Average Error: 0.5 → 0.5
Time: 13.1s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(1, \sin x, -\frac{\sin y}{\sqrt[3]{16}} \cdot \frac{1}{\sqrt[3]{16} \cdot \sqrt[3]{16}}\right) + \sqrt{2} \cdot \left(\frac{1}{\sqrt[3]{16} \cdot \sqrt[3]{16}} \cdot \left(\left(-\frac{\sin y}{\sqrt[3]{16}}\right) + \frac{\sin y}{\sqrt[3]{16}}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]
\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{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(1, \sin x, -\frac{\sin y}{\sqrt[3]{16}} \cdot \frac{1}{\sqrt[3]{16} \cdot \sqrt[3]{16}}\right) + \sqrt{2} \cdot \left(\frac{1}{\sqrt[3]{16} \cdot \sqrt[3]{16}} \cdot \left(\left(-\frac{\sin y}{\sqrt[3]{16}}\right) + \frac{\sin y}{\sqrt[3]{16}}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}
double f(double x, double y) {
        double r191747 = 2.0;
        double r191748 = sqrt(r191747);
        double r191749 = x;
        double r191750 = sin(r191749);
        double r191751 = y;
        double r191752 = sin(r191751);
        double r191753 = 16.0;
        double r191754 = r191752 / r191753;
        double r191755 = r191750 - r191754;
        double r191756 = r191748 * r191755;
        double r191757 = r191750 / r191753;
        double r191758 = r191752 - r191757;
        double r191759 = r191756 * r191758;
        double r191760 = cos(r191749);
        double r191761 = cos(r191751);
        double r191762 = r191760 - r191761;
        double r191763 = r191759 * r191762;
        double r191764 = r191747 + r191763;
        double r191765 = 3.0;
        double r191766 = 1.0;
        double r191767 = 5.0;
        double r191768 = sqrt(r191767);
        double r191769 = r191768 - r191766;
        double r191770 = r191769 / r191747;
        double r191771 = r191770 * r191760;
        double r191772 = r191766 + r191771;
        double r191773 = r191765 - r191768;
        double r191774 = r191773 / r191747;
        double r191775 = r191774 * r191761;
        double r191776 = r191772 + r191775;
        double r191777 = r191765 * r191776;
        double r191778 = r191764 / r191777;
        return r191778;
}
↓
double f(double x, double y) {
        double r191779 = 2.0;
        double r191780 = sqrt(r191779);
        double r191781 = 1.0;
        double r191782 = x;
        double r191783 = sin(r191782);
        double r191784 = y;
        double r191785 = sin(r191784);
        double r191786 = 16.0;
        double r191787 = cbrt(r191786);
        double r191788 = r191785 / r191787;
        double r191789 = r191787 * r191787;
        double r191790 = r191781 / r191789;
        double r191791 = r191788 * r191790;
        double r191792 = -r191791;
        double r191793 = fma(r191781, r191783, r191792);
        double r191794 = r191780 * r191793;
        double r191795 = -r191788;
        double r191796 = r191795 + r191788;
        double r191797 = r191790 * r191796;
        double r191798 = r191780 * r191797;
        double r191799 = r191794 + r191798;
        double r191800 = r191783 / r191786;
        double r191801 = r191785 - r191800;
        double r191802 = cos(r191782);
        double r191803 = cos(r191784);
        double r191804 = r191802 - r191803;
        double r191805 = r191801 * r191804;
        double r191806 = fma(r191799, r191805, r191779);
        double r191807 = 3.0;
        double r191808 = 5.0;
        double r191809 = sqrt(r191808);
        double r191810 = r191807 - r191809;
        double r191811 = r191810 / r191779;
        double r191812 = 1.0;
        double r191813 = r191809 - r191812;
        double r191814 = r191813 / r191779;
        double r191815 = fma(r191814, r191802, r191812);
        double r191816 = fma(r191811, r191803, r191815);
        double r191817 = r191806 / r191816;
        double r191818 = r191817 / r191807;
        return r191818;
}

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. Simplified0.5

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

  4. Applied add-cube-cbrt0.5

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

  5. Applied add-sqr-sqrt32.3

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

  6. Applied times-frac32.3

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

  7. Applied add-sqr-sqrt48.5

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

  8. Applied prod-diff48.5

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

  9. Applied distribute-lft-in48.5

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

  10. Simplified32.3

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020002 +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))))))