\right)}}\]

13. Simplified 0.2

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

15. Applied associate-*l* 0.2

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

16. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))