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

Average Error: 0.5 → 0.4

Time: 39.4s

Precision: binary64

Cost: 85568

Math

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

↓

\[\frac{2 + \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)} \]

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
 (/
  (+
   2.0
   (*
    (*
     (fma -0.0625 (sin y) (sin x))
     (* (sqrt 2.0) (fma -0.0625 (sin x) (sin y))))
    (- (cos x) (cos y))))
  (*
   3.0
   (+
    (+ 1.0 (/ (cos x) (* 0.5 (+ 1.0 (sqrt 5.0)))))
    (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))

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 (2.0 + ((fma(-0.0625, sin(y), sin(x)) * (sqrt(2.0) * fma(-0.0625, sin(x), sin(y)))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (cos(x) / (0.5 * (1.0 + sqrt(5.0))))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
}

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(Float64(2.0 + Float64(Float64(fma(-0.0625, sin(y), sin(x)) * Float64(sqrt(2.0) * fma(-0.0625, sin(x), sin(y)))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) / Float64(0.5 * Float64(1.0 + sqrt(5.0))))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))))
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[(2.0 + N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] / N[(0.5 * N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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. Applied egg-rr 0.5

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

3. Applied egg-rr 0.4

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(\sqrt{5} + 1\right)}\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

4. Taylor expanded in x around inf 0.4

   \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(\sqrt{5} + 1\right)}\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

5. Simplified 0.4

   \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(\sqrt{5} + 1\right)}\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]
   Proof
   (*.f64 (fma.f64 -1/16 (sin.f64 y) (sin.f64 x)) (*.f64 (sqrt.f64 2) (fma.f64 -1/16 (sin.f64 x) (sin.f64 y)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/16 (sin.f64 y)) (sin.f64 x))) (*.f64 (sqrt.f64 2) (fma.f64 -1/16 (sin.f64 x) (sin.f64 y)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 1/16)) (sin.f64 y)) (sin.f64 x)) (*.f64 (sqrt.f64 2) (fma.f64 -1/16 (sin.f64 x) (sin.f64 y)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 1/16 (sin.f64 y)))) (sin.f64 x)) (*.f64 (sqrt.f64 2) (fma.f64 -1/16 (sin.f64 x) (sin.f64 y)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (sin.f64 x) (neg.f64 (*.f64 1/16 (sin.f64 y))))) (*.f64 (sqrt.f64 2) (fma.f64 -1/16 (sin.f64 x) (sin.f64 y)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (sin.f64 x) (*.f64 1/16 (sin.f64 y)))) (*.f64 (sqrt.f64 2) (fma.f64 -1/16 (sin.f64 x) (sin.f64 y)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sin.f64 x) (*.f64 1/16 (sin.f64 y))) (*.f64 (sqrt.f64 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/16 (sin.f64 x)) (sin.f64 y))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sin.f64 x) (*.f64 1/16 (sin.f64 y))) (*.f64 (sqrt.f64 2) (+.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 1/16)) (sin.f64 x)) (sin.f64 y)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sin.f64 x) (*.f64 1/16 (sin.f64 y))) (*.f64 (sqrt.f64 2) (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 1/16 (sin.f64 x)))) (sin.f64 y)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sin.f64 x) (*.f64 1/16 (sin.f64 y))) (*.f64 (sqrt.f64 2) (Rewrite<= +-commutative_binary64 (+.f64 (sin.f64 y) (neg.f64 (*.f64 1/16 (sin.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (sin.f64 x) (*.f64 1/16 (sin.f64 y))) (*.f64 (sqrt.f64 2) (Rewrite<= sub-neg_binary64 (-.f64 (sin.f64 y) (*.f64 1/16 (sin.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 y) (*.f64 1/16 (sin.f64 x)))) (-.f64 (sin.f64 x) (*.f64 1/16 (sin.f64 y))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r*_binary64 (*.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (*.f64 1/16 (sin.f64 x))) (-.f64 (sin.f64 x) (*.f64 1/16 (sin.f64 y)))))): 32 points increase in error, 50 points decrease in error
   (*.f64 (sqrt.f64 2) (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 (sin.f64 x) (*.f64 1/16 (sin.f64 y))) (-.f64 (sin.f64 y) (*.f64 1/16 (sin.f64 x)))))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.4

   \[\leadsto \frac{2 + \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	73024

\[\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)} \]

Alternative 2
Error	0.5
Cost	72896

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

Alternative 3
Error	0.4
Cost	72640

\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \left(3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \]

Alternative 4
Error	0.4
Cost	72512

\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \cos x \cdot \left(\sqrt{11.25} + -1.5\right)\right)} \]

Alternative 5
Error	12.2
Cost	66888

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_2 := \cos x - \cos y\\ t_3 := 2 + t_2 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right)\\ \mathbf{if}\;x \leq -224589722.5804572:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(t_1 + \left(1 + \cos x \cdot \frac{t_0}{2}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00011324836066801668:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_2 \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + t_0 \cdot \left(1.5 + x \cdot \left(x \cdot -0.75\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\right) + t_1\right)}\\ \end{array} \]

Alternative 6
Error	12.2
Cost	66760

\[\begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := 3 \cdot \left(t_0 + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := 2 + t_2 \cdot \left(t_3 \cdot \left(\sin x \cdot \sqrt{2}\right)\right)\\ \mathbf{if}\;x \leq -224589722.5804572:\\ \;\;\;\;\frac{t_4}{t_1}\\ \mathbf{elif}\;x \leq 0.00011324836066801668:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\right) + t_0\right)}\\ \end{array} \]

Alternative 7
Error	12.1
Cost	66632

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right)\\ t_2 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ \mathbf{if}\;x \leq -224589722.5804572:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(t_2 + \left(1 + \cos x \cdot \frac{t_0}{2}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00011324836066801668:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_0\right) + \cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\right) + t_2\right)}\\ \end{array} \]

Alternative 8
Error	12.8
Cost	66504

\[\begin{array}{l} t_0 := 1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\\ t_1 := \cos x - \cos y\\ t_2 := \frac{2 + t_1 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;x \leq -224589722.5804572:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.8674719066651668 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(-0.0625 \cdot \sin y + x \cdot 1.00390625\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	12.8
Cost	66504

\[\begin{array}{l} t_0 := 1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\\ t_1 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_2 := \cos x - \cos y\\ t_3 := 2 + t_2 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right)\\ \mathbf{if}\;x \leq -224589722.5804572:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(t_1 + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{elif}\;x \leq 1.8674719066651668 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(-0.0625 \cdot \sin y + x \cdot 1.00390625\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(t_0 + t_1\right)}\\ \end{array} \]

Alternative 10
Error	13.9
Cost	60360

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{2 + t_1 \cdot \left({\sin x}^{2} \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right)}{t_0}\\ \mathbf{if}\;x \leq -224589722.5804572:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.8674719066651668 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(-0.0625 \cdot \sin y + x \cdot 1.00390625\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	13.4
Cost	60040

\[\begin{array}{l} t_0 := 1 + \sqrt{5}\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot t_0}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{if}\;y \leq -30.6555554063952:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.594510597872792 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(7.5 + 1.5 \cdot \frac{\cos x \cdot 4}{t_0}\right) + \sqrt{5} \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	13.9
Cost	60040

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(1 + \sqrt{5}\right)}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{2 + t_1 \cdot \left({\sin x}^{2} \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right)}{t_0}\\ \mathbf{if}\;x \leq -262194576.8389305:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.8674719066651668 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	14.2
Cost	59400

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 1 + \sqrt{5}\\ \mathbf{if}\;x \leq -4.744104858132617 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{\left(7.5 + \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right) - \sqrt{11.25}}\\ \mathbf{elif}\;x \leq 1.8674719066651668 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{3 + \mathsf{fma}\left(\cos y, 4.5 - \sqrt{11.25}, \frac{6}{t_1}\right)}{\mathsf{fma}\left(-0.0625, \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right), 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot \left(\cos x + -1\right)\right)\right)}{\left(7.5 + 1.5 \cdot \frac{\cos x \cdot 4}{t_1}\right) + \sqrt{5} \cdot -1.5}\\ \end{array} \]

Alternative 14
Error	14.2
Cost	46984

\[\begin{array}{l} t_0 := 1 + \sqrt{5}\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -4.744104858132617 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_1 \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{\left(7.5 + \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right) - \sqrt{11.25}}\\ \mathbf{elif}\;x \leq 1.8674719066651668 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + 6 \cdot \frac{1}{t_0}\right)}{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_1 \cdot \left(\cos x + -1\right)\right)\right)}{\left(7.5 + 1.5 \cdot \frac{\cos x \cdot 4}{t_0}\right) + \sqrt{5} \cdot -1.5}\\ \end{array} \]

Alternative 15
Error	14.2
Cost	46856

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -4.744104858132617 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{\left(7.5 + \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right) - \sqrt{11.25}}\\ \mathbf{elif}\;x \leq 1.8674719066651668 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + \cos y \cdot \left(4.5 - \sqrt{11.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot \left(\cos x + -1\right)\right)\right)}{\left(7.5 + 1.5 \cdot \frac{\cos x \cdot 4}{1 + \sqrt{5}}\right) + \sqrt{5} \cdot -1.5}\\ \end{array} \]

Alternative 16
Error	14.2
Cost	46728

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -4.744104858132617 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{\left(7.5 + \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right) - \sqrt{11.25}}\\ \mathbf{elif}\;x \leq 1.8674719066651668 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + \cos y \cdot \left(4.5 - \sqrt{11.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(7.5 + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right) - \sqrt{11.25}}{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot \left(\cos x + -1\right)\right)\right)}}\\ \end{array} \]

Alternative 17
Error	25.7
Cost	46336

\[\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{\left(7.5 + \cos x \cdot \left(-1.5 + \sqrt{5} \cdot 1.5\right)\right) - \sqrt{11.25}} \]

Alternative 18
Error	25.7
Cost	40384

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 + \cos \left(x + x\right) \cdot -0.5\right)\right)\right)}{\left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 7.5\right) + \sqrt{5} \cdot -1.5} \]

Alternative 19
Error	38.1
Cost	39936

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(7.5 + 1.5 \cdot \left(\sqrt{5} + -1\right)\right) + \sqrt{5} \cdot -1.5} \]

Alternative 20
Error	38.1
Cost	13376

\[\frac{2}{\left(\sqrt{5} \cdot 1.5 + 6\right) - \sqrt{11.25}} \]

Reproduce

herbie shell --seed 2022316 
(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))))))