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

Average Error: 0.73% → 0.6%

Time: 34.7s

Precision: binary64

Cost: 85440

Math

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

↓

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos y - \cos x\right)\right) \cdot \left(\frac{\sin x}{16} - \sin y\right), 2\right)}{\cos x \cdot \frac{6}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]

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
 (/
  (fma
   (sqrt 2.0)
   (*
    (* (- (sin x) (/ (sin y) 16.0)) (- (cos y) (cos x)))
    (- (/ (sin x) 16.0) (sin y)))
   2.0)
  (+
   (* (cos x) (/ 6.0 (+ (sqrt 5.0) 1.0)))
   (fma (cos y) (/ (/ 4.0 (+ (sqrt 5.0) 3.0)) 0.6666666666666666) 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) {
	return fma(sqrt(2.0), (((sin(x) - (sin(y) / 16.0)) * (cos(y) - cos(x))) * ((sin(x) / 16.0) - sin(y))), 2.0) / ((cos(x) * (6.0 / (sqrt(5.0) + 1.0))) + fma(cos(y), ((4.0 / (sqrt(5.0) + 3.0)) / 0.6666666666666666), 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)
	return Float64(fma(sqrt(2.0), Float64(Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(cos(y) - cos(x))) * Float64(Float64(sin(x) / 16.0) - sin(y))), 2.0) / Float64(Float64(cos(x) * Float64(6.0 / Float64(sqrt(5.0) + 1.0))) + fma(cos(y), Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) / 0.6666666666666666), 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_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision] - N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(6.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.73

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

2. Simplified 0.71

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

   [Start] 0.73	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied egg-rr 0.62

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

4. Simplified 0.62

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   Proof

   [Start] 0.62	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   associate-/r* [=>] 0.62	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   metadata-eval [=>] 0.62	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

5. Applied egg-rr 0.6

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

6. Final simplification 0.6

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos y - \cos x\right)\right) \cdot \left(\frac{\sin x}{16} - \sin y\right), 2\right)}{\cos x \cdot \frac{6}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]

Alternatives

Alternative 1
Error	0.61%
Cost	78912

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

Alternative 2
Error	0.61%
Cost	72640

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

Alternative 3
Error	18.8%
Cost	67145

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0003 \lor \neg \left(x \leq 0.042\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_0 \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - t_1\right) + \cos y \cdot \frac{-1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_1 + -0.5\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	18.82%
Cost	66761

\[\begin{array}{l} t_0 := \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0003 \lor \neg \left(x \leq 0.024\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot t_0}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - t_1\right) + \cos y \cdot \frac{-1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_1 + -0.5\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	18.95%
Cost	66633

\[\begin{array}{l} t_0 := \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-5} \lor \neg \left(x \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot t_0}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right) + \cos y \cdot \frac{-1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_1 - \cos y \cdot \left(t_1 + -1.5\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	19.05%
Cost	66505

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0003 \lor \neg \left(x \leq 0.013\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right) + \cos y \cdot \frac{-1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right)}\\ \end{array} \]

Alternative 7
Error	19.08%
Cost	66377

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00105 \lor \neg \left(x \leq 0.013\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) - \cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right)}\\ \end{array} \]

Alternative 8
Error	20.75%
Cost	65928

\[\begin{array}{l} t_0 := \frac{\cos y}{\sqrt{5} + 3}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;\frac{1}{3 \cdot \left(\cos x \cdot \left(t_1 + -0.5\right) + \left(1 + \cos y \cdot \left(1.5 - t_1\right)\right)\right)} \cdot \left(2 + \sqrt{2} \cdot \left(0.0625 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.013:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(6, t_0, \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot t_0 + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 9
Error	20.72%
Cost	65928

\[\begin{array}{l} t_0 := \sqrt{5} + 1\\ t_1 := \frac{\cos y}{\sqrt{5} + 3}\\ \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, 0.0625 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right), 2\right)}{\cos x \cdot \frac{6}{t_0} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}\\ \mathbf{elif}\;x \leq 0.013:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(6, t_1, \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot t_1 + 6 \cdot \frac{\cos x}{t_0}\right)}\\ \end{array} \]

Alternative 10
Error	20.72%
Cost	65928

\[\begin{array}{l} t_0 := \sqrt{5} + 1\\ t_1 := \sqrt{5} + 3\\ t_2 := \frac{\cos y}{t_1}\\ \mathbf{if}\;x \leq -0.00076:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, 0.0625 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right), 2\right)}{\cos x \cdot \frac{6}{t_0} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{t_1}}{0.6666666666666666}, 3\right)}\\ \mathbf{elif}\;x \leq 0.013:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(6, t_2, \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot t_2 + 6 \cdot \frac{\cos x}{t_0}\right)}\\ \end{array} \]

Alternative 11
Error	20.79%
Cost	59976

\[\begin{array}{l} t_0 := 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{3 \cdot \left(\cos x \cdot \left(t_1 + -0.5\right) + \left(1 + \cos y \cdot \left(1.5 - t_1\right)\right)\right)} \cdot \left(2 + \sqrt{2} \cdot \left(0.0625 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{-1.5 \cdot \left(1 - \sqrt{5}\right) + \left(3 + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(t_0 + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 12
Error	20.87%
Cost	59721

\[\begin{array}{l} t_0 := \sqrt{5} + 3\\ \mathbf{if}\;y \leq -0.046 \lor \neg \left(y \leq 29500\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t_0} - 1.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(\frac{6}{t_0} + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 13
Error	20.81%
Cost	53641

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0003 \lor \neg \left(x \leq 0.013\right):\\ \;\;\;\;\frac{1}{3 \cdot \left(\cos x \cdot \left(t_0 + -0.5\right) + \left(1 + \cos y \cdot \left(1.5 - t_0\right)\right)\right)} \cdot \left(2 + \sqrt{2} \cdot \left(0.0625 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{\sqrt{5} + 3} - 1.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	21%
Cost	53385

\[\begin{array}{l} t_0 := \sqrt{5} + 3\\ \mathbf{if}\;y \leq -0.046 \lor \neg \left(y \leq 29500\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t_0} - 1.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{3 + 1.5 \cdot \left(\frac{4}{t_0} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 15
Error	21.47%
Cost	46856

\[\begin{array}{l} t_0 := \cos x \cdot \left(1 - \sqrt{5}\right)\\ t_1 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ t_2 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) - t_0\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(t_2 - \cos y \cdot \left(t_2 + -1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_1}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} + t_0\right)\right)}\\ \end{array} \]

Alternative 16
Error	21.38%
Cost	46856

\[\begin{array}{l} t_0 := \cos x \cdot \left(1 - \sqrt{5}\right)\\ t_1 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) - t_0\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 6 \cdot \left(\frac{\cos y}{\sqrt{5} + 3} + \frac{1}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_1}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} + t_0\right)\right)}\\ \end{array} \]

Alternative 17
Error	21.37%
Cost	46856

\[\begin{array}{l} t_0 := 1 - \sqrt{5}\\ t_1 := \cos x \cdot t_0\\ t_2 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - -0.0625 \cdot \left(\sqrt{2} \cdot t_2\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) - t_1\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{-1.5 \cdot t_0 + \left(3 + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_2}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} + t_1\right)\right)}\\ \end{array} \]

Alternative 18
Error	21.35%
Cost	46856

\[\begin{array}{l} t_0 := 1 - \sqrt{5}\\ t_1 := 2 - -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + 0.5 \cdot \left(t_2 - \cos x \cdot t_0\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{-1.5 \cdot t_0 + \left(3 + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot t_2\right)}\\ \end{array} \]

Alternative 19
Error	21.34%
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} + 3\\ t_1 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_1}{3 + 1.5 \cdot \left(\frac{4}{t_0} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{-1.5 \cdot \left(1 - \sqrt{5}\right) + \left(3 + 6 \cdot \frac{\cos y}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 20
Error	38.52%
Cost	46729

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-6} \lor \neg \left(x \leq 1.6 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) - \cos x \cdot \left(1 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{6}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{6}{\sqrt{5} + 3}, 3\right)}\\ \end{array} \]

Alternative 21
Error	38.53%
Cost	46729

\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-8} \lor \neg \left(x \leq 2.3 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} + \cos x \cdot \left(1 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{6}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{6}{\sqrt{5} + 3}, 3\right)}\\ \end{array} \]

Alternative 22
Error	38.52%
Cost	46728

\[\begin{array}{l} t_0 := \cos x \cdot \left(1 - \sqrt{5}\right)\\ t_1 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) - t_0\right)}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{6}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{6}{\sqrt{5} + 3}, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_1}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} + t_0\right)\right)}\\ \end{array} \]

Alternative 23
Error	57.43%
Cost	20288

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

Alternative 24
Error	56.87%
Cost	20288

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

Alternative 25
Error	59.36%
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023088 
(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))))))