Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, D

Average Error: 1.3 → 0.3

Time: 12.2s

Precision: 64

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

Math

\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

↓

\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{{\left(\cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)\right)}^{3} \cdot \frac{1}{3}}\]

C

double f(double x, double y, double z, double t) {
        double r894451 = 1.0;
        double r894452 = 3.0;
        double r894453 = r894451 / r894452;
        double r894454 = x;
        double r894455 = y;
        double r894456 = 27.0;
        double r894457 = r894455 * r894456;
        double r894458 = r894454 / r894457;
        double r894459 = r894452 * r894458;
        double r894460 = z;
        double r894461 = 2.0;
        double r894462 = r894460 * r894461;
        double r894463 = r894459 / r894462;
        double r894464 = t;
        double r894465 = sqrt(r894464);
        double r894466 = r894463 * r894465;
        double r894467 = acos(r894466);
        double r894468 = r894453 * r894467;
        return r894468;
}

↓

double f(double x, double y, double z, double t) {
        double r894469 = 1.0;
        double r894470 = cbrt(r894469);
        double r894471 = r894470 * r894470;
        double r894472 = 3.0;
        double r894473 = cbrt(r894472);
        double r894474 = r894473 * r894473;
        double r894475 = r894471 / r894474;
        double r894476 = x;
        double r894477 = r894472 * r894476;
        double r894478 = z;
        double r894479 = 2.0;
        double r894480 = r894478 * r894479;
        double r894481 = y;
        double r894482 = 27.0;
        double r894483 = r894481 * r894482;
        double r894484 = r894480 * r894483;
        double r894485 = r894477 / r894484;
        double r894486 = t;
        double r894487 = sqrt(r894486);
        double r894488 = r894485 * r894487;
        double r894489 = acos(r894488);
        double r894490 = 3.0;
        double r894491 = pow(r894489, r894490);
        double r894492 = r894469 / r894472;
        double r894493 = r894491 * r894492;
        double r894494 = cbrt(r894493);
        double r894495 = r894475 * r894494;
        return r894495;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.3
Target	1.2
Herbie	0.3

\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

Derivation

1. Initial program 1.3

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt 1.3

   \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

4. Applied add-cube-cbrt 1.3

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

5. Applied times-frac 0.4

   \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{3}}\right)} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

6. Applied associate-*l* 0.3

   \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)}\]
7. Using strategy rm

8. Applied associate-*r/ 0.4

   \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{3 \cdot x}{y \cdot 27}}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\]

9. Applied associate-/l/ 0.3

   \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\color{blue}{\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)}} \cdot \sqrt{t}\right)\right)\]
10. Using strategy rm

11. Applied add-cbrt-cube 1.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)}}\right)\]

12. Applied cbrt-undiv 0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\color{blue}{\sqrt[3]{\frac{1}{3}}} \cdot \sqrt[3]{\left(\cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)}\right)\]

13. Applied cbrt-unprod 0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\sqrt[3]{\frac{1}{3} \cdot \left(\left(\cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)\right)}}\]

14. Simplified 0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{\color{blue}{{\left(\cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)\right)}^{3} \cdot \frac{1}{3}}}\]

15. Final simplification 0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{{\left(\cos^{-1} \left(\frac{3 \cdot x}{\left(z \cdot 2\right) \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)\right)}^{3} \cdot \frac{1}{3}}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3)

  (* (/ 1 3) (acos (* (/ (* 3 (/ x (* y 27))) (* z 2)) (sqrt t)))))