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

Percentage Accurate: 99.3% → 99.3%

Time: 45.3s

Precision: binary64

Cost: 85440

Math

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

↓

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

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
 (/
  (+
   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
    (fma
     (cos x)
     (+ -0.5 (* (sqrt 5.0) 0.5))
     (/ (cos y) (fma 0.5 (sqrt 5.0) 1.5)))))))

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 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))))) / (3.0 * (1.0 + fma(cos(x), (-0.5 + (sqrt(5.0) * 0.5)), (cos(y) / fma(0.5, sqrt(5.0), 1.5)))));
}

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(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))))) / Float64(3.0 * Float64(1.0 + fma(cos(x), Float64(-0.5 + Float64(sqrt(5.0) * 0.5)), Float64(cos(y) / fma(0.5, sqrt(5.0), 1.5))))))
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[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] / N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}} \]
   Step-by-step derivation

   [Start] 99.2%	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   associate-*l* [=>] 99.2%	\[ \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   associate-+l+ [=>] 99.2%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]
   *-commutative [=>] 99.2%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]
   div-sub [=>] 99.2%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]
   metadata-eval [=>] 99.2%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - \color{blue}{0.5}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]
   *-commutative [=>] 99.2%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)\right)} \]
   div-sub [=>] 99.2%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)}\right)\right)} \]
   metadata-eval [=>] 99.2%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(\color{blue}{1.5} - \frac{\sqrt{5}}{2}\right)\right)\right)} \]

3. Applied egg-rr 99.1%

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   Step-by-step derivation

   [Start] 99.2%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)} \]
   flip-- [=>] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{1.5 \cdot 1.5 - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}}\right)\right)} \]
   metadata-eval [=>] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]
   div-inv [=>] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]
   metadata-eval [=>] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]
   div-inv [=>] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]
   metadata-eval [=>] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]
   div-inv [=>] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]
   metadata-eval [=>] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

4. Simplified 99.3%

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

   [Start] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]
   swap-sqr [=>] 99.1%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]
   rem-square-sqrt [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]
   cancel-sign-sub-inv [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]
   metadata-eval [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]
   metadata-eval [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]
   metadata-eval [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]
   metadata-eval [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{1}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]
   +-commutative [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\color{blue}{\sqrt{5} \cdot 0.5 + 1.5}}\right)\right)} \]
   *-commutative [<=] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\color{blue}{0.5 \cdot \sqrt{5}} + 1.5}\right)\right)} \]
   fma-def [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}}\right)\right)} \]

5. Applied egg-rr 99.4%

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

   [Start] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)} \]
   fma-def [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} - 0.5, \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)}\right)} \]
   sub-neg [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, \color{blue}{\frac{\sqrt{5}}{2} + \left(-0.5\right)}, \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)} \]
   div-inv [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(-0.5\right), \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)} \]
   metadata-eval [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, \sqrt{5} \cdot \color{blue}{0.5} + \left(-0.5\right), \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)} \]
   metadata-eval [=>] 99.3%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, \sqrt{5} \cdot 0.5 + \color{blue}{-0.5}, \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)} \]
   un-div-inv [=>] 99.4%	\[ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, \sqrt{5} \cdot 0.5 + -0.5, \color{blue}{\frac{\cos y}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}}\right)\right)} \]

6. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	85440

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

Alternative 2
Accuracy	99.3%
Cost	72896

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{2 + \sqrt{2} \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)\right)} \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	72896

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)\right)} \end{array} \]

Alternative 4
Accuracy	81.0%
Cost	72644

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 2 + \left(t_1 \cdot t_0\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ t_3 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + t_3, \cos y \cdot \left(1.5 - t_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0026:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_3} + \cos x \cdot \left(t_3 - 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	81.1%
Cost	66761

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x - \cos y\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0255 \lor \neg \left(x \leq 0.0026\right):\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_1\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 6
Accuracy	80.9%
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.0007 \lor \neg \left(x \leq 0.0026\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y + x \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_1} + \cos x \cdot \left(t_1 - 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	81.0%
Cost	66505

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)\right)\\ \mathbf{if}\;x \leq -0.0007 \lor \neg \left(x \leq 0.0026\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y + x \cdot -0.0625\right)\right)}{t_1}\\ \end{array} \]

Alternative 8
Accuracy	79.2%
Cost	60361

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)\right)\\ t_2 := \sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\\ \mathbf{if}\;x \leq -0.0007 \lor \neg \left(x \leq 0.0026\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot -0.0625\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y + x \cdot -0.0625\right)\right)}{t_1}\\ \end{array} \]

Alternative 9
Accuracy	79.1%
Cost	60233

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)\right)\\ t_2 := \sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\\ \mathbf{if}\;x \leq -0.0007 \lor \neg \left(x \leq 0.00162\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot -0.0625\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{t_1}\\ \end{array} \]

Alternative 10
Accuracy	79.1%
Cost	60233

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\\ \mathbf{if}\;x \leq -0.00046 \lor \neg \left(x \leq 5.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	79.1%
Cost	60105

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)\right)\\ \mathbf{if}\;x \leq -0.0007 \lor \neg \left(x \leq 0.0026\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(0.0625 + \cos x \cdot -0.0625\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{t_1}\\ \end{array} \]

Alternative 12
Accuracy	79.0%
Cost	53513

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x \cdot \left(t_0 - 0.5\right)\\ \mathbf{if}\;y \leq -9 \cdot 10^{-5} \lor \neg \left(y \leq 2.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(y + \sin x \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + t_1\right) - t_0\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	79.0%
Cost	53513

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)\right)\\ \mathbf{if}\;x \leq -0.0007 \lor \neg \left(x \leq 0.00082\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(0.0625 + \cos x \cdot -0.0625\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t_1}\\ \end{array} \]

Alternative 14
Accuracy	59.9%
Cost	46784

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(t_0 - 0.5\right)\right) - t_0\right)\right)} \end{array} \]

Alternative 15
Accuracy	43.0%
Cost	46656

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(t_0 - 0.5\right)\right) - t_0\right)\right)} \end{array} \]

Alternative 16
Accuracy	40.6%
Cost	39488

\[\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{6} \]

Alternative 17
Accuracy	32.7%
Cost	34240

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

Alternative 18
Accuracy	32.5%
Cost	33856

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

Alternative 19
Accuracy	32.5%
Cost	27584

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

Alternative 20
Accuracy	30.4%
Cost	13760

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

Reproduce

herbie shell --seed 2023263 
(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))))))