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

Average Error: 0.5 → 0.4

Time: 48.4s

Precision: binary64

Cost: 85440

Math

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

↓

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{\sqrt{5} + 1}\right)} \]

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
 (/
  (fma
   (sqrt 2.0)
   (*
    (- (cos x) (cos y))
    (* (- (sin x) (/ (sin y) 16.0)) (- (sin y) (/ (sin x) 16.0))))
   2.0)
  (+
   3.0
   (fma
    (cos y)
    (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 0.6666666666666666)
    (* (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 fma(sqrt(2.0), ((cos(x) - cos(y)) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))), 2.0) / (3.0 + fma(cos(y), ((4.0 / (3.0 + sqrt(5.0))) / 0.6666666666666666), (cos(x) * (6.0 / (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(fma(sqrt(2.0), Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))), 2.0) / Float64(3.0 + fma(cos(y), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 0.6666666666666666), Float64(cos(x) * Float64(6.0 / Float64(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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $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 0.5

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]
   Proof
   (/.f64 (fma.f64 (sqrt.f64 2) (*.f64 (-.f64 (cos.f64 x) (cos.f64 y)) (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)))) 2) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 (sqrt.f64 2) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2)) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 1 points increase in error, 4 points decrease in error
   (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16)) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)))) (-.f64 (cos.f64 x) (cos.f64 y)))) 2) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 5 points increase in error, 4 points decrease in error
   (/.f64 (+.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16)))) (-.f64 (cos.f64 x) (cos.f64 y))) 2) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 3 points increase in error, 5 points decrease in error
   (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y))))) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2/3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) (Rewrite<= metadata-eval (/.f64 2 3))) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 3 (sqrt.f64 5)) 3) 2)) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3)) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) -1)) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (/.f64 (*.f64 (cos.f64 x) (+.f64 (sqrt.f64 5) (Rewrite<= metadata-eval (neg.f64 1)))) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (/.f64 (*.f64 (cos.f64 x) (Rewrite<= sub-neg_binary64 (-.f64 (sqrt.f64 5) 1))) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x))) 2/3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (/.f64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x)) (Rewrite<= metadata-eval (/.f64 2 3)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x)) 3) 2))))): 4 points increase in error, 17 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (-.f64 (sqrt.f64 5) 1) (cos.f64 x)) 2) 3))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (fma.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3) (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 y) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) 3)) (*.f64 3 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))))))): 9 points increase in error, 9 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 y) (/.f64 (-.f64 3 (sqrt.f64 5)) 2)) 3)) (*.f64 3 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)))))): 3 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 3 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))) 3) (*.f64 3 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (+.f64 (*.f64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)) 3) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) 3))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 3 (Rewrite<= distribute-rgt-in_binary64 (*.f64 3 (+.f64 (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)) (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))))))): 10 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 3 (*.f64 3 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (+.f64 (Rewrite<= metadata-eval (*.f64 3 1)) (*.f64 3 (+.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (Rewrite<= distribute-lft-in_binary64 (*.f64 3 (+.f64 1 (+.f64 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x)) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y))))))): 30 points increase in error, 15 points decrease in error
   (/.f64 (+.f64 2 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) 16))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) 16))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 3 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 (/.f64 (-.f64 (sqrt.f64 5) 1) 2) (cos.f64 x))) (*.f64 (/.f64 (-.f64 3 (sqrt.f64 5)) 2) (cos.f64 y)))))): 9 points increase in error, 9 points decrease in error

3. Applied egg-rr 0.4

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

4. Simplified 0.4

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

5. Applied egg-rr 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]

6. Simplified 0.4

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

7. Final simplification 0.4

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{\sqrt{5} + 1}\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	85440

\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{\sqrt{5} + 1}\right)} \]

Alternative 2
Error	0.4
Cost	79040

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

Alternative 3
Error	0.4
Cost	79040

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

Alternative 4
Error	0.4
Cost	78912

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

Alternative 5
Error	0.4
Cost	78912

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

Alternative 6
Error	0.4
Cost	73024

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

Alternative 7
Error	0.4
Cost	72896

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

Alternative 8
Error	0.5
Cost	72768

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

Alternative 9
Error	12.4
Cost	66504

\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \frac{2 + \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 + -0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{if}\;x \leq -0.00055:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	13.5
Cost	66056

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x + -1\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{\sqrt{5} + 1}\right)}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{\left(\sqrt{5} + -1\right) \cdot 1.5 + \left(3 + \cos y \cdot \left(4.5 - \sqrt{5} \cdot 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	13.6
Cost	66056

\[\begin{array}{l} t_0 := 3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{\sqrt{5} + 1}\right)\\ t_1 := \frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x + -1\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right), 2\right)}{t_0}\\ \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1680000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	13.5
Cost	53512

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

Alternative 13
Error	13.9
Cost	52744

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\mathsf{fma}\left(\cos x, -1.5 + \sqrt{11.25}, 7.5 - \sqrt{11.25}\right)}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{\left(\sqrt{5} + -1\right) \cdot 1.5 + \left(3 + \cos y \cdot \left(4.5 - \sqrt{5} \cdot 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	13.9
Cost	52744

\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_2 := 7.5 - \sqrt{11.25}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\cos x \cdot t_0, 1.5, t_2\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t_0 \cdot 1.5 + \left(3 + \cos y \cdot \left(4.5 - \sqrt{5} \cdot 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\cos x, -1.5 + \sqrt{11.25}, t_2\right)}\\ \end{array} \]

Alternative 15
Error	13.9
Cost	46856

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{\left(7.5 + 1.5 \cdot \left(\cos x \cdot \frac{4}{\sqrt{5} + 1}\right)\right) + \sqrt{5} \cdot -1.5}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{\left(\sqrt{5} + -1\right) \cdot 1.5 + \left(3 + \cos y \cdot \left(4.5 - \sqrt{5} \cdot 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(7.5 - \sqrt{11.25}\right) + \cos x \cdot \left(-1.5 + \sqrt{11.25}\right)}\\ \end{array} \]

Alternative 16
Error	24.4
Cost	46724

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ \mathbf{if}\;x \leq -0.00082:\\ \;\;\;\;\frac{t_0}{\left(7.5 + 1.5 \cdot \left(\cos x \cdot \frac{4}{\sqrt{5} + 1}\right)\right) + \sqrt{5} \cdot -1.5}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(x \cdot x\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(7.5 - \sqrt{11.25}\right) + \cos x \cdot \left(-1.5 + \sqrt{11.25}\right)}\\ \end{array} \]

Alternative 17
Error	24.4
Cost	46472

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{7.5 + \left(\cos x \cdot \left(-1.5 + \sqrt{11.25}\right) - \sqrt{11.25}\right)}\\ \mathbf{if}\;x \leq -0.0006:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(x \cdot x\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	24.4
Cost	46472

\[\begin{array}{l} t_0 := \cos x \cdot \left(-1.5 + \sqrt{11.25}\right)\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ \mathbf{if}\;x \leq -0.00048:\\ \;\;\;\;\frac{t_1}{7.5 + \left(t_0 - \sqrt{11.25}\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(x \cdot x\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\left(7.5 - \sqrt{11.25}\right) + t_0}\\ \end{array} \]

Alternative 19
Error	36.9
Cost	45636

\[\begin{array}{l} t_0 := \cos x + -1\\ \mathbf{if}\;x \leq -9.5:\\ \;\;\;\;0.3333333333333333 + \log \left({\left(e^{-0.010416666666666666}\right)}^{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t_0\right)\right)}\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(x \cdot x\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right)\right)\\ \end{array} \]

Alternative 20
Error	36.9
Cost	40904

\[\begin{array}{l} t_0 := 0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(x \cdot x\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 21
Error	38.0
Cost	20288

\[0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right)\right) \]

Alternative 22
Error	38.0
Cost	64

\[0.3333333333333333 \]

Reproduce

herbie shell --seed 2022318 
(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))))))