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

Average Accuracy: 99.2% → 99.4%

Time: 43.8s

Precision: binary64

Cost: 85312

Math

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

↓

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

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)
   (*
    (* (- (cos x) (cos y)) (- (/ (sin y) 16.0) (sin x)))
    (- (/ (sin x) 16.0) (sin y)))
   2.0)
  (+
   (fma (cos y) (/ (- 3.0 (sqrt 5.0)) 0.6666666666666666) 3.0)
   (* (cos x) (/ 6.0 (+ (sqrt 5.0) 1.0))))))

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), (((cos(x) - cos(y)) * ((sin(y) / 16.0) - sin(x))) * ((sin(x) / 16.0) - sin(y))), 2.0) / (fma(cos(y), ((3.0 - sqrt(5.0)) / 0.6666666666666666), 3.0) + (cos(x) * (6.0 / (sqrt(5.0) + 1.0))));
}

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

Derivation

1. Initial program 99.2%

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

2. Simplified 99.3%

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

   [Start] 99.2	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied egg-rr 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \left(\left(\sqrt{5} + -1\right) \cdot 1.5\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   Proof

   [Start] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   div-inv [=>] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot \frac{1}{0.6666666666666666}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   associate-*l* [=>] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \left(\left(\sqrt{5} + -1\right) \cdot \frac{1}{0.6666666666666666}\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   metadata-eval [=>] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \left(\left(\sqrt{5} + -1\right) \cdot \color{blue}{1.5}\right) + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

4. Applied egg-rr 99.4%

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

   [Start] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \left(\left(\sqrt{5} + -1\right) \cdot 1.5\right) + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   *-commutative [=>] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\left(1.5 \cdot \left(\sqrt{5} + -1\right)\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   flip-+ [=>] 98.9	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \left(1.5 \cdot \color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1}{\sqrt{5} - -1}}\right) + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   add-sqr-sqrt [<=] 99.2	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \left(1.5 \cdot \frac{\color{blue}{5} - -1 \cdot -1}{\sqrt{5} - -1}\right) + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   metadata-eval [=>] 99.2	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \left(1.5 \cdot \frac{5 - \color{blue}{1}}{\sqrt{5} - -1}\right) + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   metadata-eval [=>] 99.2	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \left(1.5 \cdot \frac{\color{blue}{4}}{\sqrt{5} - -1}\right) + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   metadata-eval [<=] 99.2	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \left(1.5 \cdot \frac{\color{blue}{\sqrt{16}}}{\sqrt{5} - -1}\right) + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   associate-*r/ [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{1.5 \cdot \sqrt{16}}{\sqrt{5} - -1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1.5 \cdot \color{blue}{4}}{\sqrt{5} - -1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} - -1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

5. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	85312

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

Alternative 2
Accuracy	99.4%
Cost	79040

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

Alternative 3
Accuracy	99.3%
Cost	72768

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

Alternative 4
Accuracy	99.3%
Cost	72768

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

Alternative 5
Accuracy	81.2%
Cost	67144

\[\begin{array}{l} t_0 := \frac{\sin x}{16}\\ t_1 := t_0 - \sin y\\ t_2 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_3 := \sqrt{2} \cdot \sin x\\ t_4 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.076:\\ \;\;\;\;\frac{2 + \left(\cos y - \cos x\right) \cdot \left(t_3 \cdot t_1\right)}{t_2}\\ \mathbf{elif}\;x \leq 0.024:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - t_0\right) \cdot \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} - \sin x\right)\right)\right) \cdot \left(-1 + \left(\cos y + 0.5 \cdot \left(x \cdot x\right)\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - t_3 \cdot \left(\left(\cos x - \cos y\right) \cdot t_1\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_4\right) + \cos y \cdot \left(1.5 - t_4\right)\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	81.0%
Cost	66760

\[\begin{array}{l} t_0 := \frac{\sin x}{16} - \sin y\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := 3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_1\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)\\ \mathbf{if}\;x \leq -12.8:\\ \;\;\;\;\frac{2 + \left(\cos y - \cos x\right) \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0095:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(\cos y + -1\right)\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - t_2 \cdot \left(\left(\cos x - \cos y\right) \cdot t_0\right)}{t_3}\\ \end{array} \]

Alternative 7
Accuracy	81.2%
Cost	66760

\[\begin{array}{l} t_0 := \frac{\sin x}{16}\\ t_1 := t_0 - \sin y\\ t_2 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_3 := \cos x - \cos y\\ t_4 := \frac{\sqrt{5}}{2}\\ t_5 := \sqrt{2} \cdot \sin x\\ \mathbf{if}\;x \leq -0.056:\\ \;\;\;\;\frac{2 + \left(\cos y - \cos x\right) \cdot \left(t_5 \cdot t_1\right)}{t_2}\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(\left(\sin y - t_0\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - t_5 \cdot \left(t_3 \cdot t_1\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_4\right) + \cos y \cdot \left(1.5 - t_4\right)\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	81.1%
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.011 \lor \neg \left(x \leq 0.0021\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 9
Accuracy	81.1%
Cost	66504

\[\begin{array}{l} t_0 := \frac{\sin x}{16} - \sin y\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := \sqrt{2} \cdot \sin x\\ \mathbf{if}\;x \leq -0.0138:\\ \;\;\;\;\frac{2 + \left(\cos y - \cos x\right) \cdot \left(t_3 \cdot t_0\right)}{t_1}\\ \mathbf{elif}\;x \leq 0.0021:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - t_3 \cdot \left(\left(\cos x - \cos y\right) \cdot t_0\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_2\right) + \cos y \cdot \left(1.5 - t_2\right)\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	79.2%
Cost	65928

\[\begin{array}{l} t_0 := 1 - \cos y\\ t_1 := {\sin y}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.000215:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot t_0\right) \cdot t_1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t_2}{2}\right)}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\frac{\sin x}{16} - \sin y\right) \cdot \left(1 - \cos x\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 - \cos x \cdot \left(0.5 - \sqrt{1.25}\right)\right) - \sqrt{1.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(t_0 \cdot t_1\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{t_2}{0.6666666666666666}, 3\right) + \cos x \cdot \frac{6}{\sqrt{5} + 1}}\\ \end{array} \]

Alternative 11
Accuracy	79.2%
Cost	59977

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00017 \lor \neg \left(y \leq 0.00032\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\frac{\sin x}{16} - \sin y\right) \cdot \left(1 - \cos x\right)\right)}{3 \cdot \left(1 + \left(1.5 - \left(\sqrt{1.25} + \cos x \cdot \left(0.5 - \sqrt{1.25}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	79.2%
Cost	59977

\[\begin{array}{l} \mathbf{if}\;y \leq -0.000112 \lor \neg \left(y \leq 0.00028\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\frac{\sin x}{16} - \sin y\right) \cdot \left(1 - \cos x\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 - \cos x \cdot \left(0.5 - \sqrt{1.25}\right)\right) - \sqrt{1.25}\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	79.1%
Cost	59849

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-5} \lor \neg \left(y \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\frac{\sin x}{16} - \sin y\right) \cdot \left(1 - \cos x\right)\right)}{3 \cdot \left(1 + \left(1.5 + \left(\cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right) + \sqrt{5} \cdot -0.5\right)\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	78.5%
Cost	53513

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-6} \lor \neg \left(y \leq 1.65 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(-0.0625 \cdot \left(\cos x + -1\right)\right)\right)}{1 + -0.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right) + \left(\sqrt{5} + -3\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	79.1%
Cost	53512

\[\begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ t_1 := \cos x + -1\\ t_2 := \cos x \cdot \left(1 - \sqrt{5}\right)\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot t_3\right) \cdot \left(-0.0625 \cdot t_1\right)\right)}{1 + -0.5 \cdot \left(t_2 + \cos y \cdot \left(\sqrt{5} + -3\right)\right)}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{1.5 \cdot t_0 + \left(3 + \frac{6}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_1 \cdot t_3\right)\right)}{1 + \left(-0.5 \cdot t_2 + t_0 \cdot 0.5\right)}\\ \end{array} \]

Alternative 16
Accuracy	79.1%
Cost	53512

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \sqrt{2} \cdot {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + t_1 \cdot \left(-0.0625 \cdot t_0\right)\right)}{1 + -0.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right) + \cos y \cdot \left(\sqrt{5} + -3\right)\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \frac{6}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 + -0.0625 \cdot \left(t_0 \cdot t_1\right)\right) \cdot \frac{0.3333333333333333}{1 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + \frac{\cos y}{0.5 \cdot \left(\sqrt{5} + 3\right)}\right)}\\ \end{array} \]

Alternative 17
Accuracy	79.1%
Cost	53512

\[\begin{array}{l} t_0 := 1 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + \frac{\cos y}{0.5 \cdot \left(\sqrt{5} + 3\right)}\right)\\ t_1 := 2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)\\ \mathbf{if}\;x \leq -0.0108:\\ \;\;\;\;\left(t_1 \cdot 0.3333333333333333\right) \cdot \frac{1}{t_0}\\ \mathbf{elif}\;x \leq 0.00105:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{0.3333333333333333}{t_0}\\ \end{array} \]

Alternative 18
Accuracy	79.1%
Cost	53385

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-6} \lor \neg \left(x \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(-0.0625 \cdot \left(\cos x + -1\right)\right)\right)}{1 + -0.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right) + \cos y \cdot \left(\sqrt{5} + -3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \frac{6}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 19
Accuracy	78.4%
Cost	53128

\[\begin{array}{l} t_0 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot t_0\right)}{\left(\cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right) + \sqrt{5} \cdot -0.5\right) + 2.5}\\ \mathbf{elif}\;x \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \frac{6}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_0\right)}{3 - 1.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right) + \left(\sqrt{5} + -3\right)\right)}\\ \end{array} \]

Alternative 20
Accuracy	78.4%
Cost	52996

\[\begin{array}{l} t_0 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot t_0\right)}{\left(\cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right) + \sqrt{5} \cdot -0.5\right) + 2.5}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{1}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_0\right)}{3 - 1.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right) + \left(\sqrt{5} + -3\right)\right)}\\ \end{array} \]

Alternative 21
Accuracy	78.4%
Cost	46984

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \cos x \cdot \left(1 - \sqrt{5}\right) + \left(\sqrt{5} + -3\right)\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot t_2\right) \cdot \left(-0.0625 \cdot t_0\right)\right)}{1 + -0.5 \cdot t_1}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{1}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot t_2\right)\right)}{3 - 1.5 \cdot t_1}\\ \end{array} \]

Alternative 22
Accuracy	78.3%
Cost	46856

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \cos x \cdot \left(1 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\frac{1 - \cos \left(x + x\right)}{2} \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} + t_1\right)\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 - \cos y \cdot \left(-1.5 - \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot {\sin x}^{2}\right)\right)}{3 - 1.5 \cdot \left(t_1 + \left(\sqrt{5} + -3\right)\right)}\\ \end{array} \]

Alternative 23
Accuracy	78.3%
Cost	46856

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \cos x \cdot \left(1 - \sqrt{5}\right) + \left(\sqrt{5} + -3\right)\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot t_2\right) \cdot \left(-0.0625 \cdot t_0\right)\right)}{1 + -0.5 \cdot t_1}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 - \cos y \cdot \left(-1.5 - \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot t_2\right)\right)}{3 - 1.5 \cdot t_1}\\ \end{array} \]

Alternative 24
Accuracy	78.4%
Cost	46856

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := 1 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ t_3 := \sqrt{5} + -3\\ t_4 := \cos x \cdot t_1 + t_3\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot t_2\right) \cdot \left(-0.0625 \cdot t_0\right)\right)}{1 + -0.5 \cdot t_4}\\ \mathbf{elif}\;x \leq 0.00024:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t_1 \cdot -1.5 + \left(3 - 1.5 \cdot \left(\cos y \cdot t_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot t_2\right)\right)}{3 - 1.5 \cdot t_4}\\ \end{array} \]

Alternative 25
Accuracy	61.5%
Cost	46728

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \sqrt{5} + -3\\ t_2 := 1 - \sqrt{5}\\ t_3 := \cos x \cdot t_2\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\frac{1 - \cos \left(x + x\right)}{2} \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} + t_3\right)\right)}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.6666666666666666}{1 + -0.5 \cdot \left(t_2 + \cos y \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot {\sin x}^{2}\right)\right)}{3 - 1.5 \cdot \left(t_3 + t_1\right)}\\ \end{array} \]

Alternative 26
Accuracy	61.5%
Cost	40649

\[\begin{array}{l} t_0 := 1 - \sqrt{5}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-7} \lor \neg \left(x \leq 1.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\frac{1 - \cos \left(x + x\right)}{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} + \cos x \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.6666666666666666}{1 + -0.5 \cdot \left(t_0 + \cos y \cdot \left(\sqrt{5} + -3\right)\right)}\\ \end{array} \]

Alternative 27
Accuracy	41.8%
Cost	20160

\[\frac{0.6666666666666666}{1 + -0.5 \cdot \left(\left(1 - \sqrt{5}\right) + \cos y \cdot \left(\sqrt{5} + -3\right)\right)} \]

Alternative 28
Accuracy	40.0%
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023135 
(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))))))