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

Average Error: 0.5 → 0.4

Time: 38.7s

Precision: binary64

Cost: 85440

Math

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

↓

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

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 y) (/ (sin x) 16.0))
    (* (- (sin x) (/ (sin y) 16.0)) (- (cos x) (cos 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(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * (cos(x) - cos(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(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(cos(x) - cos(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[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $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 0.5

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

2. Simplified 0.5

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

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

3. Applied egg-rr 0.4

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

4. Simplified 0.4

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

   [Start] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   associate-/r* [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

5. Applied egg-rr 0.4

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

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	85312

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

Alternative 2
Error	0.5
Cost	72896

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{1 + \left(\cos x \cdot \left(t_0 + -0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \]

Alternative 3
Error	0.4
Cost	72896

\[\frac{2 + \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\right)\right)} \]

Alternative 4
Error	0.4
Cost	72896

\[\frac{2 + \frac{\sin x + \sin y \cdot -0.0625}{\frac{1}{\cos x - \cos y}} \cdot \left(\sqrt{2} \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{\sqrt{1.25} + 1.5} + \cos x \cdot \left(\sqrt{5} \cdot 0.5 + -0.5\right)\right)\right)} \]

Alternative 5
Error	0.4
Cost	72768

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

Alternative 6
Error	0.4
Cost	72768

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

Alternative 7
Error	11.7
Cost	67145

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0085 \lor \neg \left(y \leq 0.115\right):\\ \;\;\;\;\frac{2 + \frac{\sin x + \sin y \cdot -0.0625}{\frac{1}{\cos x - \cos y}} \cdot \left(\sqrt{2} \cdot \sin y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\cos x + \left(0.5 \cdot \left(y \cdot y\right) + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 8
Error	11.8
Cost	66825

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \sqrt{1.25} + 1.5\\ \mathbf{if}\;y \leq -0.0034 \lor \neg \left(y \leq 0.045\right):\\ \;\;\;\;\frac{2 + \frac{\sin x + \sin y \cdot -0.0625}{\frac{1}{\cos x - \cos y}} \cdot \left(\sqrt{2} \cdot \sin y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{t_1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot t_0\right) + y \cdot \left(\left(\sin x \cdot 1.00390625\right) \cdot t_0\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{t_1} + \cos x \cdot \left(\sqrt{5} \cdot 0.5 + -0.5\right)\right)\right)}\\ \end{array} \]

Alternative 9
Error	11.8
Cost	66761

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0075 \lor \neg \left(y \leq 0.045\right):\\ \;\;\;\;\frac{2 + \frac{\sin x + \sin y \cdot -0.0625}{\frac{1}{\cos x - \cos y}} \cdot \left(\sqrt{2} \cdot \sin y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 10
Error	11.8
Cost	66633

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \frac{1}{\sqrt{1.25} + 1.5}\\ \mathbf{if}\;y \leq -0.00076 \lor \neg \left(y \leq 0.00026\right):\\ \;\;\;\;\frac{2 + \frac{\sin x + \sin y \cdot -0.0625}{\frac{1}{\cos x - \cos y}} \cdot \left(\sqrt{2} \cdot \sin y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot t_0\right) + y \cdot \left(\left(\sin x \cdot 1.00390625\right) \cdot t_0\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 + -0.5\right) + t_1\right)\right)}\\ \end{array} \]

Alternative 11
Error	11.9
Cost	66504

\[\begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ t_1 := \sin y + \sin x \cdot -0.0625\\ t_2 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.000116:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(t_3 \cdot \left(\sin x \cdot t_1\right)\right)}{1 + \left(0.5 \cdot t_2 + 0.5 \cdot t_0\right)}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot t_3\right)}{3 \cdot \left(1 + \left(\left(\frac{\cos y}{\sqrt{1.25} + 1.5} + \sqrt{5} \cdot 0.5\right) + -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(t_1 \cdot \left(\sin x \cdot t_3\right)\right)}{1 + 0.5 \cdot \left(t_0 + t_2\right)}\\ \end{array} \]

Alternative 12
Error	12.0
Cost	66377

\[\begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-5} \lor \neg \left(x \leq 1.9 \cdot 10^{-5}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x \cdot \left(\cos x - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(t_0 + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \frac{\sqrt{2} \cdot \left(1 - \cos y\right)}{2} \cdot \left(1 - \cos \left(y + y\right)\right), 2\right)}{3 + \left(\frac{6}{\sqrt{5} + 1} + 1.5 \cdot t_0\right)}\\ \end{array} \]

Alternative 13
Error	12.0
Cost	66376

\[\begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ t_1 := \sin y + \sin x \cdot -0.0625\\ t_2 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(t_3 \cdot \left(\sin x \cdot t_1\right)\right)}{1 + \left(0.5 \cdot t_2 + 0.5 \cdot t_0\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \frac{\sqrt{2} \cdot \left(1 - \cos y\right)}{2} \cdot \left(1 - \cos \left(y + y\right)\right), 2\right)}{3 + \left(\frac{6}{\sqrt{5} + 1} + 1.5 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(t_1 \cdot \left(\sin x \cdot t_3\right)\right)}{1 + 0.5 \cdot \left(t_0 + t_2\right)}\\ \end{array} \]

Alternative 14
Error	12.9
Cost	65929

\[\begin{array}{l} t_0 := \cos x + -1\\ \mathbf{if}\;y \leq -0.000125 \lor \neg \left(y \leq 6.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{\cos x \cdot \frac{6}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot t_0\right) + y \cdot \left(\left(\sin x \cdot 1.00390625\right) \cdot t_0\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 + -0.5\right) + \frac{1}{\sqrt{1.25} + 1.5}\right)\right)}\\ \end{array} \]

Alternative 15
Error	13.0
Cost	60425

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \frac{1}{\sqrt{1.25} + 1.5}\\ \mathbf{if}\;y \leq -0.0026 \lor \neg \left(y \leq 7 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot t_0\right) + y \cdot \left(\left(\sin x \cdot 1.00390625\right) \cdot t_0\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 + -0.5\right) + t_1\right)\right)}\\ \end{array} \]

Alternative 16
Error	13.0
Cost	60104

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x + -1\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left({\sin x}^{2} \cdot t_1\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \frac{\sqrt{2} \cdot \left(1 - \cos y\right)}{2} \cdot \left(1 - \cos \left(y + y\right)\right), 2\right)}{3 + \left(\frac{6}{\sqrt{5} + 1} + 1.5 \cdot \left(\cos y \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \end{array} \]

Alternative 17
Error	13.1
Cost	53513

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-5} \lor \neg \left(y \leq 3.25 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)}\\ \end{array} \]

Alternative 18
Error	13.1
Cost	53512

\[\begin{array}{l} t_0 := 2 + \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\right)\right)}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \frac{\sqrt{2} \cdot \left(1 - \cos y\right)}{2} \cdot \left(1 - \cos \left(y + y\right)\right), 2\right)}{3 + \left(\frac{6}{\sqrt{5} + 1} + 1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t_1}{2}\right)}\\ \end{array} \]

Alternative 19
Error	13.1
Cost	53380

\[\begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ t_1 := {\sin x}^{2} \cdot \left(\cos x + -1\right)\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\right)\right)}\\ \mathbf{elif}\;x \leq 7.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \frac{\sqrt{2} \cdot \left(1 - \cos y\right)}{2} \cdot \left(1 - \cos \left(y + y\right)\right), 2\right)}{3 + \left(\frac{6}{\sqrt{5} + 1} + 1.5 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{3 + 1.5 \cdot \left(t_0 + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 20
Error	13.1
Cost	53257

\[\begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(t_0 + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \frac{\sqrt{2} \cdot \left(1 - \cos y\right)}{2} \cdot \left(1 - \cos \left(y + y\right)\right), 2\right)}{3 + \left(\frac{6}{\sqrt{5} + 1} + 1.5 \cdot t_0\right)}\\ \end{array} \]

Alternative 21
Error	13.1
Cost	53256

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ t_2 := 1.5 \cdot t_1\\ t_3 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{3 + \left(t_2 + 1.5 \cdot t_3\right)}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \frac{\sqrt{2} \cdot \left(1 - \cos y\right)}{2} \cdot \left(1 - \cos \left(y + y\right)\right), 2\right)}{3 + \left(\frac{6}{\sqrt{5} + 1} + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(t_1 + t_3\right)}\\ \end{array} \]

Alternative 22
Error	13.4
Cost	47049

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-6} \lor \neg \left(x \leq 3.65 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \frac{\sqrt{2} \cdot \left(1 - \cos y\right)}{2} \cdot \left(1 - \cos \left(y + y\right)\right), 2\right)}{3 + \left(\frac{6}{\sqrt{5} + 1} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 23
Error	13.4
Cost	46985

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-5} \lor \neg \left(x \leq 1.55 \cdot 10^{-5}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{1}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 24
Error	13.5
Cost	46857

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -9 \cdot 10^{-6} \lor \neg \left(x \leq 7.6 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \end{array} \]

Alternative 25
Error	24.1
Cost	46601

\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-7} \lor \neg \left(x \leq 2.8 \cdot 10^{-7}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.6666666666666666}{\frac{\cos y}{\sqrt{1.25} + 1.5} + \left(0.5 + \sqrt{5} \cdot 0.5\right)}\\ \end{array} \]

Alternative 26
Error	35.8
Cost	40777

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-86} \lor \neg \left(y \leq 9.6 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{0.6666666666666666}{\frac{\cos y}{\sqrt{1.25} + 1.5} + \left(0.5 + \sqrt{5} \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.5\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\sin y \cdot x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 27
Error	36.8
Cost	20032

\[\frac{0.6666666666666666}{\frac{\cos y}{\sqrt{1.25} + 1.5} + \left(0.5 + \sqrt{5} \cdot 0.5\right)} \]

Alternative 28
Error	38.0
Cost	13632

\[\frac{2}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + \left(4.5 - \sqrt{11.25}\right)\right)} \]

Reproduce

herbie shell --seed 2023073 
(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))))))