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

Average Error: 0.5 → 0.4

Time: 32.9s

Precision: binary64

Cost: 72768

Math

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

↓

\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + \cos x \cdot \frac{6}{\sqrt{5} + 1}\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (*
   3.0
   (+
    (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
    (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))

↓

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (sqrt 2.0)
    (*
     (- (cos x) (cos y))
     (* (+ (* -0.0625 (sin y)) (sin x)) (+ (sin y) (* -0.0625 (sin x)))))))
  (+
   3.0
   (+
    (* (cos y) (+ 4.5 (* (sqrt 5.0) -1.5)))
    (* (cos x) (/ 6.0 (+ (sqrt 5.0) 1.0)))))))

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 (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (((-0.0625 * sin(y)) + sin(x)) * (sin(y) + (-0.0625 * sin(x))))))) / (3.0 + ((cos(y) * (4.5 + (sqrt(5.0) * -1.5))) + (cos(x) * (6.0 / (sqrt(5.0) + 1.0)))));
}

Fortran

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

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((((-0.0625d0) * sin(y)) + sin(x)) * (sin(y) + ((-0.0625d0) * sin(x))))))) / (3.0d0 + ((cos(y) * (4.5d0 + (sqrt(5.0d0) * (-1.5d0)))) + (cos(x) * (6.0d0 / (sqrt(5.0d0) + 1.0d0)))))
end function

Java

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

↓

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (((-0.0625 * Math.sin(y)) + Math.sin(x)) * (Math.sin(y) + (-0.0625 * Math.sin(x))))))) / (3.0 + ((Math.cos(y) * (4.5 + (Math.sqrt(5.0) * -1.5))) + (Math.cos(x) * (6.0 / (Math.sqrt(5.0) + 1.0)))));
}

Python

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

↓

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (((-0.0625 * math.sin(y)) + math.sin(x)) * (math.sin(y) + (-0.0625 * math.sin(x))))))) / (3.0 + ((math.cos(y) * (4.5 + (math.sqrt(5.0) * -1.5))) + (math.cos(x) * (6.0 / (math.sqrt(5.0) + 1.0)))))

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(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(Float64(Float64(-0.0625 * sin(y)) + sin(x)) * Float64(sin(y) + Float64(-0.0625 * sin(x))))))) / Float64(3.0 + Float64(Float64(cos(y) * Float64(4.5 + Float64(sqrt(5.0) * -1.5))) + Float64(cos(x) * Float64(6.0 / Float64(sqrt(5.0) + 1.0))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
end

↓

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (((-0.0625 * sin(y)) + sin(x)) * (sin(y) + (-0.0625 * sin(x))))))) / (3.0 + ((cos(y) * (4.5 + (sqrt(5.0) * -1.5))) + (cos(x) * (6.0 / (sqrt(5.0) + 1.0)))));
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[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(4.5 + N[(N[Sqrt[5.0], $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(6.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $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 0.4

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{0.6666666666666666}, \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)\right)}} \]
   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))): 1 points increase in error, 0 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))): 0 points increase in error, 1 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, 1 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))): 1 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))): 1 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))): 1 points increase in error, 2 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))): 2 points increase in error, 4 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))): 2 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))))) (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)))): 3 points increase in error, 5 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))): 3 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 (*.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)))): 12 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)))) (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)))): 5 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)))) (+.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))): 13 points increase in error, 6 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))))))): 19 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 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 1) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)))))): 10 points increase in error, 12 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

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{{\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}^{-1}}, \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)\right)} \]

4. Taylor expanded in y around inf 0.4

   \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(-0.0625 \cdot \sin x + \sin y\right)\right)\right) + 2}{3 + \left(\cos y \cdot \left(4.5 - 1.5 \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}} \]

5. Applied egg-rr 0.5

   \[\leadsto \frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(-0.0625 \cdot \sin x + \sin y\right)\right)\right) + 2}{3 + \left(\cos y \cdot \left(4.5 - 1.5 \cdot \sqrt{5}\right) + 6 \cdot \color{blue}{\log \left(e^{\frac{\cos x}{\sqrt{5} + 1}}\right)}\right)} \]

6. Taylor expanded in x around inf 0.4

   \[\leadsto \frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(-0.0625 \cdot \sin x + \sin y\right)\right)\right) + 2}{3 + \left(\cos y \cdot \left(4.5 - 1.5 \cdot \sqrt{5}\right) + \color{blue}{6 \cdot \frac{\cos x}{\sqrt{5} + 1}}\right)} \]

7. Simplified 0.4

   \[\leadsto \frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(-0.0625 \cdot \sin x + \sin y\right)\right)\right) + 2}{3 + \left(\cos y \cdot \left(4.5 - 1.5 \cdot \sqrt{5}\right) + \color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}}\right)} \]
   Proof
   (*.f64 (cos.f64 x) (/.f64 6 (+.f64 1 (sqrt.f64 5)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 x) (/.f64 6 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 5) 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (cos.f64 x) 6) (+.f64 (sqrt.f64 5) 1))): 17 points increase in error, 16 points decrease in error
   (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) 1)) 6)): 27 points increase in error, 154 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 6 (/.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) 1)))): 0 points increase in error, 0 points decrease in error

8. Final simplification 0.4

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + \cos x \cdot \frac{6}{\sqrt{5} + 1}\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	72640

\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{6}{\frac{\sqrt{5} + 1}{\cos x}}\right)} \]

Alternative 2
Error	11.7
Cost	66632

\[\begin{array}{l} t_0 := \sqrt{5} + 1\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{t_0 \cdot 0.5}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;x \leq -0.0008919942167208584:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.767354888146388 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + \frac{6}{\frac{t_0}{\cos x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.7
Cost	66632

\[\begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_1 := \sqrt{5} + 1\\ \mathbf{if}\;x \leq -0.0008919942167208584:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 7.767354888146388 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + \frac{6}{\frac{t_1}{\cos x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(\left(1 + \frac{\cos x}{t_1 \cdot 0.5}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 4
Error	11.8
Cost	66504

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} + 1\\ t_2 := \frac{2 + t_0 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{t_1 \cdot 0.5}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;x \leq -6.90811036180353 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 7.767354888146388 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + \frac{6}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	12.7
Cost	60104

\[\begin{array}{l} t_0 := -0.0625 \cdot {\sin y}^{2}\\ t_1 := 3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + \frac{6}{\frac{\sqrt{5} + 1}{\cos x}}\right)\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;y \leq -1.4964304129107702:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\sqrt{2} \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.6690247608665462 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_2 \cdot \left(\sin x \cdot \left(-0.0625 \cdot \sin x + y \cdot 1.00390625\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_2 \cdot t_0\right)}{t_1}\\ \end{array} \]

Alternative 6
Error	12.8
Cost	59652

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + \frac{6}{\frac{\sqrt{5} + 1}{\cos x}}\right)\\ \mathbf{if}\;x \leq -0.0008919942167208584:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot t_0\right)\right)}{t_1}\\ \mathbf{elif}\;x \leq 7.767354888146388 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(\sin y \cdot \left(-0.0625 \cdot \sin y + x \cdot 1.00390625\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot \left(\cos x + -1\right)\right)\right)}{t_1}\\ \end{array} \]

Alternative 7
Error	12.8
Cost	53704

\[\begin{array}{l} t_0 := 3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + \frac{6}{\frac{\sqrt{5} + 1}{\cos x}}\right)\\ t_1 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{t_0}\\ \mathbf{if}\;x \leq -0.0008919942167208584:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.767354888146388 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(\sin y \cdot \left(-0.0625 \cdot \sin y + x \cdot 1.00390625\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	12.9
Cost	53384

\[\begin{array}{l} t_0 := \sqrt{5} + 1\\ t_1 := \cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right)\\ t_2 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(t_1 + \frac{6}{\frac{t_0}{\cos x}}\right)}\\ \mathbf{if}\;x \leq -6.90811036180353 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 7.767354888146388 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(t_1 + 6 \cdot \frac{1}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	12.9
Cost	53384

\[\begin{array}{l} t_0 := 3 + \left(\cos y \cdot \left(4.5 + \sqrt{5} \cdot -1.5\right) + \frac{6}{\frac{\sqrt{5} + 1}{\cos x}}\right)\\ t_1 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{t_0}\\ \mathbf{if}\;x \leq -0.0008919942167208584:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.767354888146388 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	13.4
Cost	46984

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} + 1\\ t_2 := \sqrt{5} \cdot -1.5\\ \mathbf{if}\;x \leq -6.90811036180353 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{\left(7.5 + \frac{\cos x \cdot 6}{t_1}\right) + t_2}\\ \mathbf{elif}\;x \leq 7.767354888146388 \cdot 10^{-13}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(\left(\sqrt{5} + -1\right) \cdot 0.5 + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(\frac{6}{\frac{t_1}{\cos x}} + 7.5\right) + t_2}\\ \end{array} \]

Alternative 11
Error	13.3
Cost	46984

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} + 1\\ t_2 := \sqrt{5} \cdot -1.5\\ \mathbf{if}\;x \leq -6.90811036180353 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{\left(7.5 + \frac{\cos x \cdot 6}{t_1}\right) + t_2}\\ \mathbf{elif}\;x \leq 7.767354888146388 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + \left(\cos y \cdot \left(4.5 + t_2\right) + 6 \cdot \frac{1}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(\frac{6}{\frac{t_1}{\cos x}} + 7.5\right) + t_2}\\ \end{array} \]

Alternative 12
Error	25.1
Cost	46464

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(7.5 + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right) + \sqrt{5} \cdot -1.5} \]

Alternative 13
Error	25.1
Cost	46464

\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\frac{6}{\frac{\sqrt{5} + 1}{\cos x}} + 7.5\right) + \sqrt{5} \cdot -1.5} \]

Alternative 14
Error	35.8
Cost	32832

\[\frac{0.6666666666666666}{1 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \frac{2}{3 + \sqrt{5}}\right)} \]

Alternative 15
Error	35.8
Cost	20416

\[\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

Reproduce

herbie shell --seed 2022300 
(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))))))