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

Percentage Accurate: 99.3% → 99.4%

Time: 32.8s

Precision: binary64

Cost: 98112

Math

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

↓

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 4.5\right)}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]

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 y) (/ (sin x) 16.0))
    (* (- (sin x) (/ (sin y) 16.0)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (fma
    (cos y)
    (/ 9.0 (fma 1.5 (sqrt 5.0) 4.5))
    (* (cos x) (/ 9.0 (fma (sqrt 5.0) 1.5 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 fma(sqrt(2.0), ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * (cos(x) - cos(y)))), 2.0) / (3.0 + fma(cos(y), (9.0 / fma(1.5, sqrt(5.0), 4.5)), (cos(x) * (9.0 / fma(sqrt(5.0), 1.5, 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(fma(sqrt(2.0), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + fma(cos(y), Float64(9.0 / fma(1.5, sqrt(5.0), 4.5)), Float64(cos(x) * Float64(9.0 / fma(sqrt(5.0), 1.5, 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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(9.0 / N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(9.0 / N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]
   Step-by-step derivation

   [Start] 99.3%	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \color{blue}{1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)}\right)} \]

4. Simplified 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \left(1.5 \cdot \sqrt{5} + -1.5\right)}\right)} \]
   Step-by-step derivation

   [Start] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right)} \]
   associate-*r* [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \color{blue}{\left(1.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x}\right)} \]
   *-commutative [<=] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \left(1.5 \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]
   sub-neg [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \left(1.5 \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right)\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \left(1.5 \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right)\right)} \]
   distribute-lft-in [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \color{blue}{\left(1.5 \cdot \sqrt{5} + 1.5 \cdot -1\right)}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \left(1.5 \cdot \sqrt{5} + \color{blue}{-1.5}\right)\right)} \]

5. Applied egg-rr 99.2%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \color{blue}{\frac{\left(1.5 \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot \sqrt{5}\right) - 2.25}{1.5 \cdot \sqrt{5} - -1.5}}\right)} \]
   Step-by-step derivation

   [Start] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \left(1.5 \cdot \sqrt{5} + -1.5\right)\right)} \]
   flip-+ [=>] 99.2%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \color{blue}{\frac{\left(1.5 \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot \sqrt{5}\right) - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}}\right)} \]
   metadata-eval [=>] 99.2%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{\left(1.5 \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot \sqrt{5}\right) - \color{blue}{2.25}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

6. Simplified 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \color{blue}{\frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}}\right)} \]
   Step-by-step derivation

   [Start] 99.2%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{\left(1.5 \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot \sqrt{5}\right) - 2.25}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   sub-neg [=>] 99.2%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{\color{blue}{\left(1.5 \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot \sqrt{5}\right) + \left(-2.25\right)}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   +-commutative [=>] 99.2%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{\color{blue}{\left(-2.25\right) + \left(1.5 \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot \sqrt{5}\right)}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   metadata-eval [=>] 99.2%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{\color{blue}{-2.25} + \left(1.5 \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot \sqrt{5}\right)}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   swap-sqr [=>] 99.2%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{-2.25 + \color{blue}{\left(1.5 \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   metadata-eval [=>] 99.2%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{-2.25 + \color{blue}{2.25} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   *-commutative [=>] 99.2%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{-2.25 + \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot 2.25}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   rem-square-sqrt [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{-2.25 + \color{blue}{5} \cdot 2.25}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{-2.25 + \color{blue}{11.25}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{\color{blue}{9}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]
   *-commutative [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{9}{\color{blue}{\sqrt{5} \cdot 1.5} - -1.5}\right)} \]
   fma-neg [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{9}{\color{blue}{\mathsf{fma}\left(\sqrt{5}, 1.5, --1.5\right)}}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, \color{blue}{1.5}\right)}\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   Step-by-step derivation

   [Start] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   flip-- [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{4.5 \cdot 4.5 - \frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25} - \frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   div-inv [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot \color{blue}{1.5}\right) \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   div-inv [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{1.5}\right)}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   div-inv [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{0.6666666666666666}}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot \color{blue}{1.5}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]

8. Simplified 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 4.5\right)}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   Step-by-step derivation

   [Start] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   swap-sqr [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot 1.5\right)}}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \color{blue}{2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   rem-square-sqrt [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{5} \cdot 2.25}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   cancel-sign-sub-inv [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25 + \left(-5\right) \cdot 2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-5} \cdot 2.25}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-11.25}}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   metadata-eval [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{9}}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   +-commutative [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\color{blue}{\sqrt{5} \cdot 1.5 + 4.5}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   *-commutative [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\color{blue}{1.5 \cdot \sqrt{5}} + 4.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]
   fma-def [=>] 99.4%	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\color{blue}{\mathsf{fma}\left(1.5, \sqrt{5}, 4.5\right)}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]

9. Final simplification 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 4.5\right)}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	98112

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 4.5\right)}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]

Alternative 2
Accuracy	99.4%
Cost	91712

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)}\right)} \]

Alternative 3
Accuracy	99.3%
Cost	85312

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \left(1.5 \cdot \sqrt{5} + -1.5\right)\right)} \]

Alternative 4
Accuracy	99.3%
Cost	73024

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

Alternative 5
Accuracy	99.3%
Cost	72896

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

Alternative 6
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 7
Accuracy	81.2%
Cost	72772

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 2 + \left(t_1 \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ t_3 := \cos x \cdot \left(t_0 - 0.5\right)\\ \mathbf{if}\;x \leq -0.043:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 + \left(t_3 + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)}\\ \mathbf{elif}\;x \leq 0.061:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 + \left(t_3 + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	81.2%
Cost	67145

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.03 \lor \neg \left(x \leq 0.072\right):\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 9
Accuracy	81.1%
Cost	66761

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.009 \lor \neg \left(x \leq 0.0105\right):\\ \;\;\;\;\frac{2 + \left(t_2 \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_2 \cdot \left(1 - \cos y\right)\right)}{t_1}\\ \end{array} \]

Alternative 10
Accuracy	81.0%
Cost	66633

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\\ \mathbf{if}\;x \leq -0.009 \lor \neg \left(x \leq 0.009\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_2}{3 \cdot \left(1 + \left(\left(t_1 + \cos y \cdot \left(1.5 - t_1\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	81.0%
Cost	66633

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\\ \mathbf{if}\;x \leq -0.009 \lor \neg \left(x \leq 0.009\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_2}{3 \cdot \left(1 + \left(\left(t_1 + \frac{\cos y}{1.5 + t_1}\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	80.9%
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x \cdot \left(t_0 - 0.5\right)\\ \mathbf{if}\;x \leq -0.009 \lor \neg \left(x \leq 0.009\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(t_1 + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 \cdot \left(1 + \left(t_1 + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	79.2%
Cost	59912

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \cos x - \cos y\\ t_2 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_3 := \frac{\sqrt{5}}{2}\\ t_4 := {\sin x}^{2} \cdot t_0\\ \mathbf{if}\;x \leq -0.009:\\ \;\;\;\;\left(2 + t_1 \cdot t_4\right) \cdot \frac{0.3333333333333333}{1 + \left(\cos y \cdot \left(1.5 - t_3\right) + \cos x \cdot \left(t_3 + -0.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.009:\\ \;\;\;\;\frac{2 + t_1 \cdot \left({\sin y}^{2} \cdot t_0\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_4 \cdot \left(\cos x + -1\right)}{t_2}\\ \end{array} \]

Alternative 14
Accuracy	79.0%
Cost	59912

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := {\sin x}^{2} \cdot t_0\\ \mathbf{if}\;x \leq -560000:\\ \;\;\;\;\left(2 + \left(\cos x - \cos y\right) \cdot t_2\right) \cdot \frac{0.3333333333333333}{1 + \left(\cos y \cdot \left(1.5 - t_1\right) + \cos x \cdot \left(t_1 + -0.5\right)\right)}\\ \mathbf{elif}\;x \leq 54:\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 15
Accuracy	79.2%
Cost	59848

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := {\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\\ \mathbf{if}\;x \leq -0.009:\\ \;\;\;\;\frac{\left(2 + \left(\cos x - \cos y\right) \cdot t_2\right) \cdot 0.3333333333333333}{1 + 0.5 \cdot \left(\cos x \cdot t_1 + \cos y \cdot t_0\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y \cdot -0.0625\right)\right), 2\right)}{3 + \left(1.5 \cdot t_1 + \cos y \cdot \left(4.5 - 1.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \end{array} \]

Alternative 16
Accuracy	79.2%
Cost	59848

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := {\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\\ \mathbf{if}\;x \leq -0.009:\\ \;\;\;\;\left(2 + \left(\cos x - \cos y\right) \cdot t_2\right) \cdot \frac{0.3333333333333333}{1 + \left(\cos y \cdot \left(1.5 - t_1\right) + \cos x \cdot \left(t_1 + -0.5\right)\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y \cdot -0.0625\right)\right), 2\right)}{3 + \left(1.5 \cdot t_0 + \cos y \cdot \left(4.5 - 1.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 17
Accuracy	62.0%
Cost	59520

\[\frac{\left(2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)\right) \cdot 0.3333333333333333}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

Alternative 18
Accuracy	61.9%
Cost	53248

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

Alternative 19
Accuracy	59.6%
Cost	46592

\[0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

Alternative 20
Accuracy	42.2%
Cost	20288

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

Alternative 21
Accuracy	42.2%
Cost	20288

\[\frac{0.6666666666666666}{1 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)} \]

Alternative 22
Accuracy	42.2%
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 23
Accuracy	40.3%
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023269 
(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))))))