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

Average Error: 0.5 → 0.5

Time: 58.2s

Precision: binary64

Cost: 93056

Math

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

↓

\[\begin{array}{l} t_0 := \sin y \cdot 0.0625\\ \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(2 \cdot \left(\left(0 - \left(-1 - \frac{{\left(t_0 - \sin x\right)}^{2}}{\sqrt{2} \cdot \left(\sin x - t_0\right)}\right)\right) - 1\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \end{array} \]

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
 (let* ((t_0 (* (sin y) 0.0625)))
   (/
    (+
     2.0
     (*
      (- (sin y) (/ (sin x) 16.0))
      (*
       (*
        2.0
        (-
         (-
          0.0
          (-
           -1.0
           (/ (pow (- t_0 (sin x)) 2.0) (* (sqrt 2.0) (- (sin x) t_0)))))
         1.0))
       (- (cos x) (cos y)))))
    (*
     3.0
     (+
      1.0
      (+
       (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))
       (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))))

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) {
	double t_0 = sin(y) * 0.0625;
	return (2.0 + ((sin(y) - (sin(x) / 16.0)) * ((2.0 * ((0.0 - (-1.0 - (pow((t_0 - sin(x)), 2.0) / (sqrt(2.0) * (sin(x) - t_0))))) - 1.0)) * (cos(x) - cos(y))))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) + -1.0) / 2.0)) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)))));
}

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
    real(8) :: t_0
    t_0 = sin(y) * 0.0625d0
    code = (2.0d0 + ((sin(y) - (sin(x) / 16.0d0)) * ((2.0d0 * ((0.0d0 - ((-1.0d0) - (((t_0 - sin(x)) ** 2.0d0) / (sqrt(2.0d0) * (sin(x) - t_0))))) - 1.0d0)) * (cos(x) - cos(y))))) / (3.0d0 * (1.0d0 + ((cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0)) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))))
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) {
	double t_0 = Math.sin(y) * 0.0625;
	return (2.0 + ((Math.sin(y) - (Math.sin(x) / 16.0)) * ((2.0 * ((0.0 - (-1.0 - (Math.pow((t_0 - Math.sin(x)), 2.0) / (Math.sqrt(2.0) * (Math.sin(x) - t_0))))) - 1.0)) * (Math.cos(x) - Math.cos(y))))) / (3.0 * (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0)) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)))));
}

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):
	t_0 = math.sin(y) * 0.0625
	return (2.0 + ((math.sin(y) - (math.sin(x) / 16.0)) * ((2.0 * ((0.0 - (-1.0 - (math.pow((t_0 - math.sin(x)), 2.0) / (math.sqrt(2.0) * (math.sin(x) - t_0))))) - 1.0)) * (math.cos(x) - math.cos(y))))) / (3.0 * (1.0 + ((math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0)) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))))

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)
	t_0 = Float64(sin(y) * 0.0625)
	return Float64(Float64(2.0 + Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(2.0 * Float64(Float64(0.0 - Float64(-1.0 - Float64((Float64(t_0 - sin(x)) ^ 2.0) / Float64(sqrt(2.0) * Float64(sin(x) - t_0))))) - 1.0)) * Float64(cos(x) - cos(y))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))))
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)
	t_0 = sin(y) * 0.0625;
	tmp = (2.0 + ((sin(y) - (sin(x) / 16.0)) * ((2.0 * ((0.0 - (-1.0 - (((t_0 - sin(x)) ^ 2.0) / (sqrt(2.0) * (sin(x) - t_0))))) - 1.0)) * (cos(x) - cos(y))))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) + -1.0) / 2.0)) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)))));
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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]}, N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(0.0 - N[(-1.0 - N[(N[Power[N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $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

   \[\leadsto \color{blue}{\frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)}} \]
   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-2 [=>] 0.5	\[ \frac{2 + \color{blue}{\left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{2 + \left(\cos x - \cos y\right) \cdot \color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   rational.json-simplify-43 [=>] 0.5	\[ \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   rational.json-simplify-1 [=>] 0.5	\[ \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]
   rational.json-simplify-41 [=>] 0.5	\[ \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.5

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

5. Taylor expanded in x around inf 0.5

   \[\leadsto \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\left(2 \cdot \frac{{\left(0.0625 \cdot \sin y - \sin x\right)}^{2}}{\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot \sin y\right)}\right)} \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \]

6. Simplified 0.5

   \[\leadsto \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\left(2 \cdot \frac{{\left(\sin y \cdot 0.0625 - \sin x\right)}^{2}}{\sqrt{2} \cdot \left(\sin x - \sin y \cdot 0.0625\right)}\right)} \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \]
   Proof

   [Start] 0.5	\[ \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(2 \cdot \frac{{\left(0.0625 \cdot \sin y - \sin x\right)}^{2}}{\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot \sin y\right)}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(2 \cdot \frac{{\left(\color{blue}{\sin y \cdot 0.0625} - \sin x\right)}^{2}}{\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot \sin y\right)}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(2 \cdot \frac{{\left(\sin y \cdot 0.0625 - \sin x\right)}^{2}}{\sqrt{2} \cdot \left(\sin x - \color{blue}{\sin y \cdot 0.0625}\right)}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \]

7. Applied egg-rr 0.5

   \[\leadsto \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(2 \cdot \color{blue}{\left(\left(0 - \left(-1 - \frac{{\left(\sin y \cdot 0.0625 - \sin x\right)}^{2}}{\sqrt{2} \cdot \left(\sin x - \sin y \cdot 0.0625\right)}\right)\right) - 1\right)}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \]

8. Final simplification 0.5

   \[\leadsto \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(2 \cdot \left(\left(0 - \left(-1 - \frac{{\left(\sin y \cdot 0.0625 - \sin x\right)}^{2}}{\sqrt{2} \cdot \left(\sin x - \sin y \cdot 0.0625\right)}\right)\right) - 1\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \]

Alternatives

Alternative 1
Error	0.5
Cost	92672

\[\begin{array}{l} t_0 := \sin y \cdot 0.0625\\ \frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(2 \cdot \frac{{\left(t_0 - \sin x\right)}^{2}}{\sqrt{2} \cdot \left(\sin x - t_0\right)}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} + -1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)} \end{array} \]

Alternative 2
Error	0.5
Cost	72896

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

Alternative 3
Error	0.5
Cost	72896

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

Alternative 4
Error	0.5
Cost	72768

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

Alternative 5
Error	0.5
Cost	72768

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

Alternative 6
Error	0.5
Cost	72768

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

Alternative 7
Error	0.5
Cost	72768

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

Alternative 8
Error	12.1
Cost	66760

\[\begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_3 := \frac{2 + t_1 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(t_2 + t_0\right)\right)}\\ \mathbf{if}\;x \leq -0.0058:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{2 + \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(t_1 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + t_2\right) + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	12.1
Cost	66504

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \sqrt{5} + -1\\ t_3 := \frac{2 + t_1 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{t_2}{2} + \cos y \cdot \frac{t_0}{2}\right)\right)}\\ \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_2 + t_0 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	13.2
Cost	60104

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 3 - \sqrt{5}\\ t_3 := t_2 \cdot \cos y\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_3 + t_0 \cdot \cos x\right)\right)}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_0 + t_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos y \cdot t_2 + \cos x \cdot t_0\right)}\\ \end{array} \]

Alternative 11
Error	13.2
Cost	59844

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 3 - \sqrt{5}\\ t_3 := t_2 \cdot \cos y\\ \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_3 + t_0 \cdot \cos x\right)\right)}\\ \mathbf{elif}\;x \leq 0.000112:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_0 + t_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos y \cdot t_2 + \cos x \cdot t_0\right)}\\ \end{array} \]

Alternative 12
Error	13.2
Cost	53704

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ t_2 := 0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos y \cdot t_1 + \cos x \cdot t_0\right)}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 0.00082:\\ \;\;\;\;\frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_0 + t_1 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	13.3
Cost	53576

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos y \cdot t_0 + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin y\right) \cdot -0.0625\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\left(\sqrt{5} + t_0 \cdot \cos y\right) - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	13.4
Cost	53384

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_2 := 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(\cos y \cdot t_0 + t_1\right)}\\ \mathbf{if}\;y \leq -3 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.82 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right)}{1 + 0.5 \cdot \left(t_0 + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	13.3
Cost	53384

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ t_2 := 0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos y \cdot t_1 + \cos x \cdot t_0\right)}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_0 + t_1 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	13.7
Cost	53256

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := 0.3333333333333333 \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right)}{1 + 0.5 \cdot \left(t_0 + \cos x \cdot t_1\right)}\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(t_1 + t_0 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	13.7
Cost	46856

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 0.3333333333333333 \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right)}{1 + 0.5 \cdot \left(t_0 + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(\sqrt{5} + \left(\cos y \cdot t_0 + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	26.0
Cost	46592

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

Alternative 19
Error	38.4
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023077 
(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))))))