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

Average Error: 0.5 → 0.4

Time: 38.0s

Precision: binary64

Cost: 91584

Math

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

↓

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{0.6666666666666666}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]

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) (* -0.0625 (sin x)))
    (* (- (sin x) (* (sin y) 0.0625)) (- (cos x) (cos y))))
   2.0)
  (fma
   (cos x)
   (/ (+ (sqrt 5.0) -1.0) 0.6666666666666666)
   (fma (cos y) (* 1.5 (- 3.0 (sqrt 5.0))) 3.0))))

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) + (-0.0625 * sin(x))) * ((sin(x) - (sin(y) * 0.0625)) * (cos(x) - cos(y)))), 2.0) / fma(cos(x), ((sqrt(5.0) + -1.0) / 0.6666666666666666), fma(cos(y), (1.5 * (3.0 - sqrt(5.0))), 3.0));
}

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(-0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * Float64(cos(x) - cos(y)))), 2.0) / fma(cos(x), Float64(Float64(sqrt(5.0) + -1.0) / 0.6666666666666666), fma(cos(y), Float64(1.5 * Float64(3.0 - sqrt(5.0))), 3.0)))
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[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.5

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

2. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{0.6666666666666666}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)}} \]
   Proof

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

3. Final simplification 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{0.6666666666666666}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	78912

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

Alternative 2
Error	12.0
Cost	72776

\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := t_1 + -0.5\\ t_3 := \cos x \cdot t_2\\ t_4 := \cos y \cdot \left(1.5 - t_1\right)\\ t_5 := 1 + t_4\\ \mathbf{if}\;x \leq -0.0195:\\ \;\;\;\;\frac{0.3333333333333333}{t_4 + \left(1 + t_3\right)} \cdot t_0\\ \mathbf{elif}\;x \leq 0.09:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\left(\sin x \cdot 0.0625 - \sin y\right) \cdot \left(-1 + \left(\cos y + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)}{3 \cdot \left(t_3 + t_5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, t_2, t_5\right)}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	72768

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

Alternative 4
Error	0.4
Cost	72768

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

Alternative 5
Error	0.4
Cost	72640

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

Alternative 6
Error	12.0
Cost	67144

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos y \cdot \left(1.5 - t_0\right) + \left(1 + \cos x \cdot \left(t_0 + -0.5\right)\right)\\ t_2 := 2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\\ \mathbf{if}\;x \leq -0.0195:\\ \;\;\;\;\frac{0.3333333333333333}{t_1} \cdot t_2\\ \mathbf{elif}\;x \leq 0.07:\\ \;\;\;\;\frac{2 + \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right) \cdot \left(\left(\frac{\sin x}{16} - \sin y\right) \cdot \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} - \sin x\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{1}{3 \cdot t_1}\\ \end{array} \]

Alternative 7
Error	12.0
Cost	67144

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x \cdot \left(t_0 + -0.5\right)\\ t_2 := \cos y \cdot \left(1.5 - t_0\right)\\ t_3 := t_2 + \left(1 + t_1\right)\\ t_4 := 2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\\ \mathbf{if}\;x \leq -0.0195:\\ \;\;\;\;\frac{0.3333333333333333}{t_3} \cdot t_4\\ \mathbf{elif}\;x \leq 0.059:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\left(\sin x \cdot 0.0625 - \sin y\right) \cdot \left(-1 + \left(\cos y + \left(x \cdot x\right) \cdot 0.5\right)\right)\right)}{3 \cdot \left(t_1 + \left(1 + t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \frac{1}{3 \cdot t_3}\\ \end{array} \]

Alternative 8
Error	12.0
Cost	66760

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \cos y \cdot \left(1.5 - t_1\right) + \left(1 + \cos x \cdot \left(t_1 + -0.5\right)\right)\\ t_3 := 2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot t_0\right)\\ \mathbf{if}\;x \leq -0.017:\\ \;\;\;\;\frac{0.3333333333333333}{t_2} \cdot t_3\\ \mathbf{elif}\;x \leq 0.0295:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{1}{3 \cdot t_2}\\ \end{array} \]

Alternative 9
Error	12.0
Cost	66760

\[\begin{array}{l} t_0 := \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \cos x \cdot \left(t_1 + -0.5\right)\\ t_3 := \cos y \cdot \left(1.5 - t_1\right)\\ t_4 := t_3 + \left(1 + t_2\right)\\ t_5 := 2 + \left(\sqrt{2} \cdot \sin x\right) \cdot t_0\\ \mathbf{if}\;x \leq -0.0195:\\ \;\;\;\;\frac{0.3333333333333333}{t_4} \cdot t_5\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{2 - t_0 \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 0.0625 - x\right)\right)}{3 \cdot \left(t_2 + \left(1 + t_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \frac{1}{3 \cdot t_4}\\ \end{array} \]

Alternative 10
Error	12.1
Cost	66632

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos y \cdot \left(1.5 - t_0\right) + \left(1 + \cos x \cdot \left(t_0 + -0.5\right)\right)\\ t_2 := 2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\\ \mathbf{if}\;x \leq -0.0025:\\ \;\;\;\;\frac{0.3333333333333333}{t_1} \cdot t_2\\ \mathbf{elif}\;x \leq 0.0155:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)}{1 + \left(-0.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right)\right) + \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{1}{3 \cdot t_1}\\ \end{array} \]

Alternative 11
Error	12.1
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.008 \lor \neg \left(x \leq 0.0076\right):\\ \;\;\;\;\frac{0.3333333333333333}{\cos y \cdot \left(1.5 - t_0\right) + \left(1 + \cos x \cdot \left(t_0 + -0.5\right)\right)} \cdot \left(2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)}{1 + \left(-0.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right)\right) + \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5\right)}\\ \end{array} \]

Alternative 12
Error	12.1
Cost	66377

\[\begin{array}{l} t_0 := \cos x \cdot \left(1 - \sqrt{5}\right)\\ t_1 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -0.0046 \lor \neg \left(x \leq 0.0042\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\sin x \cdot \left(\cos x - \cos y\right)\right)\right)}{1 + -0.5 \cdot \left(t_0 - t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)}{1 + \left(-0.5 \cdot t_0 + t_1 \cdot 0.5\right)}\\ \end{array} \]

Alternative 13
Error	13.2
Cost	60296

\[\begin{array}{l} t_0 := \sqrt{5} \cdot -0.5\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \cos x + -1\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0064:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_1 \cdot t_2\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(-0.5 - t_0\right)\right) + t_0\right)\right)}\\ \mathbf{elif}\;x \leq 0.0034:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)}{1 + \left(-0.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right)\right) + \left(\cos y \cdot t_3\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t_3}{2}\right)}\\ \end{array} \]

Alternative 14
Error	14.2
Cost	60105

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-6} \lor \neg \left(x \leq 5.5 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot {\sin y}^{2}, 2\right)}{1.5 + 1.5 \cdot \mathsf{fma}\left(\frac{4}{\sqrt{5} + 3}, \cos y, \sqrt{5}\right)}\\ \end{array} \]

Alternative 15
Error	14.2
Cost	60104

\[\begin{array}{l} t_0 := \sqrt{5} \cdot -0.5\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_1 \cdot t_2\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(-0.5 - t_0\right)\right) + t_0\right)\right)}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot {\sin y}^{2}, 2\right)}{1.5 + 1.5 \cdot \mathsf{fma}\left(\frac{4}{\sqrt{5} + 3}, \cos y, \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 16
Error	14.2
Cost	59273

\[\begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-5} \lor \neg \left(x \leq 5.5 \cdot 10^{-57}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{1 + \left(-0.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right)\right) + \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot {\sin y}^{2}, 2\right)}{1.5 + 1.5 \cdot \mathsf{fma}\left(\frac{4}{\sqrt{5} + 3}, \cos y, \sqrt{5}\right)}\\ \end{array} \]

Alternative 17
Error	13.4
Cost	53513

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-6} \lor \neg \left(y \leq 7.6 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + 0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(\cos y + -1\right)\right)}{1 + \left(-0.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right)\right) + \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{3 + 1.5 \cdot \left(\frac{4}{\sqrt{5} + 3} + \frac{4}{\frac{\sqrt{5} + 1}{\cos x}}\right)}\\ \end{array} \]

Alternative 18
Error	14.3
Cost	53513

\[\begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-6} \lor \neg \left(x \leq 5.5 \cdot 10^{-57}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{1 + \left(-0.5 \cdot \left(\cos x \cdot \left(1 - \sqrt{5}\right)\right) + t_0 \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{\left(\sqrt{5} + -1\right) \cdot 1.5 + \left(3 - t_0 \cdot -1.5\right)}\\ \end{array} \]

Alternative 19
Error	14.7
Cost	46985

\[\begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-6} \lor \neg \left(x \leq 5.5 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{3 + 1.5 \cdot \left(\frac{4}{\sqrt{5} + 3} + \frac{4}{\frac{\sqrt{5} + 1}{\cos x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{\left(\sqrt{5} + -1\right) \cdot 1.5 + \left(3 - \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot -1.5\right)}\\ \end{array} \]

Alternative 20
Error	14.7
Cost	46857

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-6} \lor \neg \left(x \leq 5.5 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{3 + 1.5 \cdot \left(\frac{4}{\sqrt{5} + 3} + \cos x \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t_0 \cdot 1.5 + \left(3 - \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot -1.5\right)}\\ \end{array} \]

Alternative 21
Error	14.7
Cost	46856

\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} - t_1\right)\right)}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + 4 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + -1.5 \cdot \left(\left(\sqrt{5} + -3\right) - t_1\right)}\\ \end{array} \]

Alternative 22
Error	14.7
Cost	46856

\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\\ t_1 := \sqrt{5} + -1\\ t_2 := \cos x \cdot t_1\\ \mathbf{if}\;x \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} - t_2\right)\right)}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t_1 \cdot 1.5 + \left(3 - \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + -1.5 \cdot \left(\left(\sqrt{5} + -3\right) - t_2\right)}\\ \end{array} \]

Alternative 23
Error	14.7
Cost	46856

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := \cos x \cdot t_1\\ t_3 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_3}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} - t_2\right)\right)}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t_1 \cdot 1.5 + \left(3 - \left(\cos y \cdot t_0\right) \cdot -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.0625 \cdot \left(\sqrt{2} \cdot t_3\right)}{3 + \left(1.5 \cdot t_0 + 1.5 \cdot t_2\right)}\\ \end{array} \]

Alternative 24
Error	25.4
Cost	46729

\[\begin{array}{l} \mathbf{if}\;y \leq -25500000000000 \lor \neg \left(y \leq 1.25 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left({\sin x}^{2} \cdot \sqrt{0.0078125}\right)}{3 + -1.5 \cdot \left(\left(\sqrt{5} + -3\right) - \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 25
Error	14.7
Cost	46729

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-5} \lor \neg \left(x \leq 5.5 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{3 + -1.5 \cdot \left(\left(\sqrt{5} + -3\right) - \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 26
Error	14.7
Cost	46728

\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(3 - \left(\sqrt{5} - t_1\right)\right)}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + -1.5 \cdot \left(\left(\sqrt{5} + -3\right) - t_1\right)}\\ \end{array} \]

Alternative 27
Error	37.0
Cost	46336

\[\frac{2 + \left(\cos x + -1\right) \cdot \left({\sin x}^{2} \cdot \sqrt{0.0078125}\right)}{3 + -1.5 \cdot \left(\left(\sqrt{5} + -3\right) - \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

Alternative 28
Error	37.1
Cost	20160

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

Alternative 29
Error	38.3
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023060 
(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))))))