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

Average Error: 0.72% → 0.73%

Time: 35.9s

Precision: binary64

Cost: 91840

Math

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

↓

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

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)))
    (* (- (cos y) (cos x)) (- (/ (sin x) 16.0) (sin y)))))
  (*
   3.0
   (+
    1.0
    (log
     (exp
      (fma
       (cos x)
       (+ -0.5 (* (sqrt 5.0) 0.5))
       (/ (cos y) (+ 1.5 (sqrt 1.25))))))))))

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))) * ((cos(y) - cos(x)) * ((sin(x) / 16.0) - sin(y))))) / (3.0 * (1.0 + log(exp(fma(cos(x), (-0.5 + (sqrt(5.0) * 0.5)), (cos(y) / (1.5 + sqrt(1.25))))))));
}

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(cos(y) - cos(x)) * Float64(Float64(sin(x) / 16.0) - sin(y))))) / Float64(3.0 * Float64(1.0 + log(exp(fma(cos(x), Float64(-0.5 + Float64(sqrt(5.0) * 0.5)), Float64(cos(y) / Float64(1.5 + sqrt(1.25)))))))))
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[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision] - N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[Log[N[Exp[N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] / N[(1.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.72

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

2. Simplified 0.72

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

   [Start] 0.72	\[ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   associate-*l* [=>] 0.72	\[ \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied egg-rr 0.68

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

4. Applied egg-rr 0.73

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \color{blue}{\log \left(e^{\mathsf{fma}\left(\cos x, \sqrt{5} \cdot 0.5 + -0.5, \frac{\cos y}{1.5 + \sqrt{1.25}}\right)}\right)}\right)} \]

5. Final simplification 0.73

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\cos y - \cos x\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\right)}{3 \cdot \left(1 + \log \left(e^{\mathsf{fma}\left(\cos x, -0.5 + \sqrt{5} \cdot 0.5, \frac{\cos y}{1.5 + \sqrt{1.25}}\right)}\right)\right)} \]

Alternatives

Alternative 1
Error	0.76%
Cost	72768

\[0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y \cdot 0.0625 - \sin x\right) \cdot \left(\sin x \cdot 0.0625 - \sin y\right)\right)\right)}{\frac{\cos y}{1.5 + \sqrt{1.25}} + \left(1 + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)} \]

Alternative 2
Error	0.68%
Cost	72768

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

Alternative 3
Error	0.68%
Cost	72768

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

Alternative 4
Error	17.67%
Cost	67016

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 1.5 + \sqrt{1.25}\\ t_2 := \frac{\sin x}{16} - \sin y\\ t_3 := 2 + \left(\left(\cos y - \cos x\right) \cdot t_2\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ \mathbf{if}\;x \leq -0.055:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(1 + \left(\frac{\cos y}{t_1} + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.26:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} - \sin x\right)\right) \cdot \left(\left(1 - \left(\cos y + 0.5 \cdot \left(x \cdot x\right)\right)\right) \cdot t_2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - t_0\right) + \cos y \cdot \frac{-1}{t_1}\right)\right)}\\ \end{array} \]

Alternative 5
Error	17.66%
Cost	67016

\[\begin{array}{l} t_0 := 1.5 + \sqrt{1.25}\\ t_1 := 3 \cdot \left(1 + \left(\frac{\cos y}{t_0} + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)\\ t_2 := \frac{\sin x}{16} - \sin y\\ t_3 := 2 + \left(\left(\cos y - \cos x\right) \cdot t_2\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ \mathbf{if}\;x \leq -0.052:\\ \;\;\;\;\frac{t_3}{t_1}\\ \mathbf{elif}\;x \leq 0.114:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} - \sin x\right)\right) \cdot \left(\left(1 - \left(\cos y + 0.5 \cdot \left(x \cdot x\right)\right)\right) \cdot t_2\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right) + \cos y \cdot \frac{-1}{t_0}\right)\right)}\\ \end{array} \]

Alternative 6
Error	17.73%
Cost	66632

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \left(\cos y - \cos x\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\\ t_2 := 2 + t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\\ t_3 := 1.5 + \sqrt{1.25}\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 + \left(\frac{\cos y}{t_3} + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.041:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + t_0\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - t_0\right) + \cos y \cdot \frac{-1}{t_3}\right)\right)}\\ \end{array} \]

Alternative 7
Error	17.72%
Cost	66632

\[\begin{array}{l} t_0 := \left(\cos y - \cos x\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\\ t_1 := 2 + t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\\ t_2 := 1.5 + \sqrt{1.25}\\ t_3 := 3 \cdot \left(1 + \left(\frac{\cos y}{t_2} + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)\\ \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;\frac{t_1}{t_3}\\ \mathbf{elif}\;x \leq 0.029:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right) + \cos y \cdot \frac{-1}{t_2}\right)\right)}\\ \end{array} \]

Alternative 8
Error	18.12%
Cost	66504

\[\begin{array}{l} t_0 := 2 + \left(\left(\cos y - \cos x\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ t_1 := 1.5 + \sqrt{1.25}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(1 + \left(\frac{\cos y}{t_1} + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \sqrt{5} + -1, 3 - 1.5 \cdot \left(\cos y \cdot \frac{-4}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right) + \cos y \cdot \frac{-1}{t_1}\right)\right)}\\ \end{array} \]

Alternative 9
Error	17.98%
Cost	66504

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \left(\cos y - \cos x\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\\ t_2 := 2 + t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\\ t_3 := 1.5 + \sqrt{1.25}\\ t_4 := \frac{\cos y}{t_3}\\ \mathbf{if}\;x \leq -0.00033:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 + \left(t_4 + \cos x \cdot \left(-0.5 + t_0\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_1}{3 \cdot \left(1 + \left(-0.5 + \left(t_4 + t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 - \left(\cos x \cdot \left(0.5 - \frac{\sqrt{5}}{2}\right) + \cos y \cdot \frac{-1}{t_3}\right)\right)}\\ \end{array} \]

Alternative 10
Error	18.14%
Cost	66377

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-5} \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos y - \cos x\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \sqrt{5} + -1, 3 - 1.5 \cdot \left(\cos y \cdot \frac{-4}{3 + \sqrt{5}}\right)\right)}\\ \end{array} \]

Alternative 11
Error	18.12%
Cost	66377

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-5} \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos y - \cos x\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + \sqrt{1.25}} + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \sqrt{5} + -1, 3 - 1.5 \cdot \left(\cos y \cdot \frac{-4}{3 + \sqrt{5}}\right)\right)}\\ \end{array} \]

Alternative 12
Error	19.68%
Cost	59977

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-5} \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\frac{\sin x}{16} - \sin y\right) \cdot \left(1 - \cos x\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \sqrt{5} + -1, 3 - 1.5 \cdot \left(\cos y \cdot \frac{-4}{3 + \sqrt{5}}\right)\right)}\\ \end{array} \]

Alternative 13
Error	20.08%
Cost	59849

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-5} \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos y - \cos x\right) \cdot \left(\frac{\sin x}{16} - \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\left(1.5 - \cos x \cdot \left(0.5 + \sqrt{5} \cdot -0.5\right)\right) - \sqrt{1.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \sqrt{5} + -1, 3 - 1.5 \cdot \left(\cos y \cdot \frac{-4}{3 + \sqrt{5}}\right)\right)}\\ \end{array} \]

Alternative 14
Error	20.32%
Cost	59529

\[\begin{array}{l} t_0 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -0.000135 \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{3 + 1.5 \cdot \left(\frac{4}{t_0} - \cos x \cdot \left(1 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \sqrt{5} + -1, 3 - 1.5 \cdot \left(\cos y \cdot \frac{-4}{t_0}\right)\right)}\\ \end{array} \]

Alternative 15
Error	20.32%
Cost	46985

\[\begin{array}{l} t_0 := 1 - \sqrt{5}\\ t_1 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-5} \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{3 + 1.5 \cdot \left(\frac{4}{t_1} - \cos x \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{t_0 \cdot -1.5 + \left(3 - 1.5 \cdot \left(\cos y \cdot \frac{-4}{t_1}\right)\right)}\\ \end{array} \]

Alternative 16
Error	20.4%
Cost	46857

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0001 \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\left(1 - \cos \left(y + y\right)\right) \cdot \frac{\sqrt{2} \cdot \left(\cos y + -1\right)}{2}\right)}{\left(1 - \sqrt{5}\right) \cdot -1.5 + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 17
Error	20.45%
Cost	46857

\[\begin{array}{l} t_0 := \sqrt{5} \cdot -0.5\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-5} \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\left(2.5 - \cos x \cdot \left(0.5 + t_0\right)\right) + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\left(1 - \cos \left(y + y\right)\right) \cdot \frac{\sqrt{2} \cdot \left(\cos y + -1\right)}{2}\right)}{\left(1 - \sqrt{5}\right) \cdot -1.5 + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 18
Error	20.38%
Cost	46857

\[\begin{array}{l} t_0 := \sqrt{5} \cdot -0.5\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-5} \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\left(2.5 - \cos x \cdot \left(0.5 + t_0\right)\right) + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{6}{1 + \sqrt{5}} + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 19
Error	20.3%
Cost	46857

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-5} \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{3 + 1.5 \cdot \left(\frac{4}{3 + \sqrt{5}} - \cos x \cdot \left(1 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(\cos y + -1\right)\right)\right)}{\frac{6}{1 + \sqrt{5}} + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 20
Error	20.39%
Cost	46729

\[\begin{array}{l} \mathbf{if}\;x \leq -0.000118 \lor \neg \left(x \leq 1.85 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\left(1 - \cos \left(y + y\right)\right) \cdot \frac{\sqrt{2} \cdot \left(\cos y + -1\right)}{2}\right)}{\left(1 - \sqrt{5}\right) \cdot -1.5 + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \end{array} \]

Alternative 21
Error	20.41%
Cost	46728

\[\begin{array}{l} t_0 := 1 - \sqrt{5}\\ t_1 := 2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\left(1 - \cos \left(y + y\right)\right) \cdot \frac{\sqrt{2} \cdot \left(\cos y + -1\right)}{2}\right)}{t_0 \cdot -1.5 + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} + \cos x \cdot t_0\right)\right)}\\ \end{array} \]

Alternative 22
Error	37.93%
Cost	46601

\[\begin{array}{l} \mathbf{if}\;y \leq -780 \lor \neg \left(y \leq 6.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\left(1 - \cos \left(y + y\right)\right) \cdot \frac{\sqrt{2} \cdot \left(\cos y + -1\right)}{2}\right)}{\left(1 - \sqrt{5}\right) \cdot -1.5 + \left(3 + -1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} + -3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{0.0078125}\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 23
Error	39.43%
Cost	40512

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

Alternative 24
Error	58.63%
Cost	20416

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

Alternative 25
Error	58.67%
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023102 
(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))))))