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

Average Error: 0.5 → 0.4

Time: 37.7s

Precision: binary64

Cost: 111168

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\sqrt{5} + -1}\\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(\cos x, t_0 \cdot \left(t_0 \cdot 1.5\right), \mathsf{fma}\left(\cos y, 1.5 \cdot \frac{4}{\sqrt{5} + 3}, 3\right)\right)} \end{array} \]

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 (sqrt (+ (sqrt 5.0) -1.0))))
   (/
    (fma
     (sqrt 2.0)
     (*
      (* (- (cos x) (cos y)) (- (sin x) (* (sin y) 0.0625)))
      (+ (sin y) (* -0.0625 (sin x))))
     2.0)
    (fma
     (cos x)
     (* t_0 (* t_0 1.5))
     (fma (cos y) (* 1.5 (/ 4.0 (+ (sqrt 5.0) 3.0))) 3.0)))))

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 = sqrt((sqrt(5.0) + -1.0));
	return fma(sqrt(2.0), (((cos(x) - cos(y)) * (sin(x) - (sin(y) * 0.0625))) * (sin(y) + (-0.0625 * sin(x)))), 2.0) / fma(cos(x), (t_0 * (t_0 * 1.5)), fma(cos(y), (1.5 * (4.0 / (sqrt(5.0) + 3.0))), 3.0));
}

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 = sqrt(Float64(sqrt(5.0) + -1.0))
	return Float64(fma(sqrt(2.0), Float64(Float64(Float64(cos(x) - cos(y)) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * Float64(sin(y) + Float64(-0.0625 * sin(x)))), 2.0) / fma(cos(x), Float64(t_0 * Float64(t_0 * 1.5)), fma(cos(y), Float64(1.5 * Float64(4.0 / Float64(sqrt(5.0) + 3.0))), 3.0)))
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[Sqrt[N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 * N[(t$95$0 * 1.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $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 0.4

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{0.6666666666666666}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)}} \]
   Proof

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

3. Applied egg-rr 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{\sqrt{5} + -1} \cdot \left(\sqrt{\sqrt{5} + -1} \cdot 1.5\right)}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]

4. Applied egg-rr 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \sqrt{\sqrt{5} + -1} \cdot \left(\sqrt{\sqrt{5} + -1} \cdot 1.5\right), \mathsf{fma}\left(\cos y, 1.5 \cdot \color{blue}{\frac{4}{\sqrt{5} + 3}}, 3\right)\right)} \]

5. Final simplification 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(\cos x, \sqrt{\sqrt{5} + -1} \cdot \left(\sqrt{\sqrt{5} + -1} \cdot 1.5\right), \mathsf{fma}\left(\cos y, 1.5 \cdot \frac{4}{\sqrt{5} + 3}, 3\right)\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	111040

\[\begin{array}{l} t_0 := \sqrt{\sqrt{5} + -1}\\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(\cos x, t_0 \cdot \left(t_0 \cdot 1.5\right), \mathsf{fma}\left(\cos y, -1.5 \cdot \left(\sqrt{5} + -3\right), 3\right)\right)} \end{array} \]

Alternative 2
Error	0.4
Cost	91712

\[\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{1.5 + \sqrt{1.25}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right)\right)} \]

Alternative 3
Error	0.5
Cost	72768

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

Alternative 4
Error	0.4
Cost	72768

\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(\left(\cos x \cdot \left(1 - \sqrt{5}\right)\right) \cdot -1.5 - \frac{\cos y}{\sqrt{5} + 3} \cdot -6\right)} \]

Alternative 5
Error	0.5
Cost	72640

\[\frac{2 - \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\right)}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - \sqrt{1.25}\right) - \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)} \]

Alternative 6
Error	11.7
Cost	67145

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.047 \lor \neg \left(x \leq 0.47\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} - \sin x\right)\right) \cdot \left(t_0 \cdot \left(-1 + \left(\cos y + 0.5 \cdot \left(x \cdot x\right)\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_1 + -0.5\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	11.8
Cost	66633

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.0138 \lor \neg \left(x \leq 0.47\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 0.0625 - x\right)\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 8
Error	11.8
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.04 \lor \neg \left(x \leq 0.47\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot t_1\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 + -0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 0.0625 - x\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 9
Error	12.9
Cost	66376

\[\begin{array}{l} t_0 := \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.0255:\\ \;\;\;\;\frac{2 + \left(\cos y - \cos x\right) \cdot \left(\sqrt{2} \cdot \left(0.0625 \cdot {\sin x}^{2}\right)\right)}{t_1}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;\frac{2 - \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right) \cdot \left(\left(\sin y - t_0\right) \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 0.0625 - x\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(t_0 - \sin y\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right) - \sqrt{1.25}\right)\right)}\\ \end{array} \]

Alternative 10
Error	12.8
Cost	60744

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.041:\\ \;\;\;\;\frac{2 + \left(\cos y - \cos x\right) \cdot \left(\sqrt{2} \cdot \left(0.0625 \cdot {\sin x}^{2}\right)\right)}{t_0}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;\frac{2 - \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 0.0625 - x\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{t_0}\\ \end{array} \]

Alternative 11
Error	12.8
Cost	60616

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \sqrt{5} + -1\\ t_2 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.0034:\\ \;\;\;\;\frac{2 + \left(\cos y - \cos x\right) \cdot \left(\sqrt{2} \cdot \left(0.0625 \cdot {\sin x}^{2}\right)\right)}{t_2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;\frac{2 - \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 0.0625 - x\right)\right)\right)}{3 \cdot \left(\left(1 - t_1 \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.25\right)\right) - \cos y \cdot \frac{\sqrt{5} + -3}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{t_2}\\ \end{array} \]

Alternative 12
Error	13.0
Cost	60104

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\cos y - \cos x\right) \cdot \left(\sqrt{2} \cdot \left(0.0625 \cdot {\sin x}^{2}\right)\right)}{t_0}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(x \cdot \left(\sin y \cdot -1.00390625\right) + 0.0625 \cdot {\sin y}^{2}\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_1 - \cos y \cdot \left(t_1 + -1.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{t_0}\\ \end{array} \]

Alternative 13
Error	13.1
Cost	59780

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\cos y - \cos x\right) \cdot \left(\sqrt{2} \cdot \left(0.0625 \cdot t_0\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(x \cdot \left(\sin y \cdot -1.00390625\right) + 0.0625 \cdot {\sin y}^{2}\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_2 - \cos y \cdot \left(t_2 + -1.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(0.0625 \cdot \left(t_0 \cdot \left(1 - \cos x\right)\right)\right)}{3 \cdot \left(\cos x \cdot \left(t_1 + -0.5\right) + \left(1 + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	13.1
Cost	53769

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7} \lor \neg \left(x \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(0.0625 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)}{3 \cdot \left(\cos x \cdot \left(t_1 + -0.5\right) + \left(1 + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(x \cdot \left(\sin y \cdot -1.00390625\right) + 0.0625 \cdot {\sin y}^{2}\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_0 - \cos y \cdot \left(t_0 + -1.5\right)\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	13.1
Cost	53513

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7} \lor \neg \left(x \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(0.0625 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)}{3 \cdot \left(\cos x \cdot \left(t_0 + -0.5\right) + \left(1 + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{\left(\sqrt{5} + -1\right) \cdot 1.5 + \left(3 + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}\\ \end{array} \]

Alternative 16
Error	13.5
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_2 := \sqrt{2} \cdot 0.0625\\ t_3 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + t_2 \cdot t_3}{3 + 1.5 \cdot \left(3 + \left(t_1 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(0.5 - \frac{\cos \left(y + y\right)}{2}\right) \cdot \left(\cos y + -1\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_0 - \cos y \cdot \left(t_0 + -1.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot t_3\right)}{1 - -0.5 \cdot \left(\left(3 - \sqrt{5}\right) + t_1\right)}\\ \end{array} \]

Alternative 17
Error	13.5
Cost	46856

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := \cos x \cdot t_1\\ t_3 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_3}{3 + 1.5 \cdot \left(3 + \left(t_2 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t_1 \cdot 1.5 + \left(3 + 1.5 \cdot \left(\cos y \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot t_3\right)}{1 - -0.5 \cdot \left(t_0 + t_2\right)}\\ \end{array} \]

Alternative 18
Error	13.5
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos x \cdot t_0\\ t_2 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_2}{3 + 1.5 \cdot \left(3 + \left(t_1 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t_0 \cdot 1.5 + \left(3 + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot t_2\right)}{1 - -0.5 \cdot \left(\left(3 - \sqrt{5}\right) + t_1\right)}\\ \end{array} \]

Alternative 19
Error	13.5
Cost	46856

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_1}{3 + 1.5 \cdot \left(t_0 - \cos x \cdot \frac{-4}{\sqrt{5} + 1}\right)}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t_2 \cdot 1.5 + \left(3 + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{1 - -0.5 \cdot \left(t_0 + \cos x \cdot t_2\right)}\\ \end{array} \]

Alternative 20
Error	13.6
Cost	46729

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \sqrt{2} \cdot 0.0625\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7} \lor \neg \left(x \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + t_1 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{3 - 1.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right) + \left(\sqrt{5} + -3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(0.5 - \frac{\cos \left(y + y\right)}{2}\right) \cdot \left(\cos y + -1\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_0 - \cos y \cdot \left(t_0 + -1.5\right)\right)\right)\right)}\\ \end{array} \]

Alternative 21
Error	13.6
Cost	46729

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \sqrt{2} \cdot 0.0625\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-7} \lor \neg \left(x \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + t_1 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{3 + 1.5 \cdot \left(3 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(0.5 - \frac{\cos \left(y + y\right)}{2}\right) \cdot \left(\cos y + -1\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_0 - \cos y \cdot \left(t_0 + -1.5\right)\right)\right)\right)}\\ \end{array} \]

Alternative 22
Error	26.1
Cost	40640

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left(\left(0.5 - \frac{\cos \left(y + y\right)}{2}\right) \cdot \left(\cos y + -1\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_0 - \cos y \cdot \left(t_0 + -1.5\right)\right)\right)\right)} \end{array} \]

Alternative 23
Error	38.3
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 24
Error	38.3
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023040 
(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))))))