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

Average Error: 0.5 → 0.5

Time: 11.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r208500 = 2.0;
        double r208501 = sqrt(r208500);
        double r208502 = x;
        double r208503 = sin(r208502);
        double r208504 = y;
        double r208505 = sin(r208504);
        double r208506 = 16.0;
        double r208507 = r208505 / r208506;
        double r208508 = r208503 - r208507;
        double r208509 = r208501 * r208508;
        double r208510 = r208503 / r208506;
        double r208511 = r208505 - r208510;
        double r208512 = r208509 * r208511;
        double r208513 = cos(r208502);
        double r208514 = cos(r208504);
        double r208515 = r208513 - r208514;
        double r208516 = r208512 * r208515;
        double r208517 = r208500 + r208516;
        double r208518 = 3.0;
        double r208519 = 1.0;
        double r208520 = 5.0;
        double r208521 = sqrt(r208520);
        double r208522 = r208521 - r208519;
        double r208523 = r208522 / r208500;
        double r208524 = r208523 * r208513;
        double r208525 = r208519 + r208524;
        double r208526 = r208518 - r208521;
        double r208527 = r208526 / r208500;
        double r208528 = r208527 * r208514;
        double r208529 = r208525 + r208528;
        double r208530 = r208518 * r208529;
        double r208531 = r208517 / r208530;
        return r208531;
}

↓

double f(double x, double y) {
        double r208532 = 2.0;
        double r208533 = sqrt(r208532);
        double r208534 = x;
        double r208535 = sin(r208534);
        double r208536 = r208535 * r208535;
        double r208537 = y;
        double r208538 = sin(r208537);
        double r208539 = 16.0;
        double r208540 = r208538 / r208539;
        double r208541 = r208540 * r208540;
        double r208542 = r208536 - r208541;
        double r208543 = r208533 * r208542;
        double r208544 = cbrt(r208543);
        double r208545 = r208544 * r208544;
        double r208546 = r208545 * r208544;
        double r208547 = r208535 + r208540;
        double r208548 = r208546 / r208547;
        double r208549 = r208535 / r208539;
        double r208550 = r208538 - r208549;
        double r208551 = r208548 * r208550;
        double r208552 = cos(r208534);
        double r208553 = cos(r208537);
        double r208554 = r208552 - r208553;
        double r208555 = r208551 * r208554;
        double r208556 = r208532 + r208555;
        double r208557 = 3.0;
        double r208558 = 1.0;
        double r208559 = 5.0;
        double r208560 = sqrt(r208559);
        double r208561 = sqrt(r208560);
        double r208562 = sqrt(r208558);
        double r208563 = r208561 + r208562;
        double r208564 = 1.0;
        double r208565 = r208563 / r208564;
        double r208566 = r208561 - r208562;
        double r208567 = r208566 / r208532;
        double r208568 = r208567 * r208552;
        double r208569 = r208565 * r208568;
        double r208570 = r208558 + r208569;
        double r208571 = r208557 * r208557;
        double r208572 = -r208559;
        double r208573 = r208571 + r208572;
        double r208574 = r208557 + r208560;
        double r208575 = r208573 / r208574;
        double r208576 = r208575 / r208532;
        double r208577 = r208576 * r208553;
        double r208578 = r208570 + r208577;
        double r208579 = r208557 * r208578;
        double r208580 = r208556 / r208579;
        return r208580;
}

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 0.5

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

6. Applied *-un-lft-identity 0.5

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

7. Applied add-sqr-sqrt 0.5

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

8. Applied add-sqr-sqrt 0.5

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

9. Applied sqrt-prod 0.9

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

10. Applied difference-of-squares 0.5

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

11. Applied times-frac 0.5

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

12. Applied associate-*l* 0.5

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

14. Applied flip-- 0.5

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

15. Applied associate-*r/ 0.5

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

17. Applied add-cube-cbrt 0.5

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

18. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020057 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2 (* (* (* (sqrt 2) (- (sin x) (/ (sin y) 16))) (- (sin y) (/ (sin x) 16))) (- (cos x) (cos y)))) (* 3 (+ (+ 1 (* (/ (- (sqrt 5) 1) 2) (cos x))) (* (/ (- 3 (sqrt 5)) 2) (cos y))))))