Average Error: 0.5 → 0.4
Time: 35.9s
Precision: binary64
Cost: 85312
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{6}{\frac{\sqrt{5} + 1}{\cos x}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
(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)
   (*
    (- (sin y) (/ (sin x) 16.0))
    (* (- (sin x) (/ (sin y) 16.0)) (- (cos x) (cos y))))
   2.0)
  (+
   (/ 6.0 (/ (+ (sqrt 5.0) 1.0) (cos x)))
   (fma (cos y) (/ (- 3.0 (sqrt 5.0)) 0.6666666666666666) 3.0))))
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), ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * (cos(x) - cos(y)))), 2.0) / ((6.0 / ((sqrt(5.0) + 1.0) / cos(x))) + fma(cos(y), ((3.0 - sqrt(5.0)) / 0.6666666666666666), 3.0));
}

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(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(Float64(6.0 / Float64(Float64(sqrt(5.0) + 1.0) / cos(x))) + fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 0.6666666666666666), 3.0)))
end

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[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

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. Simplified0.5

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
    Proof
    (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (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 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 0 points decrease in error
    (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (Rewrite<= *-commutative_binary64 (*.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 (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 0 points decrease in error
    (/.f64 (fma.f64 (sqrt.f64 2) (Rewrite<= associate-*r*_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) (+.f64 (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 4 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)) (+.f64 (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 30 points increase in error, 0 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) (+.f64 (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 0 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) (+.f64 (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 30 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 (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 30 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 (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) (Rewrite<= metadata-eval (neg.f64 1)))) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 27 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 (*.f64 (cos.f64 x) (Rewrite<= sub-neg_binary64 (-.f64 (sqrt.f64 5) 1))) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 27 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 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x))) 2/3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 30 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 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x)) (Rewrite<= metadata-eval (/.f64 2 3))) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 25 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<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x)) 3) 2)) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 25 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 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x)) 2) 3)) (fma.f64 (cos.f64 y) (/.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)))) (+.f64 (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))) 3) (fma.f64 (cos.f64 y) (/.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)))) (+.f64 (*.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) (Rewrite<= metadata-eval (/.f64 2 3))) 3))): 30 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 (*.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 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, 30 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 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) (fma.f64 (cos.f64 y) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3)) 3))): 30 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 (*.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) (fma.f64 (cos.f64 y) (Rewrite<= *-commutative_binary64 (*.f64 3 (/.f64 (-.f64 3 (sqrt.f64 5)) 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 (*.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 y) (*.f64 3 (/.f64 (-.f64 3 (sqrt.f64 5)) 2))) 3)))): 0 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 (*.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 3 (/.f64 (-.f64 3 (sqrt.f64 5)) 2)) (cos.f64 y))) 3))): 24 points increase in error, 3 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 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 3 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)))) 3))): 3 points increase in error, 10 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 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)) 3)) 3))): 0 points increase in error, 30 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 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) (Rewrite<= +-commutative_binary64 (+.f64 3 (*.f64 (*.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<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) 3) (*.f64 (*.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)))) (+.f64 (+.f64 (*.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3) (Rewrite<= metadata-eval (*.f64 1 3))) (*.f64 (*.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)))) (+.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 3 (+.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 1))) (*.f64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)) 3))): 30 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 (*.f64 3 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))))) (*.f64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)) 3))): 0 points increase in error, 24 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 3 (+.f64 1 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)))) (Rewrite=> *-commutative_binary64 (*.f64 3 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)))))): 0 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)))) (Rewrite<= distribute-lft-in_binary64 (*.f64 3 (+.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)))))): 0 points increase in error, 19 points decrease in error

  3. Applied egg-rr0.4

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\left(\sqrt{5} + -1\right) \cdot \left(\cos x \cdot 1.5\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

  4. Applied egg-rr0.4

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\left(\cos x \cdot 1.5\right) \cdot 4}{\sqrt{5} - -1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

  5. Simplified0.4

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{6}{\frac{\sqrt{5} + 1}{\cos x}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
    Proof
    (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)) (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 (/.f64 6 (/.f64 (+.f64 (sqrt.f64 5) 1) (cos.f64 x))) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (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 (sin.f64 y) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 (/.f64 6 (/.f64 (+.f64 (sqrt.f64 5) (Rewrite<= metadata-eval (neg.f64 -1))) (cos.f64 x))) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 7 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 (sin.f64 y) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 (/.f64 6 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (sqrt.f64 5) -1)) (cos.f64 x))) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 3 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 (sin.f64 y) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 6 (cos.f64 x)) (-.f64 (sqrt.f64 5) -1))) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (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 (sin.f64 y) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) 6)) (-.f64 (sqrt.f64 5) -1)) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 7 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 (sin.f64 y) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 (/.f64 (*.f64 (cos.f64 x) (Rewrite<= metadata-eval (*.f64 3/2 4))) (-.f64 (sqrt.f64 5) -1)) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 7 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 (sin.f64 y) 16)) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 x) 3/2) 4)) (-.f64 (sqrt.f64 5) -1)) (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) 3))): 0 points increase in error, 0 points decrease in error

  6. Final simplification0.4

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

Alternatives

Alternative 1
Error0.4
Cost79040
\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{3 + \mathsf{fma}\left(\sqrt{5} \cdot 1.5 + -1.5, \cos x, 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)} \]
Alternative 2
Error0.4
Cost79040
\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{3 + \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 6, \left(\cos x \cdot 1.5\right) \cdot \left(\sqrt{5} + -1\right)\right)} \]
Alternative 3
Error0.4
Cost72768
\[\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \]
Alternative 4
Error11.9
Cost67016
\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} + -1\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.105:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.07:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{\sqrt{5} + 3} + 1.5 \cdot \left(\cos x \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_3 \cdot t_1\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \end{array} \]
Alternative 5
Error11.9
Cost66888
\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} + -1\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0125:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0132:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_1 \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)}{3 + \left(t_2 \cdot \left(1.5 + \left(x \cdot x\right) \cdot -0.75\right) + \cos y \cdot \frac{6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_3 \cdot t_1\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \end{array} \]
Alternative 6
Error11.9
Cost66632
\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := t_2 \cdot t_1\\ t_4 := 3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)\\ \mathbf{if}\;x \leq -0.0215:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_2 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.043:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot t_3}{t_4}\\ \end{array} \]
Alternative 7
Error12.0
Cost66632
\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} + -1\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.006:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00315:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{\sqrt{5} + 3} + 1.5 \cdot \left(\cos x \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_3 \cdot t_1\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)}\\ \end{array} \]
Alternative 8
Error12.1
Cost66377
\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\\ \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 1.28 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_0 \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_1 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]
Alternative 9
Error12.1
Cost66376
\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_3 \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(t_1 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_3 \cdot t_2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right) + t_1\right)\right)}\\ \end{array} \]
Alternative 10
Error13.1
Cost60105
\[\begin{array}{l} t_0 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 0.0013\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{\sqrt{5} + 3} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot t_0\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]
Alternative 11
Error13.2
Cost60105
\[\begin{array}{l} \mathbf{if}\;x \leq -0.009 \lor \neg \left(x \leq 106\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{\sqrt{5} + 3} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(-0.5 + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]
Alternative 12
Error13.3
Cost59785
\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sqrt{5} + 3\\ \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 106\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t_1} + 1.5 \cdot \left(\cos x \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, t_0, 3 + \frac{\cos y \cdot 6}{t_1}\right)}\\ \end{array} \]
Alternative 13
Error13.7
Cost59400
\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\\ t_1 := \sqrt{5} + 3\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;\frac{t_0}{\left(3 + \frac{6}{t_1}\right) + 1.5 \cdot \left(\cos x \cdot t_2\right)}\\ \mathbf{elif}\;x \leq 106:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, t_2, 3 + \frac{\cos y \cdot 6}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \frac{\cos x \cdot 4}{\sqrt{5} + 1}\right)}\\ \end{array} \]
Alternative 14
Error13.8
Cost46857
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 106\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}\\ \end{array} \]
Alternative 15
Error13.8
Cost46857
\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 106\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)}{3 + 1.5 \cdot \left(t_1 + \cos x \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_0 + \left(3 + 1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \end{array} \]
Alternative 16
Error13.8
Cost46857
\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 106\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_0 + \left(3 + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}\\ \end{array} \]
Alternative 17
Error13.7
Cost46856
\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x + -1\\ t_3 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_2 \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)}{3 + 1.5 \cdot \left(t_1 + \cos x \cdot t_3\right)}\\ \mathbf{elif}\;x \leq 106:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_3 + \left(3 + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left({\sin x}^{2} \cdot t_2\right)}{3 + 1.5 \cdot \left(t_1 + \frac{4}{\frac{\sqrt{5} + 1}{\cos x}}\right)}\\ \end{array} \]
Alternative 18
Error13.7
Cost46856
\[\begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x + -1\\ t_3 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_2 \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)}{3 + 1.5 \cdot \left(t_1 + \cos x \cdot t_3\right)}\\ \mathbf{elif}\;x \leq 106:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_3 + \left(3 + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left({\sin x}^{2} \cdot t_2\right)}{3 + 1.5 \cdot \left(t_1 + \frac{\cos x \cdot 4}{\sqrt{5} + 1}\right)}\\ \end{array} \]
Alternative 19
Error13.7
Cost46856
\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\\ t_1 := \sqrt{5} + 3\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;\frac{t_0}{\left(3 + \frac{6}{t_1}\right) + 1.5 \cdot \left(\cos x \cdot t_2\right)}\\ \mathbf{elif}\;x \leq 106:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_2 + \left(3 + 6 \cdot \frac{\cos y}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \frac{\cos x \cdot 4}{\sqrt{5} + 1}\right)}\\ \end{array} \]
Alternative 20
Error25.4
Cost40384
\[\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]
Alternative 21
Error37.8
Cost39296
\[\frac{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{\mathsf{fma}\left(1.5, 2, 3\right)} \]
Alternative 22
Error43.3
Cost33728
\[\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]
Alternative 23
Error43.4
Cost27200
\[\frac{2 + -0.0625 \cdot \left(\left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)\right)} \]
Alternative 24
Error43.9
Cost20928
\[\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)\right)} \]

Error

Reproduce

herbie shell --seed 2022343 
(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))))))