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

Average Error: 0.5 → 0.4

Time: 40.0s

Precision: binary64

Cost: 85312

Math

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

↓

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (*
   3.0
   (+
    (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
    (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))

↓

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (sqrt 2.0) (+ (sin x) (* (sin y) -0.0625)))
   (* (- (cos x) (cos y)) (+ (sin y) (* (sin x) -0.0625)))
   2.0)
  (*
   3.0
   (+
    (fma (cos x) (+ -0.5 (* (sqrt 5.0) 0.5)) 1.0)
    (/ (cos y) (+ 1.5 (sqrt 1.25)))))))

C

double code(double x, double y) {
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
}

↓

double code(double x, double y) {
	return fma((sqrt(2.0) * (sin(x) + (sin(y) * -0.0625))), ((cos(x) - cos(y)) * (sin(y) + (sin(x) * -0.0625))), 2.0) / (3.0 * (fma(cos(x), (-0.5 + (sqrt(5.0) * 0.5)), 1.0) + (cos(y) / (1.5 + sqrt(1.25)))));
}

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(Float64(sqrt(2.0) * Float64(sin(x) + Float64(sin(y) * -0.0625))), Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) + Float64(sin(x) * -0.0625))), 2.0) / Float64(3.0 * Float64(fma(cos(x), Float64(-0.5 + Float64(sqrt(5.0) * 0.5)), 1.0) + Float64(cos(y) / Float64(1.5 + sqrt(1.25))))))
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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] / N[(1.5 + N[Sqrt[1.25], $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 0.5

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right)\right)}} \]
   Proof

   [Start] 0.5	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   +-commutative [=>] 0.5	\[ \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   associate-*l* [=>] 0.5	\[ \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   fma-def [=>] 0.5	\[ \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   +-commutative [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]
   *-commutative [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]
   fma-def [=>] 0.5	\[ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

3. Applied egg-rr 0.4

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

4. Applied egg-rr 0.4

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

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
Cost	72768

\[0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)} \]

Alternative 2
Error	0.5
Cost	72768

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

Alternative 3
Error	0.4
Cost	72768

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

Alternative 4
Error	12.0
Cost	67144

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sqrt{2} \cdot \sin x\\ t_2 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_3 := \cos x - \cos y\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_4 \cdot t_1\right)}{3 \cdot \left(t_2 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.11:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_4\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(t_2 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_3 \cdot t_4\right) \cdot t_1}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	12.1
Cost	66760

\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := 3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_1\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)\\ t_3 := \cos x - \cos y\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0082:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_4 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_4 \cdot \left(1 - \cos y\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_3 \cdot t_4\right) \cdot t_0}{t_2}\\ \end{array} \]

Alternative 6
Error	12.0
Cost	66760

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := \sqrt{2} \cdot \sin x\\ t_4 := \cos x \cdot \left(-0.5 + t_0\right)\\ \mathbf{if}\;x \leq -0.031:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_2 \cdot t_3\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_2 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(t_4 + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot t_2\right) \cdot t_3}{3 \cdot \left(1 + \left(t_4 + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	12.1
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 1 - \cos y\\ \mathbf{if}\;x \leq -0.00032 \lor \neg \left(x \leq 0.046\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(t_1 \cdot {\sin y}^{2}\right) \cdot -0.0625 + x \cdot \left(\sin y \cdot \left(t_1 \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	12.1
Cost	66504

\[\begin{array}{l} t_0 := 1 - \cos y\\ t_1 := 2 + \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := \cos x \cdot \left(-0.5 + t_2\right)\\ \mathbf{if}\;x \leq -0.00072:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(1 + \left(t_3 + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(t_0 \cdot {\sin y}^{2}\right) \cdot -0.0625 + x \cdot \left(\sin y \cdot \left(t_0 \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(1 + \left(t_3 + \cos y \cdot \left(1.5 - t_2\right)\right)\right)}\\ \end{array} \]

Alternative 9
Error	12.1
Cost	66504

\[\begin{array}{l} t_0 := 1.5 + \sqrt{1.25}\\ t_1 := \left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\\ t_2 := 2 + t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\\ t_3 := \frac{\sqrt{5}}{2}\\ t_4 := \cos x \cdot \left(-0.5 + t_3\right)\\ \mathbf{if}\;x \leq -0.00072:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 + \left(t_4 + \cos y \cdot \frac{1}{t_0}\right)\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_1}{3 \cdot \left(1 + \left(-0.5 + \left(\frac{\cos y}{t_0} + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 + \left(t_4 + \cos y \cdot \left(1.5 - t_3\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	12.1
Cost	66504

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{2} \cdot \sin x\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := t_0 \cdot t_3\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_3 \cdot t_1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_4}{3 \cdot \left(1 + \left(-0.5 + \left(\frac{\cos y}{1.5 + \sqrt{1.25}} + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_4 \cdot t_1}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_2\right) + \cos y \cdot \left(1.5 - t_2\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	13.3
Cost	65928

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 1 - \cos y\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.00019:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(t_1 \cdot {\sin y}^{2}\right) \cdot -0.0625 + x \cdot \left(\sin y \cdot \left(t_1 \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(t_2 \cdot \left(\cos x + -1\right)\right), 2\right)}{\frac{\cos x \cdot t_0}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}\\ \end{array} \]

Alternative 12
Error	13.3
Cost	60424

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\\ t_3 := 1 - \cos y\\ t_4 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.00052:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_4 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(t_3 \cdot {\sin y}^{2}\right) \cdot -0.0625 + x \cdot \left(\sin y \cdot \left(t_3 \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_1 + t_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_0 \cdot \left(t_4 \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + t_1\right) + t_2\right)}\\ \end{array} \]

Alternative 13
Error	13.3
Cost	60232

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\\ t_3 := \cos x - \cos y\\ t_4 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_4 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + \left(t_3 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_1 + t_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_0 \cdot \left(t_4 \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + t_1\right) + t_2\right)}\\ \end{array} \]

Alternative 14
Error	13.3
Cost	60168

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := {\sin x}^{2}\\ t_2 := 1 - \cos y\\ \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(t_2 \cdot {\sin y}^{2}\right) \cdot -0.0625 + x \cdot \left(\sin y \cdot \left(t_2 \cdot 1.00390625\right)\right)\right)}{1 + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + \left(-0.5 + \sqrt{1.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_0 \cdot \left(t_1 \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 15
Error	13.3
Cost	59976

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.0013:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + \left(-0.5 + \sqrt{1.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_0 \cdot \left(t_1 \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 16
Error	13.3
Cost	59976

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + \left(-0.5 + \sqrt{1.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_0 \cdot \left(t_1 \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 17
Error	13.9
Cost	59780

\[\begin{array}{l} t_0 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{+26}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-20}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \end{array} \]

Alternative 18
Error	13.6
Cost	53513

\[\begin{array}{l} t_0 := \sqrt{5} \cdot -0.5\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{-6} \lor \neg \left(y \leq 1.15 \cdot 10^{-20}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + \cos y \cdot \left(1.5 + t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(-0.5 + \sqrt{1.25}\right)\right) + t_0}\\ \end{array} \]

Alternative 19
Error	13.7
Cost	53513

\[\begin{array}{l} t_0 := \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-5} \lor \neg \left(x \leq 2.95 \cdot 10^{-30}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + t_1\right) + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_1 + t_0\right)\right)\right)}\\ \end{array} \]

Alternative 20
Error	13.9
Cost	53513

\[\begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+26} \lor \neg \left(y \leq 1.15 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 21
Error	13.7
Cost	53512

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\\ t_2 := {\sin x}^{2} \cdot \left(\cos x + -1\right)\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-30}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_0 + t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot t_2}{1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + t_1\right)}\\ \end{array} \]

Alternative 22
Error	13.8
Cost	46984

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} \cdot -0.5\\ \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\left(2.5 + \cos x \cdot \left(-0.5 + \sqrt{1.25}\right)\right) + t_1}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + t_1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) + t_1}\\ \end{array} \]

Alternative 23
Error	13.9
Cost	46728

\[\begin{array}{l} t_0 := \sqrt{5} \cdot -0.5\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_2 := -0.5 + \sqrt{1.25}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{\left(2.5 + \cos x \cdot t_2\right) + t_0}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{1 + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) + t_0}\\ \end{array} \]

Alternative 24
Error	25.4
Cost	46464

\[0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(-0.5 + \sqrt{1.25}\right)\right) + \sqrt{5} \cdot -0.5} \]

Alternative 25
Error	25.5
Cost	40512

\[0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\frac{\sqrt{2} \cdot \left(\cos x + -1\right)}{2} \cdot \left(1 - \cos \left(x + x\right)\right)\right)}{\left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) + 2.5\right) + \sqrt{5} \cdot -0.5} \]

Alternative 26
Error	37.9
Cost	40064

\[0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\left(-0.5 + \sqrt{5} \cdot 0.5\right) + 2.5\right) + \sqrt{5} \cdot -0.5} \]

Alternative 27
Error	37.9
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023027 
(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))))))