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

Average Error: 0.5 → 0.4

Time: 33.5s

Precision: binary64

Cost: 72640

Math

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

↓

\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(0.0625 \cdot \sin x - \sin y\right) \cdot \left(\sin y \cdot 0.0625 - \sin x\right)\right)\right)}{3 - \left(\cos y \cdot \frac{-6}{3 + \sqrt{5}} - \frac{\cos x}{0.16666666666666666 + \sqrt{0.1388888888888889}}\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   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))))))

↓

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (sqrt 2.0)
    (*
     (- (cos x) (cos y))
     (* (- (* 0.0625 (sin x)) (sin y)) (- (* (sin y) 0.0625) (sin x))))))
  (-
   3.0
   (-
    (* (cos y) (/ -6.0 (+ 3.0 (sqrt 5.0))))
    (/ (cos x) (+ 0.16666666666666666 (sqrt 0.1388888888888889)))))))

C

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 (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (((0.0625 * sin(x)) - sin(y)) * ((sin(y) * 0.0625) - sin(x)))))) / (3.0 - ((cos(y) * (-6.0 / (3.0 + sqrt(5.0)))) - (cos(x) / (0.16666666666666666 + sqrt(0.1388888888888889)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / (3.0d0 * ((1.0d0 + (((sqrt(5.0d0) - 1.0d0) / 2.0d0) * cos(x))) + (((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y))))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * (((0.0625d0 * sin(x)) - sin(y)) * ((sin(y) * 0.0625d0) - sin(x)))))) / (3.0d0 - ((cos(y) * ((-6.0d0) / (3.0d0 + sqrt(5.0d0)))) - (cos(x) / (0.16666666666666666d0 + sqrt(0.1388888888888889d0)))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((1.0 + (((Math.sqrt(5.0) - 1.0) / 2.0) * Math.cos(x))) + (((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y))));
}

↓

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (((0.0625 * Math.sin(x)) - Math.sin(y)) * ((Math.sin(y) * 0.0625) - Math.sin(x)))))) / (3.0 - ((Math.cos(y) * (-6.0 / (3.0 + Math.sqrt(5.0)))) - (Math.cos(x) / (0.16666666666666666 + Math.sqrt(0.1388888888888889)))));
}

Python

def code(x, y):
	return (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / (3.0 * ((1.0 + (((math.sqrt(5.0) - 1.0) / 2.0) * math.cos(x))) + (((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y))))

↓

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (((0.0625 * math.sin(x)) - math.sin(y)) * ((math.sin(y) * 0.0625) - math.sin(x)))))) / (3.0 - ((math.cos(y) * (-6.0 / (3.0 + math.sqrt(5.0)))) - (math.cos(x) / (0.16666666666666666 + math.sqrt(0.1388888888888889)))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
end

↓

function code(x, y)
	return Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(Float64(Float64(0.0625 * sin(x)) - sin(y)) * Float64(Float64(sin(y) * 0.0625) - sin(x)))))) / Float64(3.0 - Float64(Float64(cos(y) * Float64(-6.0 / Float64(3.0 + sqrt(5.0)))) - Float64(cos(x) / Float64(0.16666666666666666 + sqrt(0.1388888888888889))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (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))));
end

↓

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (((0.0625 * sin(x)) - sin(y)) * ((sin(y) * 0.0625) - sin(x)))))) / (3.0 - ((cos(y) * (-6.0 / (3.0 + sqrt(5.0)))) - (cos(x) / (0.16666666666666666 + sqrt(0.1388888888888889)))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision] - N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(N[(N[Cos[y], $MachinePrecision] * N[(-6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] / N[(0.16666666666666666 + N[Sqrt[0.1388888888888889], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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 0.5

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   Proof

   [Start] 0.5	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   +-commutative [=>] 0.5	\[ \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   associate-*l* [=>] 0.5	\[ \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   associate-*l* [=>] 0.5	\[ \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   fma-def [=>] 0.5	\[ \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   associate-*r* [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   *-commutative [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   associate-*l* [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   distribute-rgt-in [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]
   *-commutative [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]
   +-commutative [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]
   distribute-rgt-in [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + 1 \cdot 3\right)} + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]
   metadata-eval [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \color{blue}{3}\right) + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]
   associate-+l+ [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]
   associate-*l/ [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2}} \cdot 3 + \left(3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]
   associate-*l/ [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) \cdot 3}{2}} + \left(3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]
   associate-/l* [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{\frac{2}{3}}} + \left(3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]
   *-commutative [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}}{\frac{2}{3}} + \left(3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]
   sub-neg [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}}{\frac{2}{3}} + \left(3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]
   metadata-eval [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right)}{\frac{2}{3}} + \left(3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]
   metadata-eval [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{\color{blue}{0.6666666666666666}} + \left(3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]
   +-commutative [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + 3\right)}} \]
   *-commutative [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \left(\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3\right)} \]
   associate-*r* [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \left(\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3\right)} \]
   *-commutative [<=] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \left(\color{blue}{\cos y \cdot \left(3 \cdot \frac{3 - \sqrt{5}}{2}\right)} + 3\right)} \]
   fma-def [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \color{blue}{\mathsf{fma}\left(\cos y, 3 \cdot \frac{3 - \sqrt{5}}{2}, 3\right)}} \]
   *-commutative [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot 3}, 3\right)} \]
   associate-*l/ [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \color{blue}{\frac{\left(3 - \sqrt{5}\right) \cdot 3}{2}}, 3\right)} \]
   associate-/l* [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{\frac{2}{3}}}, 3\right)} \]
   metadata-eval [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{\color{blue}{0.6666666666666666}}, 3\right)} \]

3. Applied egg-rr 15.8

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\sqrt{\cos x}}{\sqrt{5} \cdot 0.16666666666666666 + 0.16666666666666666} \cdot \sqrt{\cos x}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

4. Simplified 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\mathsf{fma}\left(\sqrt{5}, 0.16666666666666666, 0.16666666666666666\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   Proof

   [Start] 15.8	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\sqrt{\cos x}}{\sqrt{5} \cdot 0.16666666666666666 + 0.16666666666666666} \cdot \sqrt{\cos x} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   associate-*l/ [=>] 15.8	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\sqrt{\cos x} \cdot \sqrt{\cos x}}{\sqrt{5} \cdot 0.16666666666666666 + 0.16666666666666666}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   rem-square-sqrt [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\color{blue}{\cos x}}{\sqrt{5} \cdot 0.16666666666666666 + 0.16666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   fma-def [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x}{\color{blue}{\mathsf{fma}\left(\sqrt{5}, 0.16666666666666666, 0.16666666666666666\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

5. Taylor expanded in y around -inf 0.4

   \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{3 + \left(\frac{\cos x}{0.16666666666666666 \cdot \sqrt{5} + 0.16666666666666666} + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]

6. Applied egg-rr 0.4

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{3 + \left(\frac{\cos x}{0.16666666666666666 \cdot \sqrt{5} + 0.16666666666666666} + 1.5 \cdot \left(\color{blue}{\frac{4}{\sqrt{5} + 3}} \cdot \cos y\right)\right)} \]

7. Applied egg-rr 0.4

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{3 + \color{blue}{\left(\frac{\cos x}{0.16666666666666666 + \sqrt{0.1388888888888889}} + \cos y \cdot \frac{6}{\sqrt{5} + 3}\right) \cdot 1}} \]

8. Simplified 0.4

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{3 + \color{blue}{\left(\frac{\cos x}{0.16666666666666666 + \sqrt{0.1388888888888889}} + \cos y \cdot \frac{6}{\sqrt{5} + 3}\right)}} \]
   Proof

   [Start] 0.4	\[ \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{3 + \left(\frac{\cos x}{0.16666666666666666 + \sqrt{0.1388888888888889}} + \cos y \cdot \frac{6}{\sqrt{5} + 3}\right) \cdot 1} \]
   *-rgt-identity [=>] 0.4	\[ \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{3 + \color{blue}{\left(\frac{\cos x}{0.16666666666666666 + \sqrt{0.1388888888888889}} + \cos y \cdot \frac{6}{\sqrt{5} + 3}\right)}} \]

9. Final simplification 0.4

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(0.0625 \cdot \sin x - \sin y\right) \cdot \left(\sin y \cdot 0.0625 - \sin x\right)\right)\right)}{3 - \left(\cos y \cdot \frac{-6}{3 + \sqrt{5}} - \frac{\cos x}{0.16666666666666666 + \sqrt{0.1388888888888889}}\right)} \]

Alternatives

Alternative 1
Error	11.6
Cost	67017

\[\begin{array}{l} \mathbf{if}\;x \leq -0.06 \lor \neg \left(x \leq 0.034\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{-2}{-1 - \sqrt{5}}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(\left(0.0625 \cdot \sin x - \sin y\right) \cdot \left(\sin y \cdot 0.0625 - \sin x\right)\right) \cdot \left(-1 + \left(\cos y + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)}{3 + \left(\frac{\cos x}{0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}} + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	11.7
Cost	66761

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.1 \lor \neg \left(x \leq 0.048\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{-2}{-1 - \sqrt{5}}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\cos y - \cos x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 0.0625 - x\right)\right)}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - t_1\right) + \cos y \cdot \left(t_1 + -1.5\right)\right)\right)}\\ \end{array} \]

Alternative 3
Error	11.9
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-6} \lor \neg \left(x \leq 3.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos y - \cos x\right)\right)}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - t_0\right) + \cos y \cdot \left(t_0 + -1.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{1}{0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}} + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	11.8
Cost	66505

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-6} \lor \neg \left(x \leq 7.8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{-2}{-1 - \sqrt{5}}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{1}{0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}} + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	12.8
Cost	60233

\[\begin{array}{l} t_0 := 3 + \left(\frac{\cos x}{0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)\right)\\ \mathbf{if}\;y \leq -0.0032 \lor \neg \left(y \leq 0.0029\right):\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\cos y - \cos x\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(y \cdot 1.00390625 - 0.0625 \cdot \sin x\right)\right)\right)}{t_0}\\ \end{array} \]

Alternative 6
Error	12.9
Cost	60105

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-6} \lor \neg \left(x \leq 1.3 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\right)}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - t_0\right) + \cos y \cdot \left(t_0 + -1.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{1}{0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}} + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	12.9
Cost	53513

\[\begin{array}{l} t_0 := 0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-6} \lor \neg \left(x \leq 1.86 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 + \left(\frac{\cos x}{t_0} + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{1}{t_0} + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	13.0
Cost	53385

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-5} \lor \neg \left(y \leq 1.9\right):\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(\cos y + -1\right)\right)}{3 + \left(\frac{\cos x}{0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}} + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 + 1.5 \cdot \left(\frac{4}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 9
Error	12.9
Cost	53385

\[\begin{array}{l} t_0 := -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\\ t_1 := 0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-7} \lor \neg \left(x \leq 1.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 + \left(\frac{\cos x}{t_1} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{1}{t_1} + \left(3 + t_0\right)}\\ \end{array} \]

Alternative 10
Error	13.3
Cost	46984

\[\begin{array}{l} t_0 := 0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}\\ t_1 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 + 1.5 \cdot \left(\frac{4}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{1}{t_0} + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{3 + \left(\frac{\cos x}{t_0} + \left(3 - \sqrt{5}\right) \cdot 1.5\right)}\\ \end{array} \]

Alternative 11
Error	13.3
Cost	46857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.82 \cdot 10^{-6} \lor \neg \left(x \leq 3.75 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + \left(\frac{\cos x}{0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}} + \left(3 - \sqrt{5}\right) \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right) + 1.5 \cdot \left(\sqrt{5} + -1\right)}\\ \end{array} \]

Alternative 12
Error	13.4
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 2 + \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\\ t_2 := \sqrt{5} + -3\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{3 + -1.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right) + t_2\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\left(3 + -1.5 \cdot \left(\cos y \cdot t_2\right)\right) + 1.5 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(3 + \left(\cos x \cdot t_0 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 13
Error	13.4
Cost	46856

\[\begin{array}{l} t_0 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ t_1 := \sqrt{5} + -1\\ t_2 := \cos x \cdot t_1\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_0\right)}{3 + \left(\left(3 - \sqrt{5}\right) \cdot 1.5 + 1.5 \cdot t_2\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right) + 1.5 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 + 1.5 \cdot \left(3 + \left(t_2 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 14
Error	13.3
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} + -3\\ t_1 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.68 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 + -1.5 \cdot \left(t_0 - \frac{\cos x \cdot 4}{\sqrt{5} + 1}\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\left(3 + -1.5 \cdot \left(\cos y \cdot t_0\right)\right) + 1.5 \cdot \left(\sqrt{5} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{3 + \left(\frac{\cos x}{0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}} + \left(3 - \sqrt{5}\right) \cdot 1.5\right)}\\ \end{array} \]

Alternative 15
Error	13.3
Cost	46856

\[\begin{array}{l} t_0 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 + 1.5 \cdot \left(\frac{4}{3 + \sqrt{5}} + \cos x \cdot t_1\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right) + 1.5 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_0\right)}{3 + \left(\frac{\cos x}{0.16666666666666666 + 0.16666666666666666 \cdot \sqrt{5}} + \left(3 - \sqrt{5}\right) \cdot 1.5\right)}\\ \end{array} \]

Alternative 16
Error	25.0
Cost	46464

\[\frac{2 + \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 + -1.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right) + \left(\sqrt{5} + -3\right)\right)} \]

Alternative 17
Error	25.0
Cost	46464

\[\frac{2 + \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 + 1.5 \cdot \left(3 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)\right)} \]

Alternative 18
Error	37.6
Cost	20416

\[\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)}{6} \]

Alternative 19
Error	37.6
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2022356 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 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))))))