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

Average Error: 0.5 → 0.5

Time: 36.8s

Precision: binary64

Cost: 79552

Math

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

↓

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

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) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (+ (cos x) (cos y)) (* 2.0 (cos y)))))
  (*
   3.0
   (+
    (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
    (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))

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) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * ((cos(x) + cos(y)) - (2.0 * cos(y))))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
}

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) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * ((cos(x) + cos(y)) - (2.0d0 * 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

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.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * ((Math.cos(x) + Math.cos(y)) - (2.0 * 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))));
}

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.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * ((math.cos(x) + math.cos(y)) - (2.0 * 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))))

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(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(Float64(cos(x) + cos(y)) - Float64(2.0 * 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

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) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * ((cos(x) + cos(y)) - (2.0 * 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

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[(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[(N[Cos[x], $MachinePrecision] + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Cos[y], $MachinePrecision]), $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]

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. Applied egg-rr 0.5

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

3. Simplified 0.5

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

   [Start] 0.5	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\cos x + \cos y\right) + \left(0 - \cos y \cdot 2\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   rational_best-simplify-1 [=>] 0.5	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\cos y + \cos x\right)} + \left(0 - \cos y \cdot 2\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   rational_best-simplify-10 [=>] 0.5	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\cos y + \cos x\right) + \color{blue}{\left(-\cos y \cdot 2\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

4. Taylor expanded in y around inf 0.5

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

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	72768

\[\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y + \left(\sqrt{5} + -1\right) \cdot \cos x\right)\right)} \]

Alternative 2
Error	0.5
Cost	72768

\[\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y + \left(\sqrt{5} + -1\right) \cdot \cos x\right)\right)} \]

Alternative 3
Error	12.3
Cost	66760

\[\begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t_0 \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -95:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4800:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_1 \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	12.3
Cost	66760

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right) \cdot \left(\cos x - \cos y\right)}{t_1}\\ \mathbf{if}\;x \leq -95:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4800:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_0\right) \cdot \left(1 - \cos y\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	12.3
Cost	66632

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -95:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4800:\\ \;\;\;\;\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_0 \cdot \cos y + \left(\sqrt{5} + -1\right) \cdot \cos x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	12.3
Cost	66504

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -0.0027:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4800:\\ \;\;\;\;\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_0 \cdot \cos y + \left(\sqrt{5} + -1\right) \cdot \cos x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	13.3
Cost	60232

\[\begin{array}{l} t_0 := \sin x - \frac{\sin y}{16}\\ t_1 := 3 - \sqrt{5}\\ t_2 := 3 \cdot \left(1 + 0.5 \cdot \left(t_1 \cdot \cos y + \left(\sqrt{5} + -1\right) \cdot \cos x\right)\right)\\ \mathbf{if}\;y \leq -0.0037:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{t_2}\\ \mathbf{elif}\;y \leq 0.023:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(y + -0.0625 \cdot \sin x\right)\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t_1}{2} \cdot \cos y\right)}\\ \end{array} \]

Alternative 8
Error	13.3
Cost	60104

\[\begin{array}{l} t_0 := \left(\sqrt{5} + -1\right) \cdot \cos x\\ t_1 := \sin x - \frac{\sin y}{16}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_2 \cdot \cos y + t_0\right)\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_2 + t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t_2}{2} \cdot \cos y\right)}\\ \end{array} \]

Alternative 9
Error	13.5
Cost	59976

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{t_2}{2}\\ \mathbf{if}\;x \leq -95:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot t_3\right)}\\ \mathbf{elif}\;x \leq 4800:\\ \;\;\;\;\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_2 \cdot \cos y + t_0 \cdot \cos x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t_3 \cdot \cos y\right)}\\ \end{array} \]

Alternative 10
Error	13.5
Cost	59976

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 3 \cdot \left(1 + 0.5 \cdot \left(t_0 \cdot \cos y + \left(\sqrt{5} + -1\right) \cdot \cos x\right)\right)\\ t_2 := \sin x - \frac{\sin y}{16}\\ \mathbf{if}\;x \leq -95:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(-0.0625 \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\right)}{t_1}\\ \mathbf{elif}\;x \leq 4800:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left({\sin x}^{2} \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t_0}{2} \cdot \cos y\right)}\\ \end{array} \]

Alternative 11
Error	13.6
Cost	59912

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{t_1}{2}\\ \mathbf{if}\;x \leq -95:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot t_2\right)}\\ \mathbf{elif}\;x \leq 4800:\\ \;\;\;\;\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\left(\sqrt{5} + t_1 \cdot \cos y\right) - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t_2 \cdot \cos y\right)}\\ \end{array} \]

Alternative 12
Error	13.6
Cost	53512

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{2 + \left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{if}\;x \leq -95:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4800:\\ \;\;\;\;\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\left(\sqrt{5} + t_0 \cdot \cos y\right) - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	26.1
Cost	53184

\[\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\sqrt{5} + \left(-1 + \left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)\right)} \]

Alternative 14
Error	26.1
Cost	53184

\[\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\left(\sqrt{5} + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - 1\right)\right)} \]

Alternative 15
Error	37.2
Cost	46720

\[\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \]

Alternative 16
Error	37.2
Cost	46592

\[0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\sqrt{5} + \left(\left(3 - \sqrt{5}\right) \cdot \cos y + -1\right)\right)} \]

Alternative 17
Error	37.2
Cost	20160

\[\frac{0.6666666666666666}{1 + 0.5 \cdot \left(\sqrt{5} + \left(\left(3 - \sqrt{5}\right) \cdot \cos y + -1\right)\right)} \]

Alternative 18
Error	38.4
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023092 
(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))))))