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

Average Error: 0.5 → 0.4

Time: 24.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r148419 = 2.0;
        double r148420 = sqrt(r148419);
        double r148421 = x;
        double r148422 = sin(r148421);
        double r148423 = y;
        double r148424 = sin(r148423);
        double r148425 = 16.0;
        double r148426 = r148424 / r148425;
        double r148427 = r148422 - r148426;
        double r148428 = r148420 * r148427;
        double r148429 = r148422 / r148425;
        double r148430 = r148424 - r148429;
        double r148431 = r148428 * r148430;
        double r148432 = cos(r148421);
        double r148433 = cos(r148423);
        double r148434 = r148432 - r148433;
        double r148435 = r148431 * r148434;
        double r148436 = r148419 + r148435;
        double r148437 = 3.0;
        double r148438 = 1.0;
        double r148439 = 5.0;
        double r148440 = sqrt(r148439);
        double r148441 = r148440 - r148438;
        double r148442 = r148441 / r148419;
        double r148443 = r148442 * r148432;
        double r148444 = r148438 + r148443;
        double r148445 = r148437 - r148440;
        double r148446 = r148445 / r148419;
        double r148447 = r148446 * r148433;
        double r148448 = r148444 + r148447;
        double r148449 = r148437 * r148448;
        double r148450 = r148436 / r148449;
        return r148450;
}

↓

double f(double x, double y) {
        double r148451 = 2.0;
        double r148452 = sqrt(r148451);
        double r148453 = sqrt(r148452);
        double r148454 = x;
        double r148455 = sin(r148454);
        double r148456 = r148455 * r148455;
        double r148457 = y;
        double r148458 = sin(r148457);
        double r148459 = 16.0;
        double r148460 = r148458 / r148459;
        double r148461 = r148460 * r148460;
        double r148462 = r148456 - r148461;
        double r148463 = r148453 * r148462;
        double r148464 = r148453 * r148463;
        double r148465 = r148455 / r148459;
        double r148466 = r148458 - r148465;
        double r148467 = r148464 * r148466;
        double r148468 = cos(r148454);
        double r148469 = 3.0;
        double r148470 = pow(r148468, r148469);
        double r148471 = cos(r148457);
        double r148472 = pow(r148471, r148469);
        double r148473 = r148470 - r148472;
        double r148474 = r148467 * r148473;
        double r148475 = r148471 + r148468;
        double r148476 = r148471 * r148475;
        double r148477 = r148468 * r148468;
        double r148478 = r148476 + r148477;
        double r148479 = r148455 + r148460;
        double r148480 = r148478 * r148479;
        double r148481 = r148474 / r148480;
        double r148482 = r148451 + r148481;
        double r148483 = 3.0;
        double r148484 = 1.0;
        double r148485 = 5.0;
        double r148486 = sqrt(r148485);
        double r148487 = r148486 - r148484;
        double r148488 = r148487 / r148451;
        double r148489 = r148488 * r148468;
        double r148490 = r148484 + r148489;
        double r148491 = r148483 * r148483;
        double r148492 = -r148485;
        double r148493 = r148491 + r148492;
        double r148494 = r148483 + r148486;
        double r148495 = r148493 / r148494;
        double r148496 = r148495 / r148451;
        double r148497 = r148496 * r148471;
        double r148498 = r148490 + r148497;
        double r148499 = r148483 * r148498;
        double r148500 = r148482 / r148499;
        return r148500;
}

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

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

6. Applied flip3-- 0.5

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

7. Applied flip-- 0.5

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

8. Applied associate-*r/ 0.5

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

9. Applied associate-*l/ 0.5

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

10. Applied frac-times 0.5

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

11. Simplified 0.5

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

13. Applied add-sqr-sqrt 0.5

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

14. Applied sqrt-prod 0.5

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

15. Applied associate-*l* 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 1978988140 
(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))))))