Average Error: 0.5 → 0.5
Time: 16.9s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), \mathsf{fma}\left(0 \cdot \frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}, \frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}, \sin x - \frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\]
\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{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), \mathsf{fma}\left(0 \cdot \frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}, \frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}, \sin x - \frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}
double f(double x, double y) {
        double r221738 = 2.0;
        double r221739 = sqrt(r221738);
        double r221740 = x;
        double r221741 = sin(r221740);
        double r221742 = y;
        double r221743 = sin(r221742);
        double r221744 = 16.0;
        double r221745 = r221743 / r221744;
        double r221746 = r221741 - r221745;
        double r221747 = r221739 * r221746;
        double r221748 = r221741 / r221744;
        double r221749 = r221743 - r221748;
        double r221750 = r221747 * r221749;
        double r221751 = cos(r221740);
        double r221752 = cos(r221742);
        double r221753 = r221751 - r221752;
        double r221754 = r221750 * r221753;
        double r221755 = r221738 + r221754;
        double r221756 = 3.0;
        double r221757 = 1.0;
        double r221758 = 5.0;
        double r221759 = sqrt(r221758);
        double r221760 = r221759 - r221757;
        double r221761 = r221760 / r221738;
        double r221762 = r221761 * r221751;
        double r221763 = r221757 + r221762;
        double r221764 = r221756 - r221759;
        double r221765 = r221764 / r221738;
        double r221766 = r221765 * r221752;
        double r221767 = r221763 + r221766;
        double r221768 = r221756 * r221767;
        double r221769 = r221755 / r221768;
        return r221769;
}
↓
double f(double x, double y) {
        double r221770 = x;
        double r221771 = cos(r221770);
        double r221772 = y;
        double r221773 = cos(r221772);
        double r221774 = r221771 - r221773;
        double r221775 = sin(r221772);
        double r221776 = sin(r221770);
        double r221777 = 16.0;
        double r221778 = r221776 / r221777;
        double r221779 = r221775 - r221778;
        double r221780 = r221774 * r221779;
        double r221781 = 0.0;
        double r221782 = cbrt(r221775);
        double r221783 = cbrt(r221777);
        double r221784 = r221782 / r221783;
        double r221785 = r221781 * r221784;
        double r221786 = 3.0;
        double r221787 = pow(r221782, r221786);
        double r221788 = r221787 / r221777;
        double r221789 = r221776 - r221788;
        double r221790 = fma(r221785, r221784, r221789);
        double r221791 = 2.0;
        double r221792 = sqrt(r221791);
        double r221793 = r221790 * r221792;
        double r221794 = fma(r221780, r221793, r221791);
        double r221795 = 3.0;
        double r221796 = 5.0;
        double r221797 = sqrt(r221796);
        double r221798 = r221795 - r221797;
        double r221799 = r221798 / r221791;
        double r221800 = 1.0;
        double r221801 = r221797 - r221800;
        double r221802 = r221801 / r221791;
        double r221803 = fma(r221771, r221802, r221800);
        double r221804 = fma(r221773, r221799, r221803);
        double r221805 = r221804 * r221795;
        double r221806 = r221794 / r221805;
        return r221806;
}

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. Simplified0.4

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

  4. Applied add-cube-cbrt0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied times-frac0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied prod-diff0.4

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

  9. Applied distribute-lft-in0.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  12. Simplified0.5

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), \mathsf{fma}\left(0 \cdot \frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}, \frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}, \sin x - \frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}}\]

  13. Final simplification0.5

    \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), \mathsf{fma}\left(0 \cdot \frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}, \frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}, \sin x - \frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\]

Reproduce

herbie shell --seed 2020047 +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))))))