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

Average Error: 0.5 → 0.4

Time: 34.3s

Precision: binary64

Cost: 85312

Math

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

↓

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, 1.5 \cdot \left(\sqrt{5} + -1\right), \cos y \cdot \frac{6}{3 + \sqrt{5}}\right)} \]

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) 16.0)) (- (sin y) (/ (sin x) 16.0))))
   2.0)
  (+
   3.0
   (fma
    (cos x)
    (* 1.5 (+ (sqrt 5.0) -1.0))
    (* (cos y) (/ 6.0 (+ 3.0 (sqrt 5.0))))))))

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) / 16.0)) * (sin(y) - (sin(x) / 16.0)))), 2.0) / (3.0 + fma(cos(x), (1.5 * (sqrt(5.0) + -1.0)), (cos(y) * (6.0 / (3.0 + sqrt(5.0))))));
}

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(cos(x) - cos(y)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))), 2.0) / Float64(3.0 + fma(cos(x), Float64(1.5 * Float64(sqrt(5.0) + -1.0)), Float64(cos(y) * Float64(6.0 / Float64(3.0 + sqrt(5.0)))))))
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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $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

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]
   Proof
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (cos.f64 x) (cos.f64 y)) (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)))) 2) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2)) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 2 points increase in error, 5 points decrease in error
   (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 2 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)))) (-.f64 (cos.f64 x) (cos.f64 y))) 2) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 2 points decrease in error
   (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y))))) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) (Rewrite<= metadata-eval (/.f64 2 3))) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 3 (sqrt.f64 5)) 3) 2)) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3)) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) (Rewrite<= metadata-eval (neg.f64 1)))) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (/.f64 (*.f64 (cos.f64 x) (Rewrite<= sub-neg_binary64 (-.f64 (sqrt.f64 5) 1))) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x))) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (/.f64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x)) (Rewrite<= metadata-eval (/.f64 2 3)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x)) 3) 2))))): 5 points increase in error, 11 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x)) 2) 3))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3)) (*.f64 3 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))))))): 2 points increase in error, 4 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2)) 3)) (*.f64 3 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)))))): 4 points increase in error, 4 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))) 3) (*.f64 3 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (+.f64 (*.f64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)) 3) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (Rewrite<= distribute-rgt-in_binary64 (*.f64 3 (+.f64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))))))): 9 points increase in error, 11 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (*.f64 3 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 (Rewrite<= metadata-eval (*.f64 3 1)) (*.f64 3 (+.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (Rewrite<= distribute-lft-in_binary64 (*.f64 3 (+.f64 1 (+.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))))))): 21 points increase in error, 20 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 3 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)))))): 9 points increase in error, 8 points decrease in error

3. Applied egg-rr 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), 2\right)}{3 + \color{blue}{\left(\cos x \cdot \left(\left(\sqrt{5} + -1\right) \cdot 1.5\right) + \cos y \cdot \frac{6}{\sqrt{5} + 3}\right)}} \]

4. Simplified 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(\cos x, 1.5 \cdot \left(\sqrt{5} + -1\right), \cos y \cdot \frac{6}{\sqrt{5} + 3}\right)}} \]
   Proof
   (fma.f64 (cos.f64 x) (*.f64 3/2 (+.f64 (sqrt.f64 5) -1)) (*.f64 (cos.f64 y) (/.f64 6 (+.f64 (sqrt.f64 5) 3)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 x) (*.f64 3/2 (+.f64 (sqrt.f64 5) (Rewrite<= metadata-eval (neg.f64 1)))) (*.f64 (cos.f64 y) (/.f64 6 (+.f64 (sqrt.f64 5) 3)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 x) (*.f64 3/2 (Rewrite<= sub-neg_binary64 (-.f64 (sqrt.f64 5) 1))) (*.f64 (cos.f64 y) (/.f64 6 (+.f64 (sqrt.f64 5) 3)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (cos.f64 x) (*.f64 3/2 (-.f64 (sqrt.f64 5) 1))) (*.f64 (cos.f64 y) (/.f64 6 (+.f64 (sqrt.f64 5) 3))))): 17 points increase in error, 23 points decrease in error
   (+.f64 (*.f64 (cos.f64 x) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (sqrt.f64 5) 1) 3/2))) (*.f64 (cos.f64 y) (/.f64 6 (+.f64 (sqrt.f64 5) 3)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (cos.f64 x) (*.f64 (Rewrite=> sub-neg_binary64 (+.f64 (sqrt.f64 5) (neg.f64 1))) 3/2)) (*.f64 (cos.f64 y) (/.f64 6 (+.f64 (sqrt.f64 5) 3)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (cos.f64 x) (*.f64 (+.f64 (sqrt.f64 5) (Rewrite=> metadata-eval -1)) 3/2)) (*.f64 (cos.f64 y) (/.f64 6 (+.f64 (sqrt.f64 5) 3)))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, 1.5 \cdot \left(\sqrt{5} + -1\right), \cos y \cdot \frac{6}{3 + \sqrt{5}}\right)} \]

Alternatives

Alternative 1
Error	0.5
Cost	79104

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

Alternative 2
Error	0.4
Cost	72896

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

Alternative 3
Error	0.5
Cost	72768

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

Alternative 4
Error	0.4
Cost	72768

\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)\right)}{3 + \left(\cos y \cdot \frac{9}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \]

Alternative 5
Error	0.4
Cost	72640

\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \cos y \cdot \left(4.5 - \sqrt{11.25}\right)\right)} \]

Alternative 6
Error	11.9
Cost	67144

\[\begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{if}\;x \leq -0.056:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 0.059:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	11.9
Cost	67016

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.072:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_0\right) + \cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	11.9
Cost	66888

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} + -1\\ t_2 := \frac{2 + t_0 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{if}\;x \leq -0.0074:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 0.0067:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)\right)}{3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + t_1 \cdot \left(1.5 + \left(x \cdot x\right) \cdot -0.75\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	12.0
Cost	66632

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{t_0}\\ \mathbf{if}\;x \leq -0.017:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.013:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	12.0
Cost	66504

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \frac{2 + \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 + -0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{if}\;x \leq -0.0036:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.007:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	12.0
Cost	66504

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sqrt{2} \cdot \sin x\\ t_2 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_3 := \cos x - \cos y\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.00155:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_4 \cdot t_1\right)}{3 \cdot \left(t_2 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0046:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)}{3 \cdot \left(t_2 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_3 \cdot t_4\right) \cdot t_1}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 + -0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \]

Alternative 12
Error	13.0
Cost	60424

\[\begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.004:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{t_1}\\ \end{array} \]

Alternative 13
Error	13.0
Cost	60168

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 1 + \cos x \cdot \frac{t_0}{2}\\ \mathbf{if}\;x \leq -0.0036:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0026:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_0\right) + \cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 14
Error	13.1
Cost	60040

\[\begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;y \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.0017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 15
Error	13.1
Cost	59976

\[\begin{array}{l} t_0 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	13.3
Cost	59912

\[\begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;x \leq -0.0006:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 70:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	13.3
Cost	59912

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{t_0}\\ \mathbf{if}\;x \leq -0.00056:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 70:\\ \;\;\;\;\frac{2 + t_1 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	13.3
Cost	53640

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos x + -1\\ t_2 := {\sin x}^{2}\\ t_3 := 1 + \cos x \cdot \frac{t_0}{2}\\ \mathbf{if}\;x \leq -0.00064:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_1 \cdot t_2\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_0\right) + \cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right)\right)}\\ \mathbf{elif}\;x \leq 3.9:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(t_3 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot t_2\right)\right)}{3 \cdot \left(t_3 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 19
Error	13.3
Cost	53640

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ t_1 := \frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{t_0}\\ \mathbf{if}\;x \leq -0.0006:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.9:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	13.3
Cost	53512

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} + -1\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot t_0\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_1\right) + \cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;{\left(3 \cdot \frac{\sqrt{1.25} + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + 0.5\right)}{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot t_0\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 21
Error	13.3
Cost	53384

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right)\right)}\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;{\left(3 \cdot \frac{\sqrt{1.25} + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + 0.5\right)}{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 22
Error	13.8
Cost	53064

\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\sqrt{5} \cdot -0.5 + \left(2.5 + \cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)}{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot t_1\right)\right)}}\\ \mathbf{elif}\;x \leq 70:\\ \;\;\;\;{\left(3 \cdot \frac{\sqrt{1.25} + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + 0.5\right)}{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot t_1\right)\right)}{7.5 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \sqrt{5} \cdot -1.5\right)}\\ \end{array} \]

Alternative 23
Error	13.8
Cost	46856

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{7.5 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \sqrt{5} \cdot -1.5\right)}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 70:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 24
Error	13.8
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \sqrt{5} \cdot -0.5\\ t_2 := \cos x + -1\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{t_1 + \left(2.5 + \cos x \cdot \left(-0.5 + t_0\right)\right)}{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot t_3\right)\right)}}\\ \mathbf{elif}\;x \leq 70:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(t_0 + \cos y \cdot \left(1.5 + t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot t_3\right)\right)}{7.5 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \sqrt{5} \cdot -1.5\right)}\\ \end{array} \]

Alternative 25
Error	25.9
Cost	46464

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 7.5\right) + \sqrt{5} \cdot -1.5} \]

Alternative 26
Error	25.9
Cost	46464

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\left(7.5 + \cos x \cdot \left(1.5 \cdot \left(\sqrt{5} + -1\right)\right)\right) + \sqrt{5} \cdot -1.5} \]

Alternative 27
Error	25.9
Cost	46464

\[\frac{2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{7.5 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \sqrt{5} \cdot -1.5\right)} \]

Alternative 28
Error	38.2
Cost	39936

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\left(7.5 + 1.5 \cdot \left(\sqrt{5} + -1\right)\right) + \sqrt{5} \cdot -1.5} \]

Alternative 29
Error	38.2
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2022338 
(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))))))