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

Average Error: 0.5 → 0.4

Time: 35.3s

Precision: binary64

Cost: 72768

Math

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

↓

\[\frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{6}{\frac{\sqrt{5} + 1}{\cos x}}\right)} \]

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 y) (* (sin x) -0.0625))
     (* (- (cos x) (cos y)) (+ (sin x) (* (sin y) -0.0625))))))
  (+
   3.0
   (+
    (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0))))
    (/ 6.0 (/ (+ (sqrt 5.0) 1.0) (cos x)))))))

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(y) + (sin(x) * -0.0625)) * ((cos(x) - cos(y)) * (sin(x) + (sin(y) * -0.0625)))))) / (3.0 + ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (6.0 / ((sqrt(5.0) + 1.0) / cos(x)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / (3.0d0 * ((1.0d0 + (((sqrt(5.0d0) - 1.0d0) / 2.0d0) * cos(x))) + (((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y))))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((sin(y) + (sin(x) * (-0.0625d0))) * ((cos(x) - cos(y)) * (sin(x) + (sin(y) * (-0.0625d0))))))) / (3.0d0 + ((1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))) + (6.0d0 / ((sqrt(5.0d0) + 1.0d0) / cos(x)))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((1.0 + (((Math.sqrt(5.0) - 1.0) / 2.0) * Math.cos(x))) + (((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y))));
}

↓

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * ((Math.cos(x) - Math.cos(y)) * (Math.sin(x) + (Math.sin(y) * -0.0625)))))) / (3.0 + ((1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) + (6.0 / ((Math.sqrt(5.0) + 1.0) / Math.cos(x)))));
}

Python

def code(x, y):
	return (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / (3.0 * ((1.0 + (((math.sqrt(5.0) - 1.0) / 2.0) * math.cos(x))) + (((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y))))

↓

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.sin(y) + (math.sin(x) * -0.0625)) * ((math.cos(x) - math.cos(y)) * (math.sin(x) + (math.sin(y) * -0.0625)))))) / (3.0 + ((1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))) + (6.0 / ((math.sqrt(5.0) + 1.0) / math.cos(x)))))

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(sqrt(2.0) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(x) + Float64(sin(y) * -0.0625)))))) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) + Float64(6.0 / Float64(Float64(sqrt(5.0) + 1.0) / cos(x))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (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))));
end

↓

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((sin(y) + (sin(x) * -0.0625)) * ((cos(x) - cos(y)) * (sin(x) + (sin(y) * -0.0625)))))) / (3.0 + ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (6.0 / ((sqrt(5.0) + 1.0) / cos(x)))));
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[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 / N[(N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 0.5

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   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. Applied egg-rr 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

4. Simplified 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   Proof

   [Start] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   associate-/r* [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

5. Taylor expanded in y around inf 0.4

   \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}} \]

6. Applied egg-rr 0.4

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) + \color{blue}{\frac{6}{\frac{\sqrt{5} + 1}{\cos x}}}\right)} \]

7. Final simplification 0.4

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{6}{\frac{\sqrt{5} + 1}{\cos x}}\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	72768

\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \]

Alternative 2
Error	12.2
Cost	67017

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)\\ \mathbf{if}\;x \leq -0.0072 \lor \neg \left(x \leq 0.055\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot t_0\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{t_1}\\ \end{array} \]

Alternative 3
Error	12.2
Cost	66633

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0034 \lor \neg \left(x \leq 0.0019\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 4
Error	12.2
Cost	66633

\[\begin{array}{l} t_0 := \left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\\ t_1 := 3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)\\ \mathbf{if}\;x \leq -0.0072 \lor \neg \left(x \leq 0.029\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot t_0}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right)}{t_1}\\ \end{array} \]

Alternative 5
Error	12.3
Cost	66505

\[\begin{array}{l} t_0 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.00065 \lor \neg \left(x \leq 8.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_0 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{6}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 6
Error	12.3
Cost	66377

\[\begin{array}{l} t_0 := \left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\\ \mathbf{if}\;x \leq -0.0007 \lor \neg \left(x \leq 2.4 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	13.4
Cost	65928

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.00082:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(t_2 \cdot \left(\cos x + -1\right)\right), 2\right)}{\cos x \cdot \frac{6}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{t_0}{0.6666666666666666}, 3\right)}\\ \end{array} \]

Alternative 8
Error	13.4
Cost	60232

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} + -1\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.0007:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot t_3\right) \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\cos y \cdot t_0 + \cos x \cdot t_2\right)}\\ \end{array} \]

Alternative 9
Error	13.6
Cost	59780

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(1.5 \cdot t_0, \cos y, \frac{6}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot t_2\right) \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\cos y \cdot t_0 + \cos x \cdot t_1\right)}\\ \end{array} \]

Alternative 10
Error	13.5
Cost	59400

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos y \cdot t_1\\ t_3 := {\sin x}^{2}\\ t_4 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_3 \cdot t_0\right)\right)}{3 + \left(1.5 \cdot t_2 + 1.5 \cdot t_4\right)}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(1.5 \cdot t_1, \cos y, \frac{6}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot t_3\right) \cdot t_0\right)}{3 + 1.5 \cdot \left(t_2 + t_4\right)}\\ \end{array} \]

Alternative 11
Error	13.6
Cost	53257

\[\begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -0.00065 \lor \neg \left(x \leq 1.36 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(t_0 + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(1.5 \cdot t_0 + 6 \cdot \frac{1}{\sqrt{5} + 1}\right)}\\ \end{array} \]

Alternative 12
Error	13.6
Cost	53256

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ t_2 := 1.5 \cdot t_1\\ t_3 := {\sin x}^{2}\\ t_4 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_3 \cdot t_0\right)\right)}{3 + \left(t_2 + 1.5 \cdot t_4\right)}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(t_2 + 6 \cdot \frac{1}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot t_3\right) \cdot t_0\right)}{3 + 1.5 \cdot \left(t_1 + t_4\right)}\\ \end{array} \]

Alternative 13
Error	14.0
Cost	46985

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + 1\\ \mathbf{if}\;x \leq -0.00065 \lor \neg \left(x \leq 1.4 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{t_1} + 1.5 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot t_0\right) + 6 \cdot \frac{1}{t_1}\right)}\\ \end{array} \]

Alternative 14
Error	14.0
Cost	46857

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00086 \lor \neg \left(x \leq 4.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 15
Error	14.0
Cost	46856

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot t_1\right) \cdot t_0\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_1 \cdot t_0\right)\right)}{\left(2.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)}\\ \end{array} \]

Alternative 16
Error	25.6
Cost	46464

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

Alternative 17
Error	25.6
Cost	46336

\[0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)} \]

Alternative 18
Error	35.7
Cost	40649

\[\begin{array}{l} \mathbf{if}\;y \leq -210 \lor \neg \left(y \leq 7 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), 1.5 + 1.5 \cdot \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot 0.00390625\right) \cdot \left(y \cdot \sin x\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right) + \sqrt{5} \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 19
Error	37.0
Cost	20288

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

Alternative 20
Error	38.2
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023056 
(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))))))