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

Average Error: 0.71% → 0.57%

Time: 37.8s

Precision: binary64

Cost: 91712

Math

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

↓

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{6}{1 + \sqrt{5}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \frac{4}{\sqrt{5} + 3}, 3\right)\right)} \]

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) (* -0.0625 (sin x))))
   2.0)
  (fma
   (cos x)
   (/ 6.0 (+ 1.0 (sqrt 5.0)))
   (fma (cos y) (* 1.5 (/ 4.0 (+ (sqrt 5.0) 3.0))) 3.0))))

C

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

↓

double code(double x, double y) {
	return fma(sqrt(2.0), (((sin(x) + (sin(y) * -0.0625)) * (cos(x) - cos(y))) * (sin(y) + (-0.0625 * sin(x)))), 2.0) / fma(cos(x), (6.0 / (1.0 + sqrt(5.0))), fma(cos(y), (1.5 * (4.0 / (sqrt(5.0) + 3.0))), 3.0));
}

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
end

↓

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(cos(x) - cos(y))) * Float64(sin(y) + Float64(-0.0625 * sin(x)))), 2.0) / fma(cos(x), Float64(6.0 / Float64(1.0 + sqrt(5.0))), fma(cos(y), Float64(1.5 * Float64(4.0 / Float64(sqrt(5.0) + 3.0))), 3.0)))
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(6.0 / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.71

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

2. Simplified 0.67

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

   [Start] 0.71	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied egg-rr 0.6

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{{\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}^{-1}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]

4. Simplified 0.6

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

   [Start] 0.6	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, {\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}^{-1}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]
   unpow-1 [=>] 0.6	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{\frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]
   associate-/r* [=>] 0.6	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]
   metadata-eval [=>] 0.6	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\color{blue}{6}}{\sqrt{5} + 1}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]
   +-commutative [=>] 0.6	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{6}{\color{blue}{1 + \sqrt{5}}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]

5. Applied egg-rr 0.57

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{6}{1 + \sqrt{5}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \color{blue}{\frac{4}{\sqrt{5} + 3}}, 3\right)\right)} \]

6. Final simplification 0.57

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{6}{1 + \sqrt{5}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \frac{4}{\sqrt{5} + 3}, 3\right)\right)} \]

Alternatives

Alternative 1
Error	18.79%
Cost	72776

\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot t_1\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot t_2\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	0.67%
Cost	72768

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

Alternative 3
Error	0.62%
Cost	72768

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

Alternative 4
Error	0.71%
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 5
Error	18.87%
Cost	66372

\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.00094:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(t_3 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_3\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	18.87%
Cost	66249

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0014 \lor \neg \left(x \leq 1.22 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot t_0\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \end{array} \]

Alternative 7
Error	20.54%
Cost	60360

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ t_3 := \cos x + -1\\ \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_3\right) \cdot t_0}{3 \cdot \left(\cos y \cdot \frac{t_1}{2} + \left(1 + \frac{\cos x}{0.5 + \sqrt{5} \cdot 0.5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \left(1 + t_1 \cdot \left(\cos y \cdot 0.5\right)\right)} \cdot \left(2 + t_3 \cdot \left(t_2 \cdot t_0\right)\right)\\ \end{array} \]

Alternative 8
Error	20.55%
Cost	53513

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 1.22 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{0.3333333333333333}{\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \left(1 + t_0 \cdot \left(\cos y \cdot 0.5\right)\right)} \cdot \left(2 + \left(\cos x + -1\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot t_0\right)\right)}\\ \end{array} \]

Alternative 9
Error	20.56%
Cost	53512

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ t_3 := \cos x + -1\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_3\right) \cdot t_0}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t_1}{2}\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \left(1 + t_1 \cdot \left(\cos y \cdot 0.5\right)\right)} \cdot \left(2 + t_3 \cdot \left(t_2 \cdot t_0\right)\right)\\ \end{array} \]

Alternative 10
Error	20.53%
Cost	53512

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ t_3 := \cos x + -1\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_3\right) \cdot t_0}{3 \cdot \left(\cos y \cdot \frac{t_1}{2} + \left(1 + \frac{\cos x}{0.5 + \sqrt{5} \cdot 0.5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \left(1 + t_1 \cdot \left(\cos y \cdot 0.5\right)\right)} \cdot \left(2 + t_3 \cdot \left(t_2 \cdot t_0\right)\right)\\ \end{array} \]

Alternative 11
Error	20.56%
Cost	53385

\[\begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 1.22 \cdot 10^{-12}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(t_0 + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} + 1.5 \cdot t_0\right)}\\ \end{array} \]

Alternative 12
Error	21.14%
Cost	53128

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := 1 + \sqrt{5}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{3 + \left(\frac{6}{\sqrt{5} + 3} + 6 \cdot \frac{\cos x}{t_1}\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(\frac{6}{t_1} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \end{array} \]

Alternative 13
Error	21.24%
Cost	46984

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} + 3\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{3 + \left(\frac{6}{t_1} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(\left(\sqrt{5} + -1\right) \cdot 0.5 + 2 \cdot \frac{\cos y}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \end{array} \]

Alternative 14
Error	21.18%
Cost	46984

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := 1 + \sqrt{5}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{3 + \left(\frac{6}{\sqrt{5} + 3} + 6 \cdot \frac{\cos x}{t_1}\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{1}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \end{array} \]

Alternative 15
Error	21.17%
Cost	46984

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} + 3\\ t_2 := 1 + \sqrt{5}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{3 + \left(\frac{6}{t_1} + 6 \cdot \frac{\cos x}{t_2}\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t_1} + 6 \cdot \frac{1}{t_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \end{array} \]

Alternative 16
Error	21.27%
Cost	46857

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 1.22 \cdot 10^{-12}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 17
Error	21.25%
Cost	46856

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{3 + \left(\frac{6}{\sqrt{5} + 3} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \end{array} \]

Alternative 18
Error	37.9%
Cost	46601

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 1.22 \cdot 10^{-12}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} + -1\right), 1\right)}\\ \end{array} \]

Alternative 19
Error	56.55%
Cost	20416

\[\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 2 \cdot \frac{1}{\sqrt{5} + 3}\right)} \]

Alternative 20
Error	57.14%
Cost	20160

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

Alternative 21
Error	59.05%
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023089 
(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))))))