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

Average Accuracy: 99.3% → 99.4%

Time: 41.1s

Precision: binary64

Cost: 78912

Math

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

↓

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

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)
   (*
    (- (cos x) (cos y))
    (* (- (sin y) (* (sin x) 0.0625)) (+ (sin x) (* (sin y) -0.0625))))
   2.0)
  (+
   3.0
   (* 6.0 (+ (/ (cos y) (+ 3.0 (sqrt 5.0))) (/ (cos x) (+ (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) - (sin(x) * 0.0625)) * (sin(x) + (sin(y) * -0.0625)))), 2.0) / (3.0 + (6.0 * ((cos(y) / (3.0 + sqrt(5.0))) + (cos(x) / (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(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) + Float64(sin(y) * -0.0625)))), 2.0) / Float64(3.0 + Float64(6.0 * Float64(Float64(cos(y) / Float64(3.0 + sqrt(5.0))) + Float64(cos(x) / 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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(6.0 * N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

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

3. Applied egg-rr 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in y around inf 99.4%

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

6. Simplified 99.4%

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

   [Start] 99.4	\[ \frac{2 + \sqrt{2} \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{\sqrt{5} + 3} + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	72768

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

Alternative 2
Accuracy	81.1%
Cost	67144

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.055:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.065:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 3
Accuracy	81.0%
Cost	66760

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \cos x - \cos y\\ t_4 := t_3 \cdot t_0\\ \mathbf{if}\;x \leq -0.028:\\ \;\;\;\;\frac{2 + t_2 \cdot t_4}{t_1}\\ \mathbf{elif}\;x \leq 0.024:\\ \;\;\;\;\frac{2 + t_4 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 4
Accuracy	81.0%
Cost	66760

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 5
Accuracy	80.9%
Cost	66696

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.0028:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0033:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + 1.00390625 \cdot \left(x \cdot \sin y\right)\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 6
Accuracy	80.8%
Cost	66632

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \cos x - \cos y\\ t_4 := t_3 \cdot t_0\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + t_2 \cdot t_4}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_4 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_1 + \cos y \cdot \left(1.5 - t_1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\frac{\cos x}{0.5}}{\sqrt{5} + 1}\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	80.8%
Cost	66632

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \cos x - \cos y\\ t_4 := t_3 \cdot t_0\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + t_2 \cdot t_4}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_4 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_1 + \cos y \cdot \left(1.5 - t_1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 8
Accuracy	80.6%
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.00055 \lor \neg \left(x \leq 0.0012\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_1 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 + -0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t_1 \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{3 + 6 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} + \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 9
Accuracy	80.6%
Cost	66505

\[\begin{array}{l} t_0 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.00065 \lor \neg \left(x \leq 0.00095\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_0 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t_0 \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{3 + 6 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} + \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 10
Accuracy	80.8%
Cost	66505

\[\begin{array}{l} t_0 := \left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\\ t_1 := 1.5 + \sqrt{1.25}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 8.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{t_1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \frac{\cos y}{t_1}\right)\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	80.8%
Cost	66504

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1.5 + \sqrt{1.25}\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \cos x - \cos y\\ t_4 := t_3 \cdot t_0\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + t_2 \cdot t_4}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{t_1}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00021:\\ \;\;\;\;\frac{2 + t_4 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \frac{\cos y}{t_1}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\frac{\cos x}{0.5}}{\sqrt{5} + 1}\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	79.0%
Cost	66376

\[\begin{array}{l} t_0 := -0.0625 \cdot {\sin y}^{2}\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.0006:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00029:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t_1 \cdot t_0, 2\right)}{3 + 6 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} + \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 13
Accuracy	79.0%
Cost	65928

\[\begin{array}{l} t_0 := -0.0625 \cdot {\sin y}^{2}\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.0006:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00035:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right) + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t_1 \cdot t_0, 2\right)}{3 + 6 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} + \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 14
Accuracy	78.9%
Cost	60232

\[\begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \cos x \cdot \left(-0.5 + t_1\right)\\ \mathbf{if}\;y \leq -0.00058:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot t_0\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + t_2\right) + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(t_0 \cdot \left(1 - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{1 + \left(\cos y \cdot \left(1.5 - t_1\right) + t_2\right)}\\ \end{array} \]

Alternative 15
Accuracy	78.9%
Cost	60105

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 9.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{6}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 16
Accuracy	79.0%
Cost	60105

\[\begin{array}{l} t_0 := \sqrt{5} + 1\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-6} \lor \neg \left(x \leq 9.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\frac{\cos x}{0.5}}{t_0}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{6}{t_0}\right)}\\ \end{array} \]

Alternative 17
Accuracy	78.8%
Cost	59976

\[\begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \cos x \cdot \left(-0.5 + t_1\right)\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.00058:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot t_0\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_3 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + t_2\right) + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(t_0 \cdot \left(1 - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{1 + \left(\cos y \cdot \left(1.5 - t_1\right) + t_2\right)}\\ \end{array} \]

Alternative 18
Accuracy	78.8%
Cost	53640

\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(0.0625 + \cos x \cdot -0.0625\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \cos x \cdot \left(-0.5 + t_1\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + \left(\cos y \cdot \left(1.5 - t_1\right) + t_2\right)}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{6}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + \left(t_2 + \cos y \cdot \frac{1}{1.5 + t_1}\right)}\\ \end{array} \]

Alternative 19
Accuracy	78.8%
Cost	53513

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 10^{-5}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(0.0625 + \cos x \cdot -0.0625\right)}{1 + \left(\cos y \cdot \left(1.5 - t_0\right) + \cos x \cdot \left(-0.5 + t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{6}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 20
Accuracy	78.2%
Cost	53128

\[\begin{array}{l} t_0 := \sqrt{5} + 1\\ t_1 := 3 + \sqrt{5}\\ t_2 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot t_2}{3 + 1.5 \cdot \left(\frac{4}{t_1} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{t_1} + \frac{6}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_2\right)}{3 + \left(\frac{6}{t_1} + 6 \cdot \frac{\cos x}{t_0}\right)}\\ \end{array} \]

Alternative 21
Accuracy	78.1%
Cost	46857

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 1.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	78.2%
Cost	46857

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 3.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{3 + 1.5 \cdot \left(t_1 + \cos x \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.5 \cdot t_0 + \left(3 + 1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \end{array} \]

Alternative 23
Accuracy	78.2%
Cost	46857

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 1.65 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + \left(\frac{6}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 24
Accuracy	78.2%
Cost	46856

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{3 + \left(1.5 \cdot t_1 + 1.5 \cdot \left(\cos x \cdot t_2\right)\right)}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.5 \cdot t_2 + \left(3 + 1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + \left(\frac{6}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 25
Accuracy	78.2%
Cost	46856

\[\begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot t_1}{3 + 1.5 \cdot \left(\frac{4}{t_0} + \cos x \cdot t_2\right)}\\ \mathbf{elif}\;x \leq 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.5 \cdot t_2 + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{3 + \left(\frac{6}{t_0} + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 26
Accuracy	60.1%
Cost	46464

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

Alternative 27
Accuracy	42.4%
Cost	20160

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{0.6666666666666666}{0.5 + \left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \]

Alternative 28
Accuracy	40.5%
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023122 
(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))))))