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

Average Accuracy: 99.3% → 99.4%

Time: 50.7s

Precision: binary64

Cost: 79040

Math

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

↓

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \frac{9}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \]

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
 (/
  (fma
   (sqrt 2.0)
   (*
    (- (sin y) (/ (sin x) 16.0))
    (* (- (sin x) (/ (sin y) 16.0)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (+
    (* (cos y) (/ 9.0 (+ 4.5 (sqrt 11.25))))
    (* 1.5 (* (cos x) (+ (sqrt 5.0) -1.0)))))))

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(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * (cos(x) - cos(y)))), 2.0) / (3.0 + ((cos(y) * (9.0 / (4.5 + sqrt(11.25)))) + (1.5 * (cos(x) * (sqrt(5.0) + -1.0)))));
}

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(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 + Float64(Float64(cos(y) * Float64(9.0 / Float64(4.5 + sqrt(11.25)))) + Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))))
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[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[(N[Cos[y], $MachinePrecision] * N[(9.0 / N[(4.5 + N[Sqrt[11.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $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. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{9 \cdot \frac{1}{4.5 + \sqrt{11.25}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   Step-by-step derivation

   [Start] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   flip-- [=>] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{4.5 \cdot 4.5 - \frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   div-inv [=>] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\left(4.5 \cdot 4.5 - \frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}\right) \cdot \frac{1}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   sub-neg [=>] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\left(4.5 \cdot 4.5 + \left(-\frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}\right)\right)} \cdot \frac{1}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   metadata-eval [=>] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(\color{blue}{20.25} + \left(-\frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}\right)\right) \cdot \frac{1}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   frac-times [=>] 99.3	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(20.25 + \left(-\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5}}{0.6666666666666666 \cdot 0.6666666666666666}}\right)\right) \cdot \frac{1}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   add-sqr-sqrt [<=] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(20.25 + \left(-\frac{\color{blue}{5}}{0.6666666666666666 \cdot 0.6666666666666666}\right)\right) \cdot \frac{1}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(20.25 + \left(-\frac{5}{\color{blue}{0.4444444444444444}}\right)\right) \cdot \frac{1}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(20.25 + \left(-\color{blue}{11.25}\right)\right) \cdot \frac{1}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(20.25 + \color{blue}{-11.25}\right) \cdot \frac{1}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{9} \cdot \frac{1}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   add-sqr-sqrt [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 9 \cdot \frac{1}{4.5 + \color{blue}{\sqrt{\frac{\sqrt{5}}{0.6666666666666666}} \cdot \sqrt{\frac{\sqrt{5}}{0.6666666666666666}}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   sqrt-unprod [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 9 \cdot \frac{1}{4.5 + \color{blue}{\sqrt{\frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   frac-times [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 9 \cdot \frac{1}{4.5 + \sqrt{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5}}{0.6666666666666666 \cdot 0.6666666666666666}}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   add-sqr-sqrt [<=] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 9 \cdot \frac{1}{4.5 + \sqrt{\frac{\color{blue}{5}}{0.6666666666666666 \cdot 0.6666666666666666}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 9 \cdot \frac{1}{4.5 + \sqrt{\frac{5}{\color{blue}{0.4444444444444444}}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 9 \cdot \frac{1}{4.5 + \sqrt{\color{blue}{11.25}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

4. Simplified 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{9}{4.5 + \sqrt{11.25}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   Step-by-step derivation

   [Start] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 9 \cdot \frac{1}{4.5 + \sqrt{11.25}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   associate-*r/ [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{9 \cdot 1}{4.5 + \sqrt{11.25}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{9}}{4.5 + \sqrt{11.25}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

5. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(\cos y \cdot \frac{9}{4.5 + \sqrt{11.25}} + \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)}} \]
   Step-by-step derivation

   [Start] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{4.5 + \sqrt{11.25}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   fma-udef [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(\cos y \cdot \frac{9}{4.5 + \sqrt{11.25}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]
   div-inv [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \frac{9}{4.5 + \sqrt{11.25}} + \color{blue}{\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot \frac{1}{0.6666666666666666}}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \frac{9}{4.5 + \sqrt{11.25}} + \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot \color{blue}{1.5}\right)} \]

6. Final simplification 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \frac{9}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	72896

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

Alternative 2
Accuracy	99.3%
Cost	72768

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

Alternative 3
Accuracy	99.3%
Cost	72768

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

Alternative 4
Accuracy	81.4%
Cost	67145

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.08 \lor \neg \left(x \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 5
Accuracy	81.4%
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0037 \lor \neg \left(x \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 6
Accuracy	81.4%
Cost	66505

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00185 \lor \neg \left(x \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 7
Accuracy	79.7%
Cost	60296

\[\begin{array}{l} t_0 := \cos y \cdot \frac{9}{4.5 + \sqrt{11.25}}\\ t_1 := \mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0026:\\ \;\;\;\;\frac{t_1}{3 + \left(t_0 + t_2 \cdot \left(\cos x \cdot 1.5\right)\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + \left(t_0 + 1.5 \cdot \left(\cos x \cdot t_2\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	79.4%
Cost	59657

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0009 \lor \neg \left(x \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(\cos y \cdot \frac{9}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	79.4%
Cost	59656

\[\begin{array}{l} t_0 := \cos y \cdot \frac{9}{4.5 + \sqrt{11.25}}\\ t_1 := \mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.002:\\ \;\;\;\;\frac{t_1}{3 + \left(t_0 + t_2 \cdot \left(\cos x \cdot 1.5\right)\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + \left(t_0 + 1.5 \cdot \left(\cos x \cdot t_2\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	79.3%
Cost	53513

\[\begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{-5} \lor \neg \left(y \leq 7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(4.5 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	79.4%
Cost	53513

\[\begin{array}{l} t_0 := 3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)\\ \mathbf{if}\;x \leq -0.0009 \lor \neg \left(x \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{t_0}\\ \end{array} \]

Alternative 12
Accuracy	79.4%
Cost	53512

\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)\\ t_2 := {\sin x}^{2}\\ t_3 := \cos x + -1\\ \mathbf{if}\;x \leq -0.0004:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot t_0\right)}{t_1}\\ \mathbf{elif}\;x \leq 0.00065:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\cos x}{0.5 + \sqrt{5} \cdot 0.5}\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	78.8%
Cost	53129

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-5} \lor \neg \left(x \leq 1.8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(4.5 + 1.5 \cdot \left(\cos x \cdot t_0 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot t_0 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)}\\ \end{array} \]

Alternative 14
Accuracy	44.6%
Cost	53001

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(4.5 + 1.5 \cdot \left(\cos x \cdot t_0 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \left(\left(x \cdot x\right) \cdot -0.0625 + 0.020833333333333332 \cdot {x}^{4}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 15
Accuracy	61.3%
Cost	53001

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.041 \lor \neg \left(x \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot t_0\right), 2\right)}{3 + \left(4.5 + 1.5 \cdot \left(\cos x \cdot t_1 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot \left(\left(x \cdot x\right) \cdot -0.0625 + 0.020833333333333332 \cdot {x}^{4}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 16
Accuracy	44.3%
Cost	46985

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ \mathbf{if}\;y \leq -0.0001 \lor \neg \left(y \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(t_1 + \left(1 + 0.5 \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 1.00390625 \cdot \left(\sqrt{2} \cdot \left(y \cdot \left(x \cdot \left(1 - \cos y\right)\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + t_1\right)}\\ \end{array} \]

Alternative 17
Accuracy	39.3%
Cost	40777

\[\begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -12000000000000 \lor \neg \left(x \leq 1.56 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + 1.00390625 \cdot \left(\sqrt{2} \cdot \left(y \cdot \left(x \cdot \left(1 - \cos y\right)\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(t_0 + \left(1 + 0.5 \cdot t_1\right)\right)}\\ \end{array} \]

Alternative 18
Accuracy	39.4%
Cost	40777

\[\begin{array}{l} t_0 := 1 - \cos y\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -26.5 \lor \neg \left(x \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + 1.00390625 \cdot \left(\sqrt{2} \cdot \left(y \cdot \left(x \cdot t_0\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{t_1}\\ \end{array} \]

Alternative 19
Accuracy	39.5%
Cost	40777

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -12000000000000 \lor \neg \left(x \leq 75000000000\right):\\ \;\;\;\;\frac{2 + 1.00390625 \cdot \left(\sqrt{2} \cdot \left(y \cdot \left(x \cdot \left(1 - \cos y\right)\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{t_0}\\ \end{array} \]

Alternative 20
Accuracy	33.3%
Cost	33984

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

Alternative 21
Accuracy	34.6%
Cost	33984

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

Reproduce

herbie shell --seed 2023157 
(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))))))