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

Average Error: 0.5 → 0.5

Time: 38.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r197499 = 2.0;
        double r197500 = sqrt(r197499);
        double r197501 = x;
        double r197502 = sin(r197501);
        double r197503 = y;
        double r197504 = sin(r197503);
        double r197505 = 16.0;
        double r197506 = r197504 / r197505;
        double r197507 = r197502 - r197506;
        double r197508 = r197500 * r197507;
        double r197509 = r197502 / r197505;
        double r197510 = r197504 - r197509;
        double r197511 = r197508 * r197510;
        double r197512 = cos(r197501);
        double r197513 = cos(r197503);
        double r197514 = r197512 - r197513;
        double r197515 = r197511 * r197514;
        double r197516 = r197499 + r197515;
        double r197517 = 3.0;
        double r197518 = 1.0;
        double r197519 = 5.0;
        double r197520 = sqrt(r197519);
        double r197521 = r197520 - r197518;
        double r197522 = r197521 / r197499;
        double r197523 = r197522 * r197512;
        double r197524 = r197518 + r197523;
        double r197525 = r197517 - r197520;
        double r197526 = r197525 / r197499;
        double r197527 = r197526 * r197513;
        double r197528 = r197524 + r197527;
        double r197529 = r197517 * r197528;
        double r197530 = r197516 / r197529;
        return r197530;
}

↓

double f(double x, double y) {
        double r197531 = 2.0;
        double r197532 = sqrt(r197531);
        double r197533 = -r197532;
        double r197534 = x;
        double r197535 = sin(r197534);
        double r197536 = 2.0;
        double r197537 = pow(r197535, r197536);
        double r197538 = y;
        double r197539 = sin(r197538);
        double r197540 = pow(r197539, r197536);
        double r197541 = 16.0;
        double r197542 = r197540 / r197541;
        double r197543 = r197542 / r197541;
        double r197544 = r197537 - r197543;
        double r197545 = r197535 / r197541;
        double r197546 = r197539 - r197545;
        double r197547 = cos(r197538);
        double r197548 = r197546 * r197547;
        double r197549 = r197544 * r197548;
        double r197550 = r197533 * r197549;
        double r197551 = cos(r197534);
        double r197552 = r197544 * r197551;
        double r197553 = r197546 * r197532;
        double r197554 = r197552 * r197553;
        double r197555 = r197550 + r197554;
        double r197556 = r197539 / r197541;
        double r197557 = r197556 + r197535;
        double r197558 = r197555 / r197557;
        double r197559 = exp(r197558);
        double r197560 = log(r197559);
        double r197561 = r197560 + r197531;
        double r197562 = 3.0;
        double r197563 = 3.0;
        double r197564 = pow(r197562, r197563);
        double r197565 = 5.0;
        double r197566 = sqrt(r197565);
        double r197567 = r197566 * r197565;
        double r197568 = r197564 - r197567;
        double r197569 = r197562 * r197562;
        double r197570 = r197562 * r197566;
        double r197571 = r197569 + r197570;
        double r197572 = r197571 + r197565;
        double r197573 = r197568 / r197572;
        double r197574 = r197573 / r197531;
        double r197575 = r197547 * r197574;
        double r197576 = 1.0;
        double r197577 = r197566 - r197576;
        double r197578 = r197577 / r197531;
        double r197579 = r197578 * r197551;
        double r197580 = r197579 + r197576;
        double r197581 = r197575 + r197580;
        double r197582 = r197581 * r197562;
        double r197583 = r197561 / r197582;
        return r197583;
}

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 flip3-- 0.5

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

4. Simplified 0.5

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

5. Simplified 0.5

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

7. Applied flip-- 0.5

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

8. Applied associate-*r/ 0.5

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

9. Applied associate-*l/ 0.5

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

10. Applied associate-*l/ 0.5

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

11. Simplified 0.5

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

13. Applied add-log-exp 0.5

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

14. Simplified 0.5

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

16. Applied sub-neg 0.5

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

17. Applied distribute-lft-in 0.5

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

18. Applied distribute-lft-in 0.5

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

19. Applied distribute-lft-in 0.5

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

20. Simplified 0.5

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

21. Simplified 0.5

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

22. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019179 
(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))))))