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

Average Error: 0.5 → 0.4

Time: 34.0s

Precision: binary64

Cost: 104320

Math

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

↓

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

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)
   (pow (cbrt (* 1.5 (+ (sqrt 5.0) -1.0))) 3.0)
   (fma (cos y) (- 4.5 (sqrt 11.25)) 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), pow(cbrt((1.5 * (sqrt(5.0) + -1.0))), 3.0), fma(cos(y), (4.5 - sqrt(11.25)), 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), (cbrt(Float64(1.5 * Float64(sqrt(5.0) + -1.0))) ^ 3.0), fma(cos(y), Float64(4.5 - sqrt(11.25)), 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[Power[N[Power[N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(4.5 - N[Sqrt[11.25], $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

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{0.6666666666666666}, \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)\right)}} \]
   Proof
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (+.f64 (sin.f64 y) (*.f64 -1/16 (sin.f64 x))) (*.f64 (+.f64 (sin.f64 x) (*.f64 -1/16 (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (+.f64 (sin.f64 y) (*.f64 (Rewrite<= metadata-eval (/.f64 -1 16)) (sin.f64 x))) (*.f64 (+.f64 (sin.f64 x) (*.f64 -1/16 (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (+.f64 (sin.f64 y) (*.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) 16) (sin.f64 x))) (*.f64 (+.f64 (sin.f64 x) (*.f64 -1/16 (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (+.f64 (sin.f64 y) (Rewrite<= associate-/r/_binary64 (/.f64 (neg.f64 1) (/.f64 16 (sin.f64 x))))) (*.f64 (+.f64 (sin.f64 x) (*.f64 -1/16 (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 1 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (+.f64 (sin.f64 y) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 1) (sin.f64 x)) 16))) (*.f64 (+.f64 (sin.f64 x) (*.f64 -1/16 (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 1 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (+.f64 (sin.f64 y) (/.f64 (*.f64 (Rewrite=> metadata-eval -1) (sin.f64 x)) 16)) (*.f64 (+.f64 (sin.f64 x) (*.f64 -1/16 (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (+.f64 (sin.f64 y) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (sin.f64 x))) 16)) (*.f64 (+.f64 (sin.f64 x) (*.f64 -1/16 (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (+.f64 (sin.f64 y) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (sin.f64 x) 16)))) (*.f64 (+.f64 (sin.f64 x) (*.f64 -1/16 (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (*.f64 (+.f64 (sin.f64 x) (*.f64 -1/16 (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (+.f64 (sin.f64 x) (*.f64 (Rewrite<= metadata-eval (/.f64 -1 16)) (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (+.f64 (sin.f64 x) (*.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) 16) (sin.f64 y))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (+.f64 (sin.f64 x) (Rewrite<= associate-/r/_binary64 (/.f64 (neg.f64 1) (/.f64 16 (sin.f64 y))))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (+.f64 (sin.f64 x) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 1) (sin.f64 y)) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (+.f64 (sin.f64 x) (/.f64 (*.f64 (Rewrite=> metadata-eval -1) (sin.f64 y)) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (+.f64 (sin.f64 x) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (sin.f64 y))) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (+.f64 (sin.f64 x) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (sin.f64 y) 16)))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (cos.f64 x) (cos.f64 y)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (-.f64 (cos.f64 x) (cos.f64 y))) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 4 points increase in error, 1 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (-.f64 (cos.f64 x) (cos.f64 y))))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (-.f64 (cos.f64 x) (cos.f64 y))))) 2)) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 3 points increase in error, 3 points decrease in error
   (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (-.f64 (cos.f64 x) (cos.f64 y))))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 0 points increase in error, 8 points decrease in error
   (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.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)))) 2) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 3))): 3 points increase in error, 3 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))))) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) -1) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 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)))) (fma.f64 (cos.f64 x) (/.f64 (+.f64 (sqrt.f64 5) (Rewrite<= metadata-eval (neg.f64 1))) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 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)))) (fma.f64 (cos.f64 x) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (sqrt.f64 5) 1)) 2/3) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 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)))) (fma.f64 (cos.f64 x) (/.f64 (-.f64 (sqrt.f64 5) 1) (Rewrite<= metadata-eval (/.f64 2 3))) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 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)))) (fma.f64 (cos.f64 x) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 (sqrt.f64 5) 1) 3) 2)) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 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)))) (fma.f64 (cos.f64 x) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3)) (fma.f64 (cos.f64 y) (-.f64 9/2 (/.f64 (sqrt.f64 5) 2/3)) 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (fma.f64 (cos.f64 y) (-.f64 (Rewrite<= metadata-eval (/.f64 3 2/3)) (/.f64 (sqrt.f64 5) 2/3)) 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (fma.f64 (cos.f64 y) (-.f64 (/.f64 3 (Rewrite<= metadata-eval (/.f64 2 3))) (/.f64 (sqrt.f64 5) 2/3)) 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (fma.f64 (cos.f64 y) (-.f64 (/.f64 3 (/.f64 2 3)) (/.f64 (sqrt.f64 5) (Rewrite<= metadata-eval (/.f64 2 3)))) 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (fma.f64 (cos.f64 y) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 3 (sqrt.f64 5)) (/.f64 2 3))) 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (fma.f64 (cos.f64 y) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 3 (sqrt.f64 5)) 3) 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (fma.f64 (cos.f64 y) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3)) 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3)) 3)))): 4 points increase in error, 2 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2)) 3)) 3))): 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))) 3) 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)) 1) 3)))): 18 points increase in error, 13 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)))) (fma.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)))) 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)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 x) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) 3)) (*.f64 (+.f64 1 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))) 3)))): 9 points increase in error, 8 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<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 x) (/.f64 (-.f64 (sqrt.f64 5) 1) 2)) 3)) (*.f64 (+.f64 1 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))) 3))): 7 points increase in error, 7 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 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))) 3) (*.f64 (+.f64 1 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))) 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)))) (Rewrite<= distribute-rgt-in_binary64 (*.f64 3 (+.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) (+.f64 1 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))))))): 17 points increase in error, 21 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 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 1) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)))))): 9 points increase in error, 7 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<= +-commutative_binary64 (+.f64 1 (*.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

3. Applied egg-rr 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{{\left(\sqrt[3]{\left(\sqrt{5} + -1\right) \cdot 1.5}\right)}^{3}}, \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)\right)} \]

4. Applied egg-rr 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, {\left(\sqrt[3]{\left(\sqrt{5} + -1\right) \cdot 1.5}\right)}^{3}, \mathsf{fma}\left(\cos y, \color{blue}{4.5 + \left(-\sqrt{11.25}\right)}, 3\right)\right)} \]

5. Applied egg-rr 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, {\left(\sqrt[3]{\left(\sqrt{5} + -1\right) \cdot 1.5}\right)}^{3}, \mathsf{fma}\left(\cos y, \color{blue}{4.5 - \sqrt{11.25}}, 3\right)\right)} \]

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	72768

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

Alternative 2
Error	12.2
Cost	66888

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;x \leq -2468983532.3865147:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.258339871958961 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	12.3
Cost	66760

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \cos x - \cos y\\ t_2 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_3 := \frac{2 + t_1 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right)}{t_2}\\ \mathbf{if}\;x \leq -2468983532.3865147:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 6.258339871958961 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	12.3
Cost	66504

\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right)}{t_1}\\ \mathbf{if}\;x \leq -2468983532.3865147:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.258339871958961 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	13.0
Cost	60360

\[\begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;y \leq -0.07467157572986101:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(t_2 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \sin y\right)\right)\right)}{t_1}\\ \mathbf{elif}\;y \leq 4.063041098356003 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot \left(\sqrt{2} \cdot \left(\sin x + y \cdot -0.0625\right)\right)\right) \cdot \left(\cos x + -1\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \end{array} \]

Alternative 6
Error	13.3
Cost	60232

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{t_0}\\ \mathbf{if}\;x \leq -2468983532.3865147:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.258339871958961 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot \left(\sin y \cdot -0.0625 + x \cdot 1.00390625\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	13.0
Cost	60232

\[\begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.07467157572986101:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \sin y\right)\right)\right)}{t_1}\\ \mathbf{elif}\;y \leq 4.063041098356003 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(-0.0625 \cdot \sin x + y \cdot 1.00390625\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \end{array} \]

Alternative 8
Error	13.3
Cost	60104

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{t_0}\\ \mathbf{if}\;x \leq -2468983532.3865147:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.258339871958961 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot \left(\sin y \cdot -0.0625 + x \cdot 1.00390625\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	13.1
Cost	59912

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)\\ t_3 := \sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\\ \mathbf{if}\;y \leq -0.07467157572986101:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot t_3}{t_2}\\ \mathbf{elif}\;y \leq 4.063041098356003 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.6666666666666666 + -0.020833333333333332 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t_1, \cos x, t_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot t_3}{t_2}\\ \end{array} \]

Alternative 10
Error	13.3
Cost	59912

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{t_0}\\ \mathbf{if}\;x \leq -2468983532.3865147:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.258339871958961 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot \left(\sin y \cdot -0.0625 + x \cdot 1.00390625\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	13.1
Cost	59272

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := \frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{if}\;y \leq -0.07467157572986101:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.063041098356003 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.6666666666666666 + -0.020833333333333332 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t_1, \cos x, t_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	13.1
Cost	53512

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;y \leq -0.07467157572986101:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.063041098356003 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_0\right) + 9 \cdot \frac{1}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	13.6
Cost	46984

\[\begin{array}{l} t_0 := 4.5 + \sqrt{11.25}\\ t_1 := \sqrt{5} + -1\\ t_2 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_3 := \cos x \cdot t_1\\ \mathbf{if}\;x \leq -1.865062976175606 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_2}{1 + \left(t_3 \cdot 0.5 + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \mathbf{elif}\;x \leq 6.258339871958961 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_1 + \left(3 + 9 \cdot \frac{\cos y}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 + \left(1.5 \cdot t_3 + 9 \cdot \frac{1}{t_0}\right)}\\ \end{array} \]

Alternative 14
Error	13.6
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \left(1.5 \cdot \left(\cos x \cdot t_0\right) + 7.5\right) + \sqrt{5} \cdot -1.5\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -1.865062976175606 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot \left(0.5 + \cos \left(x + x\right) \cdot -0.5\right)\right)\right)}{t_1}\\ \mathbf{elif}\;x \leq 6.258339871958961 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_0 + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot {\sin x}^{2}\right)\right)}{t_1}\\ \end{array} \]

Alternative 15
Error	13.6
Cost	46856

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_2 := \cos x \cdot t_0\\ \mathbf{if}\;x \leq -1.865062976175606 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + \left(t_2 \cdot 0.5 + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \mathbf{elif}\;x \leq 6.258339871958961 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_0 + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\left(1.5 \cdot t_2 + 7.5\right) + \sqrt{5} \cdot -1.5}\\ \end{array} \]

Alternative 16
Error	25.7
Cost	40384

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 + \cos \left(x + x\right) \cdot -0.5\right)\right)\right)}{\left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 7.5\right) + \sqrt{5} \cdot -1.5} \]

Alternative 17
Error	36.8
Cost	20288

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

Alternative 18
Error	38.0
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2022309 
(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))))))