Average Error: 0.5 → 0.4
Time: 10.9s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}} \cdot \frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt[3]{16} \cdot \sqrt[3]{16}}\right) + \frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt[3]{16} \cdot \sqrt[3]{16}} \cdot \left(\left(-\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right) + \frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right), \sin y - \frac{\sin x}{16}, 2\right)}{3}}{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(-1, 5, 3 \cdot 3\right)}{3 + \sqrt{5}}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}\]
\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(\left(\sqrt{2} \cdot \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) + \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)\right) \cdot \left(\cos x - \cos y\right), \sin y - \frac{\sin x}{16}, 2\right)}{3}}{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(-1, 5, 3 \cdot 3\right)}{3 + \sqrt{5}}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}
double code(double x, double y) {
	return ((double) (((double) (2.0 + ((double) (((double) (((double) (((double) sqrt(2.0)) * ((double) (((double) sin(x)) - ((double) (((double) sin(y)) / 16.0)))))) * ((double) (((double) sin(y)) - ((double) (((double) sin(x)) / 16.0)))))) * ((double) (((double) cos(x)) - ((double) cos(y)))))))) / ((double) (3.0 * ((double) (((double) (1.0 + ((double) (((double) (((double) (((double) sqrt(5.0)) - 1.0)) / 2.0)) * ((double) cos(x)))))) + ((double) (((double) (((double) (3.0 - ((double) sqrt(5.0)))) / 2.0)) * ((double) cos(y))))))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) fma(((double) (((double) (((double) sqrt(2.0)) * ((double) (((double) fma(((double) (((double) cbrt(((double) sin(x)))) * ((double) cbrt(((double) sin(x)))))), ((double) cbrt(((double) sin(x)))), ((double) -(((double) (((double) (((double) cbrt(((double) sin(y)))) / ((double) cbrt(16.0)))) * ((double) (((double) (((double) cbrt(((double) sin(y)))) * ((double) cbrt(((double) sin(y)))))) / ((double) (((double) cbrt(16.0)) * ((double) cbrt(16.0)))))))))))) + ((double) (((double) (((double) (((double) cbrt(((double) sin(y)))) * ((double) cbrt(((double) sin(y)))))) / ((double) (((double) cbrt(16.0)) * ((double) cbrt(16.0)))))) * ((double) (((double) -(((double) (((double) cbrt(((double) sin(y)))) / ((double) cbrt(16.0)))))) + ((double) (((double) cbrt(((double) sin(y)))) / ((double) cbrt(16.0)))))))))))) * ((double) (((double) cos(x)) - ((double) cos(y)))))), ((double) (((double) sin(y)) - ((double) (((double) sin(x)) / 16.0)))), 2.0)) / 3.0)) / ((double) fma(((double) (((double) (((double) fma(((double) -(1.0)), 5.0, ((double) (3.0 * 3.0)))) / ((double) (3.0 + ((double) sqrt(5.0)))))) / 2.0)), ((double) cos(y)), ((double) fma(((double) (((double) (((double) sqrt(5.0)) - 1.0)) / 2.0)), ((double) cos(x)), 1.0))))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-cbrt0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied times-frac0.4

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied prod-diff0.5

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

  9. Applied distribute-lft-in0.5

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

  10. Simplified0.5

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

  11. Simplified0.5

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

  13. Applied flip--0.5

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020122 +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))))))