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

Average Error: 0.5 → 0.5

Time: 41.5s

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 r12018699 = 2.0;
        double r12018700 = sqrt(r12018699);
        double r12018701 = x;
        double r12018702 = sin(r12018701);
        double r12018703 = y;
        double r12018704 = sin(r12018703);
        double r12018705 = 16.0;
        double r12018706 = r12018704 / r12018705;
        double r12018707 = r12018702 - r12018706;
        double r12018708 = r12018700 * r12018707;
        double r12018709 = r12018702 / r12018705;
        double r12018710 = r12018704 - r12018709;
        double r12018711 = r12018708 * r12018710;
        double r12018712 = cos(r12018701);
        double r12018713 = cos(r12018703);
        double r12018714 = r12018712 - r12018713;
        double r12018715 = r12018711 * r12018714;
        double r12018716 = r12018699 + r12018715;
        double r12018717 = 3.0;
        double r12018718 = 1.0;
        double r12018719 = 5.0;
        double r12018720 = sqrt(r12018719);
        double r12018721 = r12018720 - r12018718;
        double r12018722 = r12018721 / r12018699;
        double r12018723 = r12018722 * r12018712;
        double r12018724 = r12018718 + r12018723;
        double r12018725 = r12018717 - r12018720;
        double r12018726 = r12018725 / r12018699;
        double r12018727 = r12018726 * r12018713;
        double r12018728 = r12018724 + r12018727;
        double r12018729 = r12018717 * r12018728;
        double r12018730 = r12018716 / r12018729;
        return r12018730;
}

↓

double f(double x, double y) {
        double r12018731 = 2.0;
        double r12018732 = y;
        double r12018733 = sin(r12018732);
        double r12018734 = x;
        double r12018735 = sin(r12018734);
        double r12018736 = 16.0;
        double r12018737 = r12018735 / r12018736;
        double r12018738 = r12018733 - r12018737;
        double r12018739 = cos(r12018734);
        double r12018740 = r12018739 * r12018739;
        double r12018741 = r12018740 * r12018739;
        double r12018742 = cos(r12018732);
        double r12018743 = r12018742 * r12018742;
        double r12018744 = r12018742 * r12018743;
        double r12018745 = r12018741 - r12018744;
        double r12018746 = r12018738 * r12018745;
        double r12018747 = r12018733 / r12018736;
        double r12018748 = r12018735 - r12018747;
        double r12018749 = r12018747 + r12018735;
        double r12018750 = sqrt(r12018731);
        double r12018751 = r12018749 * r12018750;
        double r12018752 = r12018751 * r12018751;
        double r12018753 = r12018751 * r12018752;
        double r12018754 = cbrt(r12018753);
        double r12018755 = r12018748 * r12018754;
        double r12018756 = r12018746 * r12018755;
        double r12018757 = r12018739 + r12018742;
        double r12018758 = r12018757 * r12018742;
        double r12018759 = r12018758 + r12018740;
        double r12018760 = r12018759 * r12018749;
        double r12018761 = r12018756 / r12018760;
        double r12018762 = r12018731 + r12018761;
        double r12018763 = 3.0;
        double r12018764 = 1.0;
        double r12018765 = 5.0;
        double r12018766 = sqrt(r12018765);
        double r12018767 = r12018766 - r12018764;
        double r12018768 = r12018767 / r12018731;
        double r12018769 = r12018739 * r12018768;
        double r12018770 = r12018764 + r12018769;
        double r12018771 = r12018763 * r12018763;
        double r12018772 = r12018771 - r12018765;
        double r12018773 = r12018766 + r12018763;
        double r12018774 = r12018772 / r12018773;
        double r12018775 = r12018774 / r12018731;
        double r12018776 = r12018775 * r12018742;
        double r12018777 = r12018770 + r12018776;
        double r12018778 = r12018763 * r12018777;
        double r12018779 = r12018762 / r12018778;
        return r12018779;
}

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))))))