Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5

Average Error: 0.5 → 0.5

Time: 41.9s

Precision: 64

Math

\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

↓

\[\frac{2 + \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x - \cos y \cdot \left(\cos y \cdot \cos y\right)\right)\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt[3]{\left(\left(\frac{\sin y}{16} + \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\left(\frac{\sin y}{16} + \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\frac{\sin y}{16} + \sin x\right) \cdot \sqrt{2}\right)\right)}\right)}{\left(\left(\cos x + \cos y\right) \cdot \cos y + \cos x \cdot \cos x\right) \cdot \left(\frac{\sin y}{16} + \sin x\right)}}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} - 1}{2}\right) + \frac{\frac{3 \cdot 3 - 5}{\sqrt{5} + 3}}{2} \cdot \cos y\right)}\]

C

double f(double x, double y) {
        double r11814453 = 2.0;
        double r11814454 = sqrt(r11814453);
        double r11814455 = x;
        double r11814456 = sin(r11814455);
        double r11814457 = y;
        double r11814458 = sin(r11814457);
        double r11814459 = 16.0;
        double r11814460 = r11814458 / r11814459;
        double r11814461 = r11814456 - r11814460;
        double r11814462 = r11814454 * r11814461;
        double r11814463 = r11814456 / r11814459;
        double r11814464 = r11814458 - r11814463;
        double r11814465 = r11814462 * r11814464;
        double r11814466 = cos(r11814455);
        double r11814467 = cos(r11814457);
        double r11814468 = r11814466 - r11814467;
        double r11814469 = r11814465 * r11814468;
        double r11814470 = r11814453 + r11814469;
        double r11814471 = 3.0;
        double r11814472 = 1.0;
        double r11814473 = 5.0;
        double r11814474 = sqrt(r11814473);
        double r11814475 = r11814474 - r11814472;
        double r11814476 = r11814475 / r11814453;
        double r11814477 = r11814476 * r11814466;
        double r11814478 = r11814472 + r11814477;
        double r11814479 = r11814471 - r11814474;
        double r11814480 = r11814479 / r11814453;
        double r11814481 = r11814480 * r11814467;
        double r11814482 = r11814478 + r11814481;
        double r11814483 = r11814471 * r11814482;
        double r11814484 = r11814470 / r11814483;
        return r11814484;
}

↓

double f(double x, double y) {
        double r11814485 = 2.0;
        double r11814486 = y;
        double r11814487 = sin(r11814486);
        double r11814488 = x;
        double r11814489 = sin(r11814488);
        double r11814490 = 16.0;
        double r11814491 = r11814489 / r11814490;
        double r11814492 = r11814487 - r11814491;
        double r11814493 = cos(r11814488);
        double r11814494 = r11814493 * r11814493;
        double r11814495 = r11814494 * r11814493;
        double r11814496 = cos(r11814486);
        double r11814497 = r11814496 * r11814496;
        double r11814498 = r11814496 * r11814497;
        double r11814499 = r11814495 - r11814498;
        double r11814500 = r11814492 * r11814499;
        double r11814501 = r11814487 / r11814490;
        double r11814502 = r11814489 - r11814501;
        double r11814503 = r11814501 + r11814489;
        double r11814504 = sqrt(r11814485);
        double r11814505 = r11814503 * r11814504;
        double r11814506 = r11814505 * r11814505;
        double r11814507 = r11814505 * r11814506;
        double r11814508 = cbrt(r11814507);
        double r11814509 = r11814502 * r11814508;
        double r11814510 = r11814500 * r11814509;
        double r11814511 = r11814493 + r11814496;
        double r11814512 = r11814511 * r11814496;
        double r11814513 = r11814512 + r11814494;
        double r11814514 = r11814513 * r11814503;
        double r11814515 = r11814510 / r11814514;
        double r11814516 = r11814485 + r11814515;
        double r11814517 = 3.0;
        double r11814518 = 1.0;
        double r11814519 = 5.0;
        double r11814520 = sqrt(r11814519);
        double r11814521 = r11814520 - r11814518;
        double r11814522 = r11814521 / r11814485;
        double r11814523 = r11814493 * r11814522;
        double r11814524 = r11814518 + r11814523;
        double r11814525 = r11814517 * r11814517;
        double r11814526 = r11814525 - r11814519;
        double r11814527 = r11814520 + r11814517;
        double r11814528 = r11814526 / r11814527;
        double r11814529 = r11814528 / r11814485;
        double r11814530 = r11814529 * r11814496;
        double r11814531 = r11814524 + r11814530;
        double r11814532 = r11814517 * r11814531;
        double r11814533 = r11814516 / r11814532;
        return r11814533;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Initial program 0.5

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
2. Using strategy rm

3. Applied flip-- 0.5

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)}\]

4. Simplified 0.4

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{3 \cdot 3 - 5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
5. Using strategy rm

6. Applied add-sqr-sqrt 0.4

   \[\leadsto \frac{2 + \left(\left(\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

7. Applied sqrt-prod 0.4

   \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

8. Applied associate-*l* 0.4

   \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
9. Using strategy rm

10. Applied flip3-- 0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\frac{{\left(\cos x\right)}^{3} - {\left(\cos y\right)}^{3}}{\cos x \cdot \cos x + \left(\cos y \cdot \cos y + \cos x \cdot \cos y\right)}}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

11. Applied flip-- 0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \color{blue}{\frac{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}{\sin x + \frac{\sin y}{16}}}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\left(\cos x\right)}^{3} - {\left(\cos y\right)}^{3}}{\cos x \cdot \cos x + \left(\cos y \cdot \cos y + \cos x \cdot \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

12. Applied associate-*r/ 0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \color{blue}{\frac{\sqrt{\sqrt{2}} \cdot \left(\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}\right)}{\sin x + \frac{\sin y}{16}}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\left(\cos x\right)}^{3} - {\left(\cos y\right)}^{3}}{\cos x \cdot \cos x + \left(\cos y \cdot \cos y + \cos x \cdot \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

13. Applied associate-*r/ 0.5

    \[\leadsto \frac{2 + \left(\color{blue}{\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}\right)\right)}{\sin x + \frac{\sin y}{16}}} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\left(\cos x\right)}^{3} - {\left(\cos y\right)}^{3}}{\cos x \cdot \cos x + \left(\cos y \cdot \cos y + \cos x \cdot \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

14. Applied associate-*l/ 0.4

    \[\leadsto \frac{2 + \color{blue}{\frac{\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}{\sin x + \frac{\sin y}{16}}} \cdot \frac{{\left(\cos x\right)}^{3} - {\left(\cos y\right)}^{3}}{\cos x \cdot \cos x + \left(\cos y \cdot \cos y + \cos x \cdot \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

15. Applied frac-times 0.4

    \[\leadsto \frac{2 + \color{blue}{\frac{\left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left({\left(\cos x\right)}^{3} - {\left(\cos y\right)}^{3}\right)}{\left(\sin x + \frac{\sin y}{16}\right) \cdot \left(\cos x \cdot \cos x + \left(\cos y \cdot \cos y + \cos x \cdot \cos y\right)\right)}}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

16. Simplified 0.4

    \[\leadsto \frac{2 + \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{\sin y}{16} + \sin x\right)\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x \cdot \left(\cos x \cdot \cos x\right) - \cos y \cdot \left(\cos y \cdot \cos y\right)\right)\right)}}{\left(\sin x + \frac{\sin y}{16}\right) \cdot \left(\cos x \cdot \cos x + \left(\cos y \cdot \cos y + \cos x \cdot \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

17. Simplified 0.4

    \[\leadsto \frac{2 + \frac{\left(\left(\sqrt{2} \cdot \left(\frac{\sin y}{16} + \sin x\right)\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x \cdot \left(\cos x \cdot \cos x\right) - \cos y \cdot \left(\cos y \cdot \cos y\right)\right)\right)}{\color{blue}{\left(\cos x \cdot \cos x + \cos y \cdot \left(\cos y + \cos x\right)\right) \cdot \left(\frac{\sin y}{16} + \sin x\right)}}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
18. Using strategy rm

19. Applied add-cbrt-cube 0.5

    \[\leadsto \frac{2 + \frac{\left(\color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(\frac{\sin y}{16} + \sin x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} + \sin x\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} + \sin x\right)\right)}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x \cdot \left(\cos x \cdot \cos x\right) - \cos y \cdot \left(\cos y \cdot \cos y\right)\right)\right)}{\left(\cos x \cdot \cos x + \cos y \cdot \left(\cos y + \cos x\right)\right) \cdot \left(\frac{\sin y}{16} + \sin x\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

20. Final simplification 0.5

    \[\leadsto \frac{2 + \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x - \cos y \cdot \left(\cos y \cdot \cos y\right)\right)\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt[3]{\left(\left(\frac{\sin y}{16} + \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\left(\frac{\sin y}{16} + \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\frac{\sin y}{16} + \sin x\right) \cdot \sqrt{2}\right)\right)}\right)}{\left(\left(\cos x + \cos y\right) \cdot \cos y + \cos x \cdot \cos x\right) \cdot \left(\frac{\sin y}{16} + \sin x\right)}}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} - 1}{2}\right) + \frac{\frac{3 \cdot 3 - 5}{\sqrt{5} + 3}}{2} \cdot \cos y\right)}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  (/ (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (- (sin y) (/ (sin x) 16.0))) (- (cos x) (cos y)))) (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))