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

Average Error: 0.72% → 0.62%

Time: 40.5s

Precision: binary64

Cost: 91584

Math

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

↓

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

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
 (/
  (fma
   (sqrt 2.0)
   (*
    (* (+ (sin x) (* (sin y) -0.0625)) (- (cos y) (cos x)))
    (- (* (sin x) 0.0625) (sin y)))
   2.0)
  (fma
   (cos x)
   (/ 6.0 (+ (sqrt 5.0) 1.0))
   (fma (cos y) (* -1.5 (+ (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) {
	return fma(sqrt(2.0), (((sin(x) + (sin(y) * -0.0625)) * (cos(y) - cos(x))) * ((sin(x) * 0.0625) - sin(y))), 2.0) / fma(cos(x), (6.0 / (sqrt(5.0) + 1.0)), fma(cos(y), (-1.5 * (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)
	return Float64(fma(sqrt(2.0), Float64(Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(cos(y) - cos(x))) * Float64(Float64(sin(x) * 0.0625) - sin(y))), 2.0) / fma(cos(x), Float64(6.0 / Float64(sqrt(5.0) + 1.0)), fma(cos(y), Float64(-1.5 * 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_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision] - N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(6.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(-1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.72

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

2. Simplified 0.73

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

   [Start] 0.72	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied egg-rr 0.73

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

4. Applied egg-rr 0.62

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{6}{\sqrt{5} - -1}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)}} \]

5. Final simplification 0.62

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

Alternatives

Alternative 1
Error	0.67%
Cost	78912

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

Alternative 2
Error	0.67%
Cost	72768

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

Alternative 3
Error	0.77%
Cost	72640

\[0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{\frac{\cos y}{1.5 + \sqrt{1.25}} + \left(1 + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right)} \]

Alternative 4
Error	0.68%
Cost	72640

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

Alternative 5
Error	18.12%
Cost	67017

\[\begin{array}{l} \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 0.065\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos y \cdot \frac{1}{1.5 + \sqrt{1.25}} - \cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)\right)}{3 + \left(-1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right) + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	18.17%
Cost	66889

\[\begin{array}{l} \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 0.035\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos y \cdot \frac{1}{1.5 + \sqrt{1.25}} - \cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos y - \cos x\right)\right)\right)}{3 + \left(-1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right) + \left(\sqrt{5} + -1\right) \cdot \left(1.5 + \left(x \cdot x\right) \cdot -0.75\right)\right)}\\ \end{array} \]

Alternative 7
Error	18.27%
Cost	66761

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 0.00031\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot t_0\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos y \cdot \frac{1}{1.5 + \sqrt{1.25}} - \cos x \cdot \left(0.5 - t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(\cos y + -1\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_1\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	18.19%
Cost	66761

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 0.00031\right):\\ \;\;\;\;\frac{2 + \left(t_1 \cdot t_0\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos y \cdot \frac{1}{1.5 + \sqrt{1.25}} - \cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 9
Error	18.27%
Cost	66505

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 0.00031\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot t_0\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos y \cdot \frac{1}{1.5 + \sqrt{1.25}} - \cos x \cdot \left(0.5 - t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right) \cdot \left(t_0 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_1\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	18.3%
Cost	66377

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \cos x \cdot \left(-0.5 + t_0\right)\\ \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 0.00031\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot t_1\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(t_2 + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right) \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(t_2 + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	19.86%
Cost	60361

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)\\ \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 0.00031\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_1 \cdot \left(1 - \cos x\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right) \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right)}{t_2}\\ \end{array} \]

Alternative 12
Error	20.11%
Cost	60233

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)\\ \mathbf{if}\;x \leq -0.00055 \lor \neg \left(x \leq 0.00031\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_1 \cdot \left(1 - \cos x\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin y \cdot 0.0625\right)\right) \cdot \left(t_1 \cdot \left(\cos y + -1\right)\right)}{t_2}\\ \end{array} \]

Alternative 13
Error	19.98%
Cost	60233

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 2.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_1 \cdot \left(1 - \cos x\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_1 \cdot \left(\cos y + -1\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_2 - \cos y \cdot \left(t_2 + -1.5\right)\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	20.14%
Cost	60105

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-6} \lor \neg \left(x \leq 1.3 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos x\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	20.58%
Cost	59977

\[\begin{array}{l} \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 1.1 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right) + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	20.85%
Cost	46857

\[\begin{array}{l} t_0 := \sqrt{5} + -3\\ \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)}{3 + \left(-1.5 \cdot t_0 - \frac{\cos x}{\sqrt{5} + 1} \cdot -6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + -1.5 \cdot \left(\cos y \cdot t_0\right)\right)}\\ \end{array} \]

Alternative 17
Error	20.97%
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\\ t_2 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(3 + \left(t_2 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{0.5 + \left(t_0 - \cos y \cdot \left(t_0 + -1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + t_2\right)}\\ \end{array} \]

Alternative 18
Error	20.9%
Cost	46856

\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\\ t_1 := \sqrt{5} + -1\\ t_2 := \cos x \cdot t_1\\ \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(3 + \left(t_2 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0001:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{1.5 \cdot t_1 + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + t_2\right)}\\ \end{array} \]

Alternative 19
Error	20.88%
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos x \cdot t_0\\ t_2 := {\sin x}^{2}\\ t_3 := \sqrt{5} + -3\\ \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot t_2\right)}{3 + 1.5 \cdot \left(3 + \left(t_1 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{1.5 \cdot t_0 + \left(3 + -1.5 \cdot \left(\cos y \cdot t_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot \left(1 - \cos x\right)\right)\right)}{3 + \left(-1.5 \cdot t_3 + 1.5 \cdot t_1\right)}\\ \end{array} \]

Alternative 20
Error	20.85%
Cost	46856

\[\begin{array}{l} t_0 := 2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)\\ t_1 := \sqrt{5} + -3\\ t_2 := -1.5 \cdot t_1\\ t_3 := \sqrt{5} + 1\\ \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{t_0}{3 + \left(t_2 - \frac{\cos x}{t_3} \cdot -6\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + -1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + \left(t_2 + \frac{6}{\frac{t_3}{\cos x}}\right)}\\ \end{array} \]

Alternative 21
Error	20.84%
Cost	46856

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} + 1\\ t_2 := \sqrt{5} + -3\\ \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot t_0\right)}{3 - 1.5 \cdot \left(\cos x \cdot \frac{-4}{t_1} + t_2\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + -1.5 \cdot \left(\cos y \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot \left(1 - \cos x\right)\right)\right)}{3 + \left(-1.5 \cdot t_2 + \frac{6}{\frac{t_1}{\cos x}}\right)}\\ \end{array} \]

Alternative 22
Error	20.96%
Cost	46729

\[\begin{array}{l} \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 6.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{\cos y}{1.5 + \sqrt{1.25}} + \left(\sqrt{1.25} + 0.5\right)}\\ \end{array} \]

Alternative 23
Error	20.95%
Cost	46728

\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(3 + \left(t_1 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{\cos y}{1.5 + \sqrt{1.25}} + \left(\sqrt{1.25} + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + t_1\right)}\\ \end{array} \]

Alternative 24
Error	40.49%
Cost	46336

\[0.3333333333333333 \cdot \frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{\cos y}{1.5 + \sqrt{1.25}} + \left(\sqrt{1.25} + 0.5\right)} \]

Alternative 25
Error	59.52%
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 26
Error	59.57%
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023101 
(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))))))