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

Average Error: 0.5 → 0.4

Time: 43.0s

Precision: binary64

Cost: 91584

Math

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

↓

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

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)
   (*
    (* (- (cos x) (cos y)) (- (sin x) (* (sin y) 0.0625)))
    (+ (sin y) (* -0.0625 (sin x))))
   2.0)
  (fma
   (cos x)
   (/ 6.0 (+ 1.0 (sqrt 5.0)))
   (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), (((cos(x) - cos(y)) * (sin(x) - (sin(y) * 0.0625))) * (sin(y) + (-0.0625 * sin(x)))), 2.0) / fma(cos(x), (6.0 / (1.0 + sqrt(5.0))), 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(Float64(cos(x) - cos(y)) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * Float64(sin(y) + Float64(-0.0625 * sin(x)))), 2.0) / fma(cos(x), Float64(6.0 / Float64(1.0 + sqrt(5.0))), 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[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(6.0 / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $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. Applied egg-rr 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{{\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}^{-1}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]

4. Simplified 0.4

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

   [Start] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, {\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}^{-1}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]
   unpow-1 [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{\frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]
   associate-/r* [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\color{blue}{6}}{\sqrt{5} + 1}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]
   +-commutative [=>] 0.4	\[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{6}{\color{blue}{1 + \sqrt{5}}}, \mathsf{fma}\left(\cos y, 1.5 \cdot \left(3 - \sqrt{5}\right), 3\right)\right)} \]

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
Cost	73024

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

Alternative 2
Error	0.4
Cost	73024

\[\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{\cos x}{0.5}}{1 + \sqrt{5}}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)} \]

Alternative 3
Error	0.5
Cost	72896

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

Alternative 4
Error	0.5
Cost	72768

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

Alternative 5
Error	0.4
Cost	72768

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

Alternative 6
Error	11.7
Cost	67144

\[\begin{array}{l} t_0 := \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)\\ t_3 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.041:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(\left(1 + \frac{\frac{\cos x}{0.5}}{1 + \sqrt{5}}\right) + t_0\right)}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_1\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(t_3 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_0 + t_3\right)}\\ \end{array} \]

Alternative 7
Error	12.0
Cost	66760

\[\begin{array}{l} t_0 := \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)\\ \mathbf{if}\;x \leq -0.003:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(\left(1 + \frac{\frac{\cos x}{0.5}}{1 + \sqrt{5}}\right) + t_0\right)}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos y \cdot \frac{1}{1.5 + \sqrt{1.25}} + \cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_0 + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \end{array} \]

Alternative 8
Error	11.7
Cost	66760

\[\begin{array}{l} t_0 := \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := 2 + t_2 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{t_4}{3 \cdot \left(\left(1 + \frac{\frac{\cos x}{0.5}}{1 + \sqrt{5}}\right) + t_0\right)}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{3 \cdot \left(t_0 + t_1\right)}\\ \end{array} \]

Alternative 9
Error	12.0
Cost	66633

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.00355 \lor \neg \left(x \leq 33000000\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	12.0
Cost	66632

\[\begin{array}{l} t_0 := \cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\\ t_1 := \cos x - \cos y\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0033:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(\sin x \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_2 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(t_0 + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot t_2\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos y \cdot \frac{1}{1.5 + \sqrt{1.25}} + t_0\right)\right)}\\ \end{array} \]

Alternative 11
Error	12.0
Cost	66632

\[\begin{array}{l} t_0 := \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)\\ \mathbf{if}\;x \leq -0.00176:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(\left(1 + \frac{\frac{\cos x}{0.5}}{1 + \sqrt{5}}\right) + t_0\right)}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_0 + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \end{array} \]

Alternative 12
Error	11.8
Cost	66505

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 + -0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)\\ \mathbf{if}\;x \leq -0.0043 \lor \neg \left(x \leq 0.19\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot t_1\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right) \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right)}{t_2}\\ \end{array} \]

Alternative 13
Error	11.8
Cost	66504

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x \cdot \left(t_0 + -0.5\right)\\ t_2 := 3 \cdot \left(1 + \left(t_1 + \cos y \cdot \left(1.5 - t_0\right)\right)\right)\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := 2 + \left(\left(\cos x - \cos y\right) \cdot t_3\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ \mathbf{if}\;x \leq -0.0045:\\ \;\;\;\;\frac{t_4}{t_2}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right) \cdot \left(t_3 \cdot \left(1 - \cos y\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{3 \cdot \left(1 + \left(\cos y \cdot \frac{1}{1.5 + \sqrt{1.25}} + t_1\right)\right)}\\ \end{array} \]

Alternative 14
Error	11.8
Cost	66504

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x \cdot \left(t_0 + -0.5\right)\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \left(\sin x \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right) \cdot \left(t_3 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(t_1 + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_3\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos y \cdot \frac{1}{1.5 + \sqrt{1.25}} + t_1\right)\right)}\\ \end{array} \]

Alternative 15
Error	12.8
Cost	60360

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.00145:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x - \sin y \cdot 0.0625\right)\right) \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 + -0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{\cos x}{0.5}}{1 + \sqrt{5}}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \end{array} \]

Alternative 16
Error	13.1
Cost	60232

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{\cos x}{0.5}}{1 + \sqrt{5}}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \end{array} \]

Alternative 17
Error	13.2
Cost	60041

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-6} \lor \neg \left(y \leq 3.3 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{\cos x}{0.5}}{1 + \sqrt{5}}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \frac{-6}{-1 - \sqrt{5}}, 4.5 + \sqrt{5} \cdot -1.5\right)}\\ \end{array} \]

Alternative 18
Error	13.0
Cost	60040

\[\begin{array}{l} t_0 := 1 + \sqrt{5}\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -2.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t_1}{2}\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(\frac{6}{t_0} + 1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{\cos x}{0.5}}{t_0}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \end{array} \]

Alternative 19
Error	13.2
Cost	59913

\[\begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-6} \lor \neg \left(y \leq 3.3 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \frac{-6}{-1 - \sqrt{5}}, 4.5 + \sqrt{5} \cdot -1.5\right)}\\ \end{array} \]

Alternative 20
Error	13.6
Cost	59401

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-6} \lor \neg \left(x \leq 33000000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \frac{-6}{-1 - \sqrt{5}}, 4.5 + \sqrt{5} \cdot -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 21
Error	13.7
Cost	53128

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 1 + \sqrt{5}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + \left(1.5 \cdot t_0 + 6 \cdot \frac{\cos x}{t_1}\right)}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(\frac{6}{t_1} + 1.5 \cdot \left(\cos y \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \frac{{\sin x}^{4}}{-1 - \cos x}\right)}{\mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, 2.5 - \sqrt{1.25}\right)}\\ \end{array} \]

Alternative 22
Error	13.7
Cost	52872

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \frac{{\sin x}^{4}}{-1 - \cos x}\right)}{\mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, 2.5 - \sqrt{1.25}\right)}\\ \end{array} \]

Alternative 23
Error	13.7
Cost	46857

\[\begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-5} \lor \neg \left(x \leq 33000000\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \frac{\cos x + -1}{\frac{2}{1 - \cos \left(x + x\right)}}\right)}{\mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, 2.5 - \sqrt{1.25}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \end{array} \]

Alternative 24
Error	13.7
Cost	46856

\[\begin{array}{l} t_0 := \cos x + -1\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot {\sin x}^{2}\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \frac{t_0}{\frac{2}{1 - \cos \left(x + x\right)}}\right)}{\mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, 2.5 - \sqrt{1.25}\right)}\\ \end{array} \]

Alternative 25
Error	25.4
Cost	46528

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

Alternative 26
Error	25.4
Cost	46336

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

Alternative 27
Error	25.4
Cost	40512

\[0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right) + \sqrt{5} \cdot -0.5} \]

Alternative 28
Error	37.9
Cost	26368

\[0.3333333333333333 + \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right) \cdot -0.010416666666666666 \]

Alternative 29
Error	37.9
Cost	20544

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

Alternative 30
Error	37.9
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2023057 
(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))))))