Average Error: 0.5 → 0.4
Time: 11.3s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \mathsf{fma}\left(1, \sin y, -\frac{\sin x}{\sqrt[3]{16}} \cdot \frac{1}{\sqrt[3]{16} \cdot \sqrt[3]{16}}\right), \left(\left(\frac{1}{\sqrt[3]{16} \cdot \sqrt[3]{16}} \cdot \left(\left(-\frac{\sin x}{\sqrt[3]{16}}\right) + \frac{\sin x}{\sqrt[3]{16}}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]
\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(\cos x - \cos y, \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \mathsf{fma}\left(1, \sin y, -\frac{\sin x}{\sqrt[3]{16}} \cdot \frac{1}{\sqrt[3]{16} \cdot \sqrt[3]{16}}\right), \left(\left(\frac{1}{\sqrt[3]{16} \cdot \sqrt[3]{16}} \cdot \left(\left(-\frac{\sin x}{\sqrt[3]{16}}\right) + \frac{\sin x}{\sqrt[3]{16}}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\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 code(double x, double y) {
	return ((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)))));
}
↓
double code(double x, double y) {
	return ((fma((cos(x) - cos(y)), fma((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))), fma(1.0, sin(y), -((sin(x) / cbrt(16.0)) * (1.0 / (cbrt(16.0) * cbrt(16.0))))), ((((1.0 / (cbrt(16.0) * cbrt(16.0))) * (-(sin(x) / cbrt(16.0)) + (sin(x) / cbrt(16.0)))) * sqrt(2.0)) * (sin(x) - (sin(y) / 16.0)))), 2.0) / fma(((3.0 - sqrt(5.0)) / 2.0), cos(y), fma(((sqrt(5.0) - 1.0) / 2.0), cos(x), 1.0))) / 3.0);
}

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. Using strategy rm

  3. Applied add-cube-cbrt0.5

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

  4. Applied *-un-lft-identity0.5

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

  5. Applied times-frac0.5

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

  6. Applied add-sqr-sqrt31.7

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

  7. Applied prod-diff31.7

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

  8. Applied distribute-lft-in31.7

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

  9. Simplified0.5

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

  10. Simplified0.5

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020078 +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))))))