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

Average Error: 0.5 → 0.5

Time: 35.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r169674 = 2.0;
        double r169675 = sqrt(r169674);
        double r169676 = x;
        double r169677 = sin(r169676);
        double r169678 = y;
        double r169679 = sin(r169678);
        double r169680 = 16.0;
        double r169681 = r169679 / r169680;
        double r169682 = r169677 - r169681;
        double r169683 = r169675 * r169682;
        double r169684 = r169677 / r169680;
        double r169685 = r169679 - r169684;
        double r169686 = r169683 * r169685;
        double r169687 = cos(r169676);
        double r169688 = cos(r169678);
        double r169689 = r169687 - r169688;
        double r169690 = r169686 * r169689;
        double r169691 = r169674 + r169690;
        double r169692 = 3.0;
        double r169693 = 1.0;
        double r169694 = 5.0;
        double r169695 = sqrt(r169694);
        double r169696 = r169695 - r169693;
        double r169697 = r169696 / r169674;
        double r169698 = r169697 * r169687;
        double r169699 = r169693 + r169698;
        double r169700 = r169692 - r169695;
        double r169701 = r169700 / r169674;
        double r169702 = r169701 * r169688;
        double r169703 = r169699 + r169702;
        double r169704 = r169692 * r169703;
        double r169705 = r169691 / r169704;
        return r169705;
}

↓

double f(double x, double y) {
        double r169706 = 2.0;
        double r169707 = sqrt(r169706);
        double r169708 = x;
        double r169709 = sin(r169708);
        double r169710 = cbrt(r169709);
        double r169711 = r169710 * r169710;
        double r169712 = y;
        double r169713 = sin(r169712);
        double r169714 = cbrt(r169713);
        double r169715 = 16.0;
        double r169716 = cbrt(r169715);
        double r169717 = cbrt(r169716);
        double r169718 = r169717 * r169717;
        double r169719 = r169718 * r169717;
        double r169720 = r169714 / r169719;
        double r169721 = 3.0;
        double r169722 = pow(r169720, r169721);
        double r169723 = -r169722;
        double r169724 = fma(r169711, r169710, r169723);
        double r169725 = r169707 * r169724;
        double r169726 = r169714 / r169716;
        double r169727 = pow(r169726, r169721);
        double r169728 = -r169727;
        double r169729 = r169728 + r169727;
        double r169730 = r169729 * r169707;
        double r169731 = r169725 + r169730;
        double r169732 = r169709 / r169715;
        double r169733 = r169713 - r169732;
        double r169734 = cos(r169708);
        double r169735 = cos(r169712);
        double r169736 = r169734 - r169735;
        double r169737 = r169733 * r169736;
        double r169738 = fma(r169731, r169737, r169706);
        double r169739 = 1.0;
        double r169740 = 3.0;
        double r169741 = r169739 / r169740;
        double r169742 = 5.0;
        double r169743 = sqrt(r169742);
        double r169744 = r169740 - r169743;
        double r169745 = r169744 / r169706;
        double r169746 = 1.0;
        double r169747 = r169743 - r169746;
        double r169748 = r169747 / r169706;
        double r169749 = fma(r169734, r169748, r169746);
        double r169750 = fma(r169735, r169745, r169749);
        double r169751 = r169741 / r169750;
        double r169752 = r169738 * r169751;
        return r169752;
}

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. Simplified 0.4

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

4. Applied *-un-lft-identity 0.4

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

5. Applied div-inv 0.5

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

6. Applied times-frac 0.5

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

7. Simplified 0.5

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

9. Applied add-cube-cbrt 0.5

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

10. Applied add-cube-cbrt 0.5

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

11. Applied times-frac 0.5

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

12. Applied add-cube-cbrt 0.5

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

13. Applied prod-diff 0.5

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

14. Applied distribute-rgt-in 0.5

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

15. Simplified 0.5

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

16. Simplified 0.5

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

18. Applied add-cube-cbrt 0.5

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

19. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019209 +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))))))