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

Average Error: 0.5 → 0.5

Time: 57.5s

Precision: binary64

Cost: 73024

Math

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

↓

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

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
    (*
     (- (sin x) (/ (sin y) 16.0))
     (*
      (- (sin y) (/ (sin x) 16.0))
      (* (sqrt 2.0) (- (+ (- -1.0 (cos y)) (cos x)) -1.0)))))
   3.0)
  (+
   1.0
   (*
    0.5
    (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (* (cos y) (- 3.0 (sqrt 5.0))))))))

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 + ((sin(x) - (sin(y) / 16.0)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (((-1.0 - cos(y)) + cos(x)) - -1.0))))) / 3.0) / (1.0 + (0.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (cos(y) * (3.0 - sqrt(5.0))))));
}

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 + ((sin(x) - (sin(y) / 16.0d0)) * ((sin(y) - (sin(x) / 16.0d0)) * (sqrt(2.0d0) * ((((-1.0d0) - cos(y)) + cos(x)) - (-1.0d0)))))) / 3.0d0) / (1.0d0 + (0.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (cos(y) * (3.0d0 - sqrt(5.0d0))))))
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.sin(x) - (Math.sin(y) / 16.0)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.sqrt(2.0) * (((-1.0 - Math.cos(y)) + Math.cos(x)) - -1.0))))) / 3.0) / (1.0 + (0.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
}

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.sin(x) - (math.sin(y) / 16.0)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.sqrt(2.0) * (((-1.0 - math.cos(y)) + math.cos(x)) - -1.0))))) / 3.0) / (1.0 + (0.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + (math.cos(y) * (3.0 - math.sqrt(5.0))))))

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(Float64(2.0 + Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * Float64(Float64(Float64(-1.0 - cos(y)) + cos(x)) - -1.0))))) / 3.0) / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))))
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 + ((sin(x) - (sin(y) / 16.0)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (((-1.0 - cos(y)) + cos(x)) - -1.0))))) / 3.0) / (1.0 + (0.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (cos(y) * (3.0 - sqrt(5.0))))));
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[(N[(2.0 + N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $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

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

   [Start] 0.5	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   rational.json-simplify-5 [<=] 0.5	\[ \frac{\color{blue}{\left(2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right) - 0}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   metadata-eval [<=] 0.5	\[ \frac{\left(2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right) - \color{blue}{\left(0 - 0\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   rational.json-simplify-45 [<=] 0.5	\[ \frac{\color{blue}{0 - \left(0 - \left(2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   rational.json-simplify-12 [<=] 0.5	\[ \frac{0 - \color{blue}{\left(-\left(2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   rational.json-simplify-12 [<=] 0.5	\[ \frac{\color{blue}{-\left(-\left(2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.5

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

5. Taylor expanded in x around inf 0.5

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

6. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\left(-1 - \cos y\right) + \cos x\right) - -1\right)\right)\right)}{3}}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]
   rational.json-simplify-1 [=>] 0.5	\[ \frac{\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\left(-1 - \cos y\right) + \cos x\right) - -1\right)\right)\right)}{3}}{1 + \color{blue}{\left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right)}} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\left(-1 - \cos y\right) + \cos x\right) - -1\right)\right)\right)}{3}}{1 + \left(0.5 \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right)} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\left(-1 - \cos y\right) + \cos x\right) - -1\right)\right)\right)}{3}}{1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) \cdot 0.5}\right)} \]
   rational.json-simplify-51 [=>] 0.5	\[ \frac{\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\left(-1 - \cos y\right) + \cos x\right) - -1\right)\right)\right)}{3}}{1 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   rational.json-simplify-15 [<=] 0.5	\[ \frac{\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\left(-1 - \cos y\right) + \cos x\right) - -1\right)\right)\right)}{3}}{1 + 0.5 \cdot \left(\color{blue}{\left(\sqrt{5} + -1\right)} \cdot \cos x + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]
   rational.json-simplify-2 [<=] 0.5	\[ \frac{\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\left(-1 - \cos y\right) + \cos x\right) - -1\right)\right)\right)}{3}}{1 + 0.5 \cdot \left(\color{blue}{\cos x \cdot \left(\sqrt{5} + -1\right)} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

7. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	72768

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

Alternative 2
Error	0.5
Cost	72768

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

Alternative 3
Error	0.5
Cost	72768

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

Alternative 4
Error	12.0
Cost	66760

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\\ t_3 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right) \cdot t_0}{t_2}\\ \mathbf{if}\;x \leq -0.048:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 0.008:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right) \cdot t_1\right) \cdot t_0}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	12.1
Cost	66632

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -38:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 0.007:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot t_1\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	12.1
Cost	66632

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -38:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 0.008:\\ \;\;\;\;\frac{\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3}}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	12.2
Cost	66504

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -38:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 0.00021:\\ \;\;\;\;\frac{\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3}}{1 + 0.5 \cdot \left(\cos y \cdot t_0 + \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	13.4
Cost	60104

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos x \cdot \frac{t_0}{2}\\ t_2 := {\sin y}^{2}\\ t_3 := 3 - \sqrt{5}\\ t_4 := \cos y \cdot \frac{t_3}{2}\\ \mathbf{if}\;y \leq -0.000215:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(-2 + \left(1 - \left(\cos y + -2\right)\right)\right) \cdot t_2\right)\right)}{3 \cdot \left(1 + \left(t_1 + t_4\right)\right)}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_3 + \cos x \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot t_2\right)\right)}{3 \cdot \left(\left(1 + t_1\right) + t_4\right)}\\ \end{array} \]

Alternative 9
Error	13.4
Cost	60104

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \sin x - \frac{\sin y}{16}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -38:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(t_2 \cdot t_1\right) \cdot \left(\cos x - 1\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_3 + \cos x \cdot t_0\right)\right)}\\ \mathbf{elif}\;x \leq 0.00048:\\ \;\;\;\;\frac{\frac{2 + t_2 \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3}}{1 + 0.5 \cdot \left(\cos y \cdot t_3 + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{t_3}{2}\right)}\\ \end{array} \]

Alternative 10
Error	13.4
Cost	60104

\[\begin{array}{l} t_0 := \sin x - \frac{\sin y}{16}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -38:\\ \;\;\;\;\frac{\frac{2 + t_0 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3}}{1 + 0.5 \cdot \left(t_1 + \cos x \cdot t_3\right)}\\ \mathbf{elif}\;x \leq 0.00047:\\ \;\;\;\;\frac{\frac{2 + t_0 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3}}{1 + 0.5 \cdot \left(\cos y \cdot t_1 + t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_3}{2}\right) + \cos y \cdot \frac{t_1}{2}\right)}\\ \end{array} \]

Alternative 11
Error	13.4
Cost	59912

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_3 \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{t_2}\\ \mathbf{elif}\;x \leq 0.000205:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos y \cdot t_0 + t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot t_3\right)\right)}{t_2}\\ \end{array} \]

Alternative 12
Error	13.4
Cost	53636

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := {\sin y}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.000116:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(-2 + \left(1 - \left(\cos y + -2\right)\right)\right) \cdot t_1\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{t_0}{2} + \cos y \cdot \frac{t_2}{2}\right)\right)}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-9}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(t_2 + t_0 \cdot \cos x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot t_1\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot t_0 + \cos y \cdot t_2\right)\right)}\\ \end{array} \]

Alternative 13
Error	13.4
Cost	53384

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot t_1 + \cos y \cdot t_0\right)\right)}\\ \mathbf{if}\;y \leq -0.000116:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-9}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(t_0 + t_1 \cdot \cos x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	13.4
Cost	53384

\[\begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 - \cos y\\ t_3 := \sqrt{5} + -1\\ t_4 := 3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot t_3 + \cos y \cdot t_1\right)\right)\\ \mathbf{if}\;y \leq -0.000116:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\sqrt{2} \cdot t_0\right) \cdot -0.0625\right)}{t_4}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-9}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(t_1 + t_3 \cdot \cos x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot t_0\right)\right)}{t_4}\\ \end{array} \]

Alternative 15
Error	13.9
Cost	46856

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(t_0 + t_1 \cdot \cos x\right)}\\ \mathbf{if}\;x \leq -38:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 0.000215:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos y \cdot t_0 + t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	25.6
Cost	46592

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

Alternative 17
Error	37.9
Cost	26496

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{6} \]

Alternative 18
Error	44.4
Cost	20288

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot {y}^{4} + -0.20833333333333334 \cdot {y}^{6}\right)\right)}{6} \]

Alternative 19
Error	44.5
Cost	20096

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

Alternative 20
Error	44.6
Cost	13568

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot {y}^{4}\right)\right)}{6} \]

Reproduce

herbie shell --seed 2023075 
(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))))))