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

Average Error: 0.5 → 0.4

Time: 37.4s

Precision: binary64

Cost: 85696

Math

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

↓

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

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) (* -0.0625 (sin x)))
    (* (/ 1.0 (/ 1.0 (- (cos x) (cos y)))) (+ (sin x) (* (sin y) -0.0625))))
   2.0)
  (/
   1.0
   (+
    3.0
    (fma
     (cos x)
     (* 1.5 (+ (sqrt 5.0) -1.0))
     (* (cos y) (/ 6.0 (+ 3.0 (sqrt 5.0)))))))))

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) + (-0.0625 * sin(x))) * ((1.0 / (1.0 / (cos(x) - cos(y)))) * (sin(x) + (sin(y) * -0.0625)))), 2.0) * (1.0 / (3.0 + fma(cos(x), (1.5 * (sqrt(5.0) + -1.0)), (cos(y) * (6.0 / (3.0 + sqrt(5.0)))))));
}

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(-0.0625 * sin(x))) * Float64(Float64(1.0 / Float64(1.0 / Float64(cos(x) - cos(y)))) * Float64(sin(x) + Float64(sin(y) * -0.0625)))), 2.0) * Float64(1.0 / Float64(3.0 + fma(cos(x), Float64(1.5 * Float64(sqrt(5.0) + -1.0)), Float64(cos(y) * Float64(6.0 / Float64(3.0 + sqrt(5.0))))))))
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[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(1.0 / N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 0.4

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{0.6666666666666666}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)}} \]
   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

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

4. Applied egg-rr 0.4

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

5. Applied egg-rr 0.4

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

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	85440

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

Alternative 2
Error	0.4
Cost	79040

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

Alternative 3
Error	0.4
Cost	79040

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

Alternative 4
Error	0.5
Cost	72768

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

Alternative 5
Error	0.4
Cost	72768

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

Alternative 6
Error	0.5
Cost	72640

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

Alternative 7
Error	12.1
Cost	72388

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \sqrt{2} \cdot \sin x\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_3\right) \cdot t_1}{3 \cdot \left(1 + \mathsf{fma}\left(1.5 - \sqrt{1.25}, \cos y, \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_3\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot t_1\right)}{t_0}\\ \end{array} \]

Alternative 8
Error	12.1
Cost	67144

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \sqrt{2} \cdot \sin x\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\sqrt{5}}{2}\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_4\right) \cdot t_1}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_3\right) + \cos y \cdot \left(1.5 - t_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_4\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_4 \cdot t_1\right)}{t_0}\\ \end{array} \]

Alternative 9
Error	12.1
Cost	66760

\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot t_3\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_2\right) + \cos y \cdot \left(1.5 - t_2\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_3 \cdot t_0\right)}{t_4}\\ \end{array} \]

Alternative 10
Error	12.2
Cost	66696

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \sqrt{2} \cdot \sin x\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\sqrt{5}}{2}\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0028:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_4\right) \cdot t_1}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_3\right) + \cos y \cdot \left(1.5 - t_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + 1.00390625 \cdot \left(\sin y \cdot x\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_4 \cdot t_1\right)}{t_0}\\ \end{array} \]

Alternative 11
Error	12.2
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0038 \lor \neg \left(x \leq 0.23\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot t_1\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 12
Error	12.2
Cost	66504

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \sqrt{2} \cdot \sin x\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\sqrt{5}}{2}\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0036:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_4\right) \cdot t_1}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_3\right) + \cos y \cdot \left(1.5 - t_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{2 + \left(t_4 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right) \cdot \left(1 - \cos y\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_4 \cdot t_1\right)}{t_0}\\ \end{array} \]

Alternative 13
Error	13.3
Cost	60361

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.47 \lor \neg \left(x \leq 0.23\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_0 \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_1\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 14
Error	13.3
Cost	60297

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.47 \lor \neg \left(x \leq 0.23\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + 1.00390625 \cdot \left(\sin y \cdot x\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 15
Error	13.4
Cost	60105

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-5} \lor \neg \left(x \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	13.4
Cost	59913

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-8} \lor \neg \left(y \leq 4.1 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\frac{4}{3 + \sqrt{5}} + \cos x \cdot t_0\right)}\\ \end{array} \]

Alternative 17
Error	13.5
Cost	53513

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-8} \lor \neg \left(y \leq 4.1 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\frac{4}{3 + \sqrt{5}} + \cos x \cdot t_0\right)}\\ \end{array} \]

Alternative 18
Error	13.9
Cost	53128

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \frac{4}{3 + \sqrt{5}}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(t_1 + \cos x \cdot t_2\right)}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.5 \cdot t_2 + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \mathsf{fma}\left(t_2, \cos x, t_1\right)}\\ \end{array} \]

Alternative 19
Error	13.9
Cost	46857

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.47 \lor \neg \left(x \leq 0.23\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\frac{4}{3 + \sqrt{5}} + \cos x \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.5 \cdot t_0 + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 20
Error	14.0
Cost	46856

\[\begin{array}{l} t_0 := 3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ t_1 := \cos x + -1\\ \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_1 \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)\right)}{t_0}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot t_1\right)\right)}{t_0}\\ \end{array} \]

Alternative 21
Error	14.0
Cost	46856

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x + -1\\ t_2 := \sqrt{5} + -1\\ t_3 := 3 + 1.5 \cdot \left(t_0 + \cos x \cdot t_2\right)\\ \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_1 \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)\right)}{t_3}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.5 \cdot t_2 + \left(3 + 1.5 \cdot \left(\cos y \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot t_1\right)\right)}{t_3}\\ \end{array} \]

Alternative 22
Error	25.7
Cost	40384

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

Alternative 23
Error	38.2
Cost	20416

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(-0.5 + \frac{\cos x}{2}\right) \cdot \left(1 - \cos \left(x + x\right)\right)\right)\right)}{6} \]

Alternative 24
Error	38.3
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023046 
(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))))))