math.cube on complex, imaginary part

Average Error: 7.4 → 0.2

Time: 2.8s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r201581 = x_re;
        double r201582 = r201581 * r201581;
        double r201583 = x_im;
        double r201584 = r201583 * r201583;
        double r201585 = r201582 - r201584;
        double r201586 = r201585 * r201583;
        double r201587 = r201581 * r201583;
        double r201588 = r201583 * r201581;
        double r201589 = r201587 + r201588;
        double r201590 = r201589 * r201581;
        double r201591 = r201586 + r201590;
        return r201591;
}

↓

double f(double x_re, double x_im) {
        double r201592 = x_re;
        double r201593 = 3.0;
        double r201594 = r201592 * r201593;
        double r201595 = x_im;
        double r201596 = r201595 * r201592;
        double r201597 = r201594 * r201596;
        double r201598 = 1.0;
        double r201599 = pow(r201597, r201598);
        double r201600 = pow(r201595, r201593);
        double r201601 = r201599 - r201600;
        return r201601;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.4
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 7.4

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

2. Simplified 7.4

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

4. Applied associate-*r* 0.2

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

6. Applied associate-*r* 0.2

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

8. Applied add-cube-cbrt 0.2

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

9. Applied associate-*l* 0.3

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

11. Applied pow1 0.3

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

12. Applied pow1 0.3

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

13. Applied pow1 0.3

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

14. Applied pow-prod-down 0.3

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

15. Applied pow1 0.3

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

16. Applied pow-prod-down 0.3

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

17. Applied pow1 0.3

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

18. Applied pow1 0.3

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

19. Applied pow-prod-down 0.3

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

20. Applied pow-prod-down 0.3

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

21. Applied pow-prod-down 0.3

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

22. Simplified 0.2

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

23. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020060 
(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)))