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

Average Error: 0.5 → 0.5

Time: 13.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r157565 = 2.0;
        double r157566 = sqrt(r157565);
        double r157567 = x;
        double r157568 = sin(r157567);
        double r157569 = y;
        double r157570 = sin(r157569);
        double r157571 = 16.0;
        double r157572 = r157570 / r157571;
        double r157573 = r157568 - r157572;
        double r157574 = r157566 * r157573;
        double r157575 = r157568 / r157571;
        double r157576 = r157570 - r157575;
        double r157577 = r157574 * r157576;
        double r157578 = cos(r157567);
        double r157579 = cos(r157569);
        double r157580 = r157578 - r157579;
        double r157581 = r157577 * r157580;
        double r157582 = r157565 + r157581;
        double r157583 = 3.0;
        double r157584 = 1.0;
        double r157585 = 5.0;
        double r157586 = sqrt(r157585);
        double r157587 = r157586 - r157584;
        double r157588 = r157587 / r157565;
        double r157589 = r157588 * r157578;
        double r157590 = r157584 + r157589;
        double r157591 = r157583 - r157586;
        double r157592 = r157591 / r157565;
        double r157593 = r157592 * r157579;
        double r157594 = r157590 + r157593;
        double r157595 = r157583 * r157594;
        double r157596 = r157582 / r157595;
        return r157596;
}

↓

double f(double x, double y) {
        double r157597 = 2.0;
        double r157598 = sqrt(r157597);
        double r157599 = x;
        double r157600 = sin(r157599);
        double r157601 = cbrt(r157600);
        double r157602 = r157601 * r157601;
        double r157603 = r157602 * r157601;
        double r157604 = y;
        double r157605 = sin(r157604);
        double r157606 = cbrt(r157605);
        double r157607 = 3.0;
        double r157608 = pow(r157606, r157607);
        double r157609 = -r157608;
        double r157610 = 16.0;
        double r157611 = r157609 / r157610;
        double r157612 = r157598 * r157611;
        double r157613 = fma(r157598, r157603, r157612);
        double r157614 = r157606 * r157606;
        double r157615 = r157606 * r157614;
        double r157616 = r157615 / r157610;
        double r157617 = r157598 * r157616;
        double r157618 = fma(r157598, r157611, r157617);
        double r157619 = r157613 + r157618;
        double r157620 = r157600 / r157610;
        double r157621 = r157605 - r157620;
        double r157622 = r157619 * r157621;
        double r157623 = cos(r157599);
        double r157624 = cos(r157604);
        double r157625 = r157623 - r157624;
        double r157626 = r157622 * r157625;
        double r157627 = r157597 + r157626;
        double r157628 = 3.0;
        double r157629 = 1.0;
        double r157630 = 5.0;
        double r157631 = sqrt(r157630);
        double r157632 = r157631 - r157629;
        double r157633 = r157632 / r157597;
        double r157634 = r157633 * r157623;
        double r157635 = r157629 + r157634;
        double r157636 = r157628 - r157631;
        double r157637 = r157636 / r157597;
        double r157638 = r157637 * r157624;
        double r157639 = r157635 + r157638;
        double r157640 = r157628 * r157639;
        double r157641 = r157627 / r157640;
        return r157641;
}

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 add-sqr-sqrt 0.5

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

4. Applied sqrt-prod 0.5

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

5. Applied associate-*l* 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied add-cube-cbrt 0.5

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

9. Applied times-frac 0.5

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

10. Applied add-cube-cbrt 0.5

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

11. Applied prod-diff 0.5

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

12. Applied distribute-lft-in 0.5

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

13. Applied distribute-lft-in 0.5

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

14. Simplified 0.5

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

15. Simplified 0.5

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

16. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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))))))