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

Percentage Accurate: 99.3% → 99.3%

Time: 23.8s

Alternatives: 35

Speedup: 1.0×

Specification

Math

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

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))))))

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))));
}

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

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))));
}

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))))

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

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

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]

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

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))))))

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))));
}

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

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))));
}

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))))

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

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

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]

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- (cos x) (cos y))
   (*
    (fma (sin x) -0.0625 (sin y))
    (* (sqrt 2.0) (fma (sin y) -0.0625 (sin x))))
   2.0)
  (fma
   (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
   3.0
   (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0))))))

C

double code(double x, double y) {
	return fma((cos(x) - cos(y)), (fma(sin(x), -0.0625, sin(y)) * (sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(cos(x) - cos(y)), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. lift-fma.f64 N/A

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. flip-- N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. associate-*r/ N/A

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

8. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]
12. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (sqrt 2.0) (- (cos x) (cos y)))
   (* (fma -0.0625 (sin x) (sin y)) (fma -0.0625 (sin y) (sin x)))
   2.0)
  (fma
   (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
   3.0
   (* (* (- 3.0 (sqrt 5.0)) 0.5) (* 3.0 (cos y))))))

C

double code(double x, double y) {
	return fma((sqrt(2.0) * (cos(x) - cos(y))), (fma(-0.0625, sin(x), sin(y)) * fma(-0.0625, sin(y), sin(x))), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, (((3.0 - sqrt(5.0)) * 0.5) * (3.0 * cos(y))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(fma(-0.0625, sin(x), sin(y)) * fma(-0.0625, sin(y), sin(x))), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * Float64(3.0 * cos(y)))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

7. Final simplification 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)} \]
8. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x)))
    (- (cos x) (cos y)))
   2.0)
  (fma
   (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)
   3.0
   (* (* 1.5 (cos y)) (- 3.0 (sqrt 5.0))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * (cos(x) - cos(y))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, ((1.5 * cos(y)) * (3.0 - sqrt(5.0))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * Float64(cos(x) - cos(y))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * Float64(3.0 - sqrt(5.0)))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r/ N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
7. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * (cos(x) - cos(y))), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * Float64(cos(x) - cos(y))), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
9. Add Preprocessing

Alternative 5: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := 3 + \sqrt{5}\\ t_3 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ t_4 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot t\_0\right) \cdot t\_1 + 2}{\mathsf{fma}\left(t\_3, 3, \frac{4 \cdot \left(1.5 \cdot \cos y\right)}{t\_2}\right)}\\ \mathbf{elif}\;x \leq 0.38:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right)\right) \cdot t\_4 + 2}{\mathsf{fma}\left(t\_3, 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 \cdot t\_1\right) \cdot t\_4 + 2}{\mathsf{fma}\left(t\_3, 3, \frac{6 \cdot \cos y}{t\_2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sin x)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (+ 3.0 (sqrt 5.0)))
        (t_3 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))
        (t_4 (fma -0.0625 (sin x) (sin y))))
   (if (<= x -3.25)
     (/
      (+ (* (* (- (sin y) (/ (sin x) 16.0)) t_0) t_1) 2.0)
      (fma t_3 3.0 (/ (* 4.0 (* 1.5 (cos y))) t_2)))
     (if (<= x 0.38)
       (/
        (+
         (*
          (*
           (fma
            (*
             (fma
              (fma -0.001388888888888889 (* x x) 0.041666666666666664)
              (* x x)
              -0.5)
             x)
            x
            (- 1.0 (cos y)))
           (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
          t_4)
         2.0)
        (fma t_3 3.0 (* (* (- 3.0 (sqrt 5.0)) 0.5) (* 3.0 (cos y)))))
       (/
        (+ (* (* t_0 t_1) t_4) 2.0)
        (fma t_3 3.0 (/ (* 6.0 (cos y)) t_2)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * sin(x);
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 + sqrt(5.0);
	double t_3 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double t_4 = fma(-0.0625, sin(x), sin(y));
	double tmp;
	if (x <= -3.25) {
		tmp = ((((sin(y) - (sin(x) / 16.0)) * t_0) * t_1) + 2.0) / fma(t_3, 3.0, ((4.0 * (1.5 * cos(y))) / t_2));
	} else if (x <= 0.38) {
		tmp = (((fma((fma(fma(-0.001388888888888889, (x * x), 0.041666666666666664), (x * x), -0.5) * x), x, (1.0 - cos(y))) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))) * t_4) + 2.0) / fma(t_3, 3.0, (((3.0 - sqrt(5.0)) * 0.5) * (3.0 * cos(y))));
	} else {
		tmp = (((t_0 * t_1) * t_4) + 2.0) / fma(t_3, 3.0, ((6.0 * cos(y)) / t_2));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * sin(x))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 + sqrt(5.0))
	t_3 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	t_4 = fma(-0.0625, sin(x), sin(y))
	tmp = 0.0
	if (x <= -3.25)
		tmp = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * t_0) * t_1) + 2.0) / fma(t_3, 3.0, Float64(Float64(4.0 * Float64(1.5 * cos(y))) / t_2)));
	elseif (x <= 0.38)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(fma(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), -0.5) * x), x, Float64(1.0 - cos(y))) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))) * t_4) + 2.0) / fma(t_3, 3.0, Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * Float64(3.0 * cos(y)))));
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * t_1) * t_4) + 2.0) / fma(t_3, 3.0, Float64(Float64(6.0 * cos(y)) / t_2)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.25], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$3 * 3.0 + N[(N[(4.0 * N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.38], N[(N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$3 * 3.0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$4), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$3 * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.25

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}}\right)} \]

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 61.0

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}\right)} \]

   9. Applied rewrites 61.0%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}\right)} \]

   if -3.25 < x < 0.38

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2}} + \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x\right) \cdot x} + \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x, x, 1 - \cos y\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   9. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if 0.38 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. flip-- N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*r/ N/A

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

   8. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sin.f64 65.3

          \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{4 \cdot \left(1.5 \cdot \cos y\right)}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.38:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := 3 + \sqrt{5}\\ t_3 := \sqrt{5} - 1\\ t_4 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ \mathbf{if}\;x \leq -0.075:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot t\_0\right) \cdot t\_1 + 2}{\mathsf{fma}\left(t\_4, 3, \frac{4 \cdot \left(1.5 \cdot \cos y\right)}{t\_2}\right)}\\ \mathbf{elif}\;x \leq 0.102:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot t\_1, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x \cdot x, -0.75\right) \cdot t\_3, x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_3\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 \cdot t\_1\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(t\_4, 3, \frac{6 \cdot \cos y}{t\_2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sin x)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (+ 3.0 (sqrt 5.0)))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)))
   (if (<= x -0.075)
     (/
      (+ (* (* (- (sin y) (/ (sin x) 16.0)) t_0) t_1) 2.0)
      (fma t_4 3.0 (/ (* 4.0 (* 1.5 (cos y))) t_2)))
     (if (<= x 0.102)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x)))
          t_1)
         2.0)
        (fma
         (* (fma 0.0625 (* x x) -0.75) t_3)
         (* x x)
         (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_3) 3.0)))
       (/
        (+ (* (* t_0 t_1) (fma -0.0625 (sin x) (sin y))) 2.0)
        (fma t_4 3.0 (/ (* 6.0 (cos y)) t_2)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * sin(x);
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 + sqrt(5.0);
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double tmp;
	if (x <= -0.075) {
		tmp = ((((sin(y) - (sin(x) / 16.0)) * t_0) * t_1) + 2.0) / fma(t_4, 3.0, ((4.0 * (1.5 * cos(y))) / t_2));
	} else if (x <= 0.102) {
		tmp = fma(sqrt(2.0), ((fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * t_1), 2.0) / fma((fma(0.0625, (x * x), -0.75) * t_3), (x * x), fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), t_3), 3.0));
	} else {
		tmp = (((t_0 * t_1) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(t_4, 3.0, ((6.0 * cos(y)) / t_2));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * sin(x))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 + sqrt(5.0))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	tmp = 0.0
	if (x <= -0.075)
		tmp = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * t_0) * t_1) + 2.0) / fma(t_4, 3.0, Float64(Float64(4.0 * Float64(1.5 * cos(y))) / t_2)));
	elseif (x <= 0.102)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * t_1), 2.0) / fma(Float64(fma(0.0625, Float64(x * x), -0.75) * t_3), Float64(x * x), fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_3), 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * t_1) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(t_4, 3.0, Float64(Float64(6.0 * cos(y)) / t_2)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.075], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$4 * 3.0 + N[(N[(4.0 * N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.102], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.0625 * N[(x * x), $MachinePrecision] + -0.75), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$4 * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0749999999999999972

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}}\right)} \]

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 60.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}\right)} \]

   9. Applied rewrites 60.5%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}\right)} \]

   if -0.0749999999999999972 < x < 0.101999999999999993

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)\right)}} \]

   if 0.101999999999999993 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. flip-- N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*r/ N/A

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

   8. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sin.f64 65.3

          \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.075:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{4 \cdot \left(1.5 \cdot \cos y\right)}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.102:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x \cdot x, -0.75\right) \cdot \left(\sqrt{5} - 1\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_2 := \cos x - \cos y\\ t_3 := 3 + \sqrt{5}\\ t_4 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot t\_0\right) \cdot t\_2 + 2}{\mathsf{fma}\left(t\_4, 3, \frac{4 \cdot \left(1.5 \cdot \cos y\right)}{t\_3}\right)}\\ \mathbf{elif}\;x \leq 0.045:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right)\right) \cdot t\_1 + 2}{\mathsf{fma}\left(t\_4, 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 \cdot t\_2\right) \cdot t\_1 + 2}{\mathsf{fma}\left(t\_4, 3, \frac{6 \cdot \cos y}{t\_3}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sin x)))
        (t_1 (fma -0.0625 (sin x) (sin y)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (+ 3.0 (sqrt 5.0)))
        (t_4 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)))
   (if (<= x -3.25)
     (/
      (+ (* (* (- (sin y) (/ (sin x) 16.0)) t_0) t_2) 2.0)
      (fma t_4 3.0 (/ (* 4.0 (* 1.5 (cos y))) t_3)))
     (if (<= x 0.045)
       (/
        (+
         (*
          (*
           (fma (* x x) -0.5 (- 1.0 (cos y)))
           (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
          t_1)
         2.0)
        (fma t_4 3.0 (* (* (- 3.0 (sqrt 5.0)) 0.5) (* 3.0 (cos y)))))
       (/
        (+ (* (* t_0 t_2) t_1) 2.0)
        (fma t_4 3.0 (/ (* 6.0 (cos y)) t_3)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * sin(x);
	double t_1 = fma(-0.0625, sin(x), sin(y));
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 + sqrt(5.0);
	double t_4 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double tmp;
	if (x <= -3.25) {
		tmp = ((((sin(y) - (sin(x) / 16.0)) * t_0) * t_2) + 2.0) / fma(t_4, 3.0, ((4.0 * (1.5 * cos(y))) / t_3));
	} else if (x <= 0.045) {
		tmp = (((fma((x * x), -0.5, (1.0 - cos(y))) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))) * t_1) + 2.0) / fma(t_4, 3.0, (((3.0 - sqrt(5.0)) * 0.5) * (3.0 * cos(y))));
	} else {
		tmp = (((t_0 * t_2) * t_1) + 2.0) / fma(t_4, 3.0, ((6.0 * cos(y)) / t_3));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * sin(x))
	t_1 = fma(-0.0625, sin(x), sin(y))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 + sqrt(5.0))
	t_4 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	tmp = 0.0
	if (x <= -3.25)
		tmp = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * t_0) * t_2) + 2.0) / fma(t_4, 3.0, Float64(Float64(4.0 * Float64(1.5 * cos(y))) / t_3)));
	elseif (x <= 0.045)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(x * x), -0.5, Float64(1.0 - cos(y))) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))) * t_1) + 2.0) / fma(t_4, 3.0, Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * Float64(3.0 * cos(y)))));
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * t_2) * t_1) + 2.0) / fma(t_4, 3.0, Float64(Float64(6.0 * cos(y)) / t_3)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -3.25], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$4 * 3.0 + N[(N[(4.0 * N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.045], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$4 * 3.0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$4 * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.25

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}}\right)} \]

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 61.0

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}\right)} \]

   9. Applied rewrites 61.0%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}\right)} \]

   if -3.25 < x < 0.044999999999999998

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 99.0

         \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \color{blue}{\cos y}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if 0.044999999999999998 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. flip-- N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*r/ N/A

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

   8. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sin.f64 65.3

          \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{4 \cdot \left(1.5 \cdot \cos y\right)}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.045:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := 3 + \sqrt{5}\\ t_3 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot t\_0\right) \cdot t\_1 + 2}{\mathsf{fma}\left(t\_3, 3, \frac{4 \cdot \left(1.5 \cdot \cos y\right)}{t\_2}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot t\_1, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 0.5\right), \sqrt{5} - 1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 \cdot t\_1\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(t\_3, 3, \frac{6 \cdot \cos y}{t\_2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sin x)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (+ 3.0 (sqrt 5.0)))
        (t_3 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)))
   (if (<= x -0.0102)
     (/
      (+ (* (* (- (sin y) (/ (sin x) 16.0)) t_0) t_1) 2.0)
      (fma t_3 3.0 (/ (* 4.0 (* 1.5 (cos y))) t_2)))
     (if (<= x 0.018)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x)))
          t_1)
         2.0)
        (*
         (fma
          (fma -0.5 (sqrt 5.0) 1.5)
          (cos y)
          (fma (fma (* x x) -0.25 0.5) (- (sqrt 5.0) 1.0) 1.0))
         3.0))
       (/
        (+ (* (* t_0 t_1) (fma -0.0625 (sin x) (sin y))) 2.0)
        (fma t_3 3.0 (/ (* 6.0 (cos y)) t_2)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * sin(x);
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 + sqrt(5.0);
	double t_3 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double tmp;
	if (x <= -0.0102) {
		tmp = ((((sin(y) - (sin(x) / 16.0)) * t_0) * t_1) + 2.0) / fma(t_3, 3.0, ((4.0 * (1.5 * cos(y))) / t_2));
	} else if (x <= 0.018) {
		tmp = fma(sqrt(2.0), ((fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * t_1), 2.0) / (fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), fma(fma((x * x), -0.25, 0.5), (sqrt(5.0) - 1.0), 1.0)) * 3.0);
	} else {
		tmp = (((t_0 * t_1) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(t_3, 3.0, ((6.0 * cos(y)) / t_2));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * sin(x))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 + sqrt(5.0))
	t_3 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	tmp = 0.0
	if (x <= -0.0102)
		tmp = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * t_0) * t_1) + 2.0) / fma(t_3, 3.0, Float64(Float64(4.0 * Float64(1.5 * cos(y))) / t_2)));
	elseif (x <= 0.018)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * t_1), 2.0) / Float64(fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), fma(fma(Float64(x * x), -0.25, 0.5), Float64(sqrt(5.0) - 1.0), 1.0)) * 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * t_1) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(t_3, 3.0, Float64(Float64(6.0 * cos(y)) / t_2)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.0102], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$3 * 3.0 + N[(N[(4.0 * N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.018], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$3 * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.010200000000000001

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}}\right)} \]

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 60.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}\right)} \]

   9. Applied rewrites 60.5%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{\left(1.5 \cdot \cos y\right) \cdot 4}{\sqrt{5} + 3}\right)} \]

   if -0.010200000000000001 < x < 0.0179999999999999986

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 0.5\right), \sqrt{5} - 1, 1\right)\right)}} \]

   if 0.0179999999999999986 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. flip-- N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*r/ N/A

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

   8. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sin.f64 65.3

          \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{4 \cdot \left(1.5 \cdot \cos y\right)}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 0.5\right), \sqrt{5} - 1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_0\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 0.5\right), \sqrt{5} - 1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (/
          (+
           (* (* (* (sqrt 2.0) (sin x)) t_0) (fma -0.0625 (sin x) (sin y)))
           2.0)
          (fma
           (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
           3.0
           (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))))
   (if (<= x -0.0102)
     t_1
     (if (<= x 0.018)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x)))
          t_0)
         2.0)
        (*
         (fma
          (fma -0.5 (sqrt 5.0) 1.5)
          (cos y)
          (fma (fma (* x x) -0.25 0.5) (- (sqrt 5.0) 1.0) 1.0))
         3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = ((((sqrt(2.0) * sin(x)) * t_0) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	double tmp;
	if (x <= -0.0102) {
		tmp = t_1;
	} else if (x <= 0.018) {
		tmp = fma(sqrt(2.0), ((fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * t_0), 2.0) / (fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), fma(fma((x * x), -0.25, 0.5), (sqrt(5.0) - 1.0), 1.0)) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_0) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))))
	tmp = 0.0
	if (x <= -0.0102)
		tmp = t_1;
	elseif (x <= 0.018)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))) * t_0), 2.0) / Float64(fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), fma(fma(Float64(x * x), -0.25, 0.5), Float64(sqrt(5.0) - 1.0), 1.0)) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0102], t$95$1, If[LessEqual[x, 0.018], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.010200000000000001 or 0.0179999999999999986 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. flip-- N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*r/ N/A

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

   8. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sin.f64 62.7

          \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 62.7%

       \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   if -0.010200000000000001 < x < 0.0179999999999999986

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 0.5\right), \sqrt{5} - 1, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 0.5\right), \sqrt{5} - 1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           (*
            (* (* (sqrt 2.0) (sin x)) (- (cos x) (cos y)))
            (fma -0.0625 (sin x) (sin y)))
           2.0)
          (fma
           (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
           3.0
           (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))))
   (if (<= x -0.0102)
     t_0
     (if (<= x 0.018)
       (/
        (+
         (*
          (fma (* (fma 0.041666666666666664 (* x x) -0.5) x) x (- 1.0 (cos y)))
          (*
           (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0))))
         2.0)
        (*
         (+
          (fma
           (- (sqrt 5.0) 1.0)
           (fma (* -0.25 x) x 0.5)
           (* (fma -0.5 (sqrt 5.0) 1.5) (cos y)))
          1.0)
         3.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = ((((sqrt(2.0) * sin(x)) * (cos(x) - cos(y))) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	double tmp;
	if (x <= -0.0102) {
		tmp = t_0;
	} else if (x <= 0.018) {
		tmp = ((fma((fma(0.041666666666666664, (x * x), -0.5) * x), x, (1.0 - cos(y))) * (((sin(x) - (sin(y) / 16.0)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0)))) + 2.0) / ((fma((sqrt(5.0) - 1.0), fma((-0.25 * x), x, 0.5), (fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(cos(x) - cos(y))) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))))
	tmp = 0.0
	if (x <= -0.0102)
		tmp = t_0;
	elseif (x <= 0.018)
		tmp = Float64(Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x * x), -0.5) * x), x, Float64(1.0 - cos(y))) * Float64(Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))) + 2.0) / Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), fma(Float64(-0.25 * x), x, 0.5), Float64(fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0102], t$95$0, If[LessEqual[x, 0.018], N[(N[(N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.25 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.010200000000000001 or 0.0179999999999999986 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. flip-- N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*r/ N/A

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

   8. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sin.f64 62.7

          \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 62.7%

       \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   if -0.010200000000000001 < x < 0.0179999999999999986

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot x} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x, x, 1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x}, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}\right) \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right)} \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right) \cdot x, x, \color{blue}{1 - \cos y}\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      14. lower-cos.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \color{blue}{\cos y}\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   8. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           (*
            (* (* (sqrt 2.0) (sin x)) (- (cos x) (cos y)))
            (fma -0.0625 (sin x) (sin y)))
           2.0)
          (fma
           (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
           3.0
           (* (* (- 3.0 (sqrt 5.0)) 0.5) (* 3.0 (cos y)))))))
   (if (<= x -0.0102)
     t_0
     (if (<= x 0.018)
       (/
        (+
         (*
          (fma (* (fma 0.041666666666666664 (* x x) -0.5) x) x (- 1.0 (cos y)))
          (*
           (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0))))
         2.0)
        (*
         (+
          (fma
           (- (sqrt 5.0) 1.0)
           (fma (* -0.25 x) x 0.5)
           (* (fma -0.5 (sqrt 5.0) 1.5) (cos y)))
          1.0)
         3.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = ((((sqrt(2.0) * sin(x)) * (cos(x) - cos(y))) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, (((3.0 - sqrt(5.0)) * 0.5) * (3.0 * cos(y))));
	double tmp;
	if (x <= -0.0102) {
		tmp = t_0;
	} else if (x <= 0.018) {
		tmp = ((fma((fma(0.041666666666666664, (x * x), -0.5) * x), x, (1.0 - cos(y))) * (((sin(x) - (sin(y) / 16.0)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0)))) + 2.0) / ((fma((sqrt(5.0) - 1.0), fma((-0.25 * x), x, 0.5), (fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(cos(x) - cos(y))) * fma(-0.0625, sin(x), sin(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * Float64(3.0 * cos(y)))))
	tmp = 0.0
	if (x <= -0.0102)
		tmp = t_0;
	elseif (x <= 0.018)
		tmp = Float64(Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x * x), -0.5) * x), x, Float64(1.0 - cos(y))) * Float64(Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))) + 2.0) / Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), fma(Float64(-0.25 * x), x, 0.5), Float64(fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0102], t$95$0, If[LessEqual[x, 0.018], N[(N[(N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.25 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.010200000000000001 or 0.0179999999999999986 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.6

         \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -0.010200000000000001 < x < 0.0179999999999999986

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot x} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x, x, 1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x}, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}\right) \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right)} \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right) \cdot x, x, \color{blue}{1 - \cos y}\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      14. lower-cos.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \color{blue}{\cos y}\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   8. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2} \cdot -0.0625\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left(t\_0 \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(t\_1 \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - 1\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{t\_2}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (pow (sin y) 2.0) -0.0625))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -2.8e+33)
     (/
      (+ (* (* t_0 (sqrt 2.0)) (- (cos x) (cos y))) 2.0)
      (fma
       (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
       3.0
       (* (* t_1 0.5) (* 3.0 (cos y)))))
     (if (<= y 5.8e-5)
       (/
        (fma
         (sqrt 2.0)
         (*
          (- (cos x) 1.0)
          (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x))))
         2.0)
        (fma 1.5 (fma t_2 (cos x) (* t_1 (cos y))) 3.0))
       (/
        (fma t_0 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (*
         (+
          (* (/ 2.0 (+ 3.0 (sqrt 5.0))) (cos y))
          (+ (* (/ t_2 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0) * -0.0625;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -2.8e+33) {
		tmp = (((t_0 * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((t_1 * 0.5) * (3.0 * cos(y))));
	} else if (y <= 5.8e-5) {
		tmp = fma(sqrt(2.0), ((cos(x) - 1.0) * (fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x)))), 2.0) / fma(1.5, fma(t_2, cos(x), (t_1 * cos(y))), 3.0);
	} else {
		tmp = fma(t_0, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + (((t_2 / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64((sin(y) ^ 2.0) * -0.0625)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -2.8e+33)
		tmp = Float64(Float64(Float64(Float64(t_0 * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(t_1 * 0.5) * Float64(3.0 * cos(y)))));
	elseif (y <= 5.8e-5)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(cos(x) - 1.0) * Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x)))), 2.0) / fma(1.5, fma(t_2, cos(x), Float64(t_1 * cos(y))), 3.0));
	else
		tmp = Float64(fma(t_0, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(Float64(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * cos(y)) + Float64(Float64(Float64(t_2 / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -2.8e+33], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(t$95$1 * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e-5], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8000000000000001e33

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 58.9

         \[\leadsto \frac{2 + \left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 58.9%

      \[\leadsto \frac{2 + \color{blue}{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -2.8000000000000001e33 < y < 5.8e-5

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) \cdot \left(\cos x - 1\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 97.5%

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

   if 5.8e-5 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 28.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 28.0

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sqrt.f64 60.8

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

   10. Applied rewrites 60.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - 1\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)\\ t_1 := {\sin y}^{2} \cdot -0.0625\\ \mathbf{if}\;y \leq -0.48:\\ \;\;\;\;\frac{\left(t\_1 \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{t\_0}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, y \cdot y, -0.041666666666666664\right), y \cdot y, 0.5\right) \cdot y, y, \cos x - 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right) \cdot \sqrt{2}, y, \sqrt{2} \cdot \sin x\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
          3.0
          (* (* (- 3.0 (sqrt 5.0)) 0.5) (* 3.0 (cos y)))))
        (t_1 (* (pow (sin y) 2.0) -0.0625)))
   (if (<= y -0.48)
     (/ (+ (* (* t_1 (sqrt 2.0)) (- (cos x) (cos y))) 2.0) t_0)
     (if (<= y 5.8e-5)
       (/
        (+
         (*
          (*
           (fma
            (*
             (fma
              (fma 0.001388888888888889 (* y y) -0.041666666666666664)
              (* y y)
              0.5)
             y)
            y
            (- (cos x) 1.0))
           (fma
            (* (fma 0.010416666666666666 (* y y) -0.0625) (sqrt 2.0))
            y
            (* (sqrt 2.0) (sin x))))
          (fma -0.0625 (sin x) (sin y)))
         2.0)
        t_0)
       (/
        (fma t_1 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (*
         (+
          (* (/ 2.0 (+ 3.0 (sqrt 5.0))) (cos y))
          (+ (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, (((3.0 - sqrt(5.0)) * 0.5) * (3.0 * cos(y))));
	double t_1 = pow(sin(y), 2.0) * -0.0625;
	double tmp;
	if (y <= -0.48) {
		tmp = (((t_1 * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / t_0;
	} else if (y <= 5.8e-5) {
		tmp = (((fma((fma(fma(0.001388888888888889, (y * y), -0.041666666666666664), (y * y), 0.5) * y), y, (cos(x) - 1.0)) * fma((fma(0.010416666666666666, (y * y), -0.0625) * sqrt(2.0)), y, (sqrt(2.0) * sin(x)))) * fma(-0.0625, sin(x), sin(y))) + 2.0) / t_0;
	} else {
		tmp = fma(t_1, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + ((((sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * Float64(3.0 * cos(y))))
	t_1 = Float64((sin(y) ^ 2.0) * -0.0625)
	tmp = 0.0
	if (y <= -0.48)
		tmp = Float64(Float64(Float64(Float64(t_1 * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / t_0);
	elseif (y <= 5.8e-5)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(fma(fma(0.001388888888888889, Float64(y * y), -0.041666666666666664), Float64(y * y), 0.5) * y), y, Float64(cos(x) - 1.0)) * fma(Float64(fma(0.010416666666666666, Float64(y * y), -0.0625) * sqrt(2.0)), y, Float64(sqrt(2.0) * sin(x)))) * fma(-0.0625, sin(x), sin(y))) + 2.0) / t_0);
	else
		tmp = Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(Float64(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * cos(y)) + Float64(Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, If[LessEqual[y, -0.48], N[(N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 5.8e-5], N[(N[(N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(y * y), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * y + N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.010416666666666666 * N[(y * y), $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * y + N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.47999999999999998

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 57.2

         \[\leadsto \frac{2 + \left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 57.2%

      \[\leadsto \frac{2 + \color{blue}{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -0.47999999999999998 < y < 5.8e-5

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(y \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + \frac{1}{96} \cdot \left({y}^{2} \cdot \sqrt{2}\right)\right) + \sin x \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2} + \frac{1}{96} \cdot \left({y}^{2} \cdot \sqrt{2}\right)\right) \cdot y} + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{96} \cdot \left({y}^{2} \cdot \sqrt{2}\right) + \frac{-1}{16} \cdot \sqrt{2}\right)} \cdot y + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\left(\color{blue}{\left({y}^{2} \cdot \sqrt{2}\right) \cdot \frac{1}{96}} + \frac{-1}{16} \cdot \sqrt{2}\right) \cdot y + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\left(\color{blue}{{y}^{2} \cdot \left(\sqrt{2} \cdot \frac{1}{96}\right)} + \frac{-1}{16} \cdot \sqrt{2}\right) \cdot y + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\left({y}^{2} \cdot \color{blue}{\left(\frac{1}{96} \cdot \sqrt{2}\right)} + \frac{-1}{16} \cdot \sqrt{2}\right) \cdot y + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} \cdot \sqrt{2}\right) + \frac{-1}{16} \cdot \sqrt{2}, y, \sin x \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   9. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right), y, \sqrt{2} \cdot \sin x\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \color{blue}{\left(\left(\cos x + {y}^{2} \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{720} \cdot {y}^{2} - \frac{1}{24}\right)\right)\right) - 1\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{720} \cdot {y}^{2} - \frac{1}{24}\right)\right) + \cos x\right)} - 1\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       2. associate--l+ N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{720} \cdot {y}^{2} - \frac{1}{24}\right)\right) + \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \left(\color{blue}{\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{720} \cdot {y}^{2} - \frac{1}{24}\right)\right) \cdot {y}^{2}} + \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{720} \cdot {y}^{2} - \frac{1}{24}\right)\right) \cdot \color{blue}{\left(y \cdot y\right)} + \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{720} \cdot {y}^{2} - \frac{1}{24}\right)\right) \cdot y\right) \cdot y} + \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{720} \cdot {y}^{2} - \frac{1}{24}\right)\right) \cdot y, y, \cos x - 1\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   12. Applied rewrites 99.2%

       \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right), y, \sqrt{2} \cdot \sin x\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, y \cdot y, -0.041666666666666664\right), y \cdot y, 0.5\right) \cdot y, y, \cos x - 1\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if 5.8e-5 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 28.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 28.0

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sqrt.f64 60.8

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

   10. Applied rewrites 60.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.48:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, y \cdot y, -0.041666666666666664\right), y \cdot y, 0.5\right) \cdot y, y, \cos x - 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right) \cdot \sqrt{2}, y, \sqrt{2} \cdot \sin x\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)\\ t_1 := {\sin y}^{2} \cdot -0.0625\\ \mathbf{if}\;y \leq -0.48:\\ \;\;\;\;\frac{\left(t\_1 \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{t\_0}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, y \cdot y, 0.5\right) \cdot y, y, \cos x - 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right) \cdot \sqrt{2}, y, \sqrt{2} \cdot \sin x\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
          3.0
          (* (* (- 3.0 (sqrt 5.0)) 0.5) (* 3.0 (cos y)))))
        (t_1 (* (pow (sin y) 2.0) -0.0625)))
   (if (<= y -0.48)
     (/ (+ (* (* t_1 (sqrt 2.0)) (- (cos x) (cos y))) 2.0) t_0)
     (if (<= y 5.8e-5)
       (/
        (+
         (*
          (*
           (fma
            (* (fma -0.041666666666666664 (* y y) 0.5) y)
            y
            (- (cos x) 1.0))
           (fma
            (* (fma 0.010416666666666666 (* y y) -0.0625) (sqrt 2.0))
            y
            (* (sqrt 2.0) (sin x))))
          (fma -0.0625 (sin x) (sin y)))
         2.0)
        t_0)
       (/
        (fma t_1 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (*
         (+
          (* (/ 2.0 (+ 3.0 (sqrt 5.0))) (cos y))
          (+ (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, (((3.0 - sqrt(5.0)) * 0.5) * (3.0 * cos(y))));
	double t_1 = pow(sin(y), 2.0) * -0.0625;
	double tmp;
	if (y <= -0.48) {
		tmp = (((t_1 * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / t_0;
	} else if (y <= 5.8e-5) {
		tmp = (((fma((fma(-0.041666666666666664, (y * y), 0.5) * y), y, (cos(x) - 1.0)) * fma((fma(0.010416666666666666, (y * y), -0.0625) * sqrt(2.0)), y, (sqrt(2.0) * sin(x)))) * fma(-0.0625, sin(x), sin(y))) + 2.0) / t_0;
	} else {
		tmp = fma(t_1, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + ((((sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * Float64(3.0 * cos(y))))
	t_1 = Float64((sin(y) ^ 2.0) * -0.0625)
	tmp = 0.0
	if (y <= -0.48)
		tmp = Float64(Float64(Float64(Float64(t_1 * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / t_0);
	elseif (y <= 5.8e-5)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(fma(-0.041666666666666664, Float64(y * y), 0.5) * y), y, Float64(cos(x) - 1.0)) * fma(Float64(fma(0.010416666666666666, Float64(y * y), -0.0625) * sqrt(2.0)), y, Float64(sqrt(2.0) * sin(x)))) * fma(-0.0625, sin(x), sin(y))) + 2.0) / t_0);
	else
		tmp = Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(Float64(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * cos(y)) + Float64(Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, If[LessEqual[y, -0.48], N[(N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 5.8e-5], N[(N[(N[(N[(N[(N[(N[(-0.041666666666666664 * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * y + N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.010416666666666666 * N[(y * y), $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * y + N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.47999999999999998

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 57.2

         \[\leadsto \frac{2 + \left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 57.2%

      \[\leadsto \frac{2 + \color{blue}{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -0.47999999999999998 < y < 5.8e-5

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(y \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + \frac{1}{96} \cdot \left({y}^{2} \cdot \sqrt{2}\right)\right) + \sin x \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2} + \frac{1}{96} \cdot \left({y}^{2} \cdot \sqrt{2}\right)\right) \cdot y} + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{96} \cdot \left({y}^{2} \cdot \sqrt{2}\right) + \frac{-1}{16} \cdot \sqrt{2}\right)} \cdot y + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\left(\color{blue}{\left({y}^{2} \cdot \sqrt{2}\right) \cdot \frac{1}{96}} + \frac{-1}{16} \cdot \sqrt{2}\right) \cdot y + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\left(\color{blue}{{y}^{2} \cdot \left(\sqrt{2} \cdot \frac{1}{96}\right)} + \frac{-1}{16} \cdot \sqrt{2}\right) \cdot y + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\left({y}^{2} \cdot \color{blue}{\left(\frac{1}{96} \cdot \sqrt{2}\right)} + \frac{-1}{16} \cdot \sqrt{2}\right) \cdot y + \sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} \cdot \sqrt{2}\right) + \frac{-1}{16} \cdot \sqrt{2}, y, \sin x \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   9. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right), y, \sqrt{2} \cdot \sin x\right)} \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \color{blue}{\left(\left(\cos x + {y}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right)\right) - 1\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) + \cos x\right)} - 1\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       2. associate--l+ N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) + \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) \cdot \color{blue}{\left(y \cdot y\right)} + \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) \cdot y\right) \cdot y} + \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) \cdot y, y, \cos x - 1\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) \cdot y}, y, \cos x - 1\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{24} \cdot {y}^{2} + \frac{1}{2}\right)} \cdot y, y, \cos x - 1\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{24}, {y}^{2}, \frac{1}{2}\right)} \cdot y, y, \cos x - 1\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{24}, \color{blue}{y \cdot y}, \frac{1}{2}\right) \cdot y, y, \cos x - 1\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{24}, \color{blue}{y \cdot y}, \frac{1}{2}\right) \cdot y, y, \cos x - 1\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       12. lower--.f64 N/A

           \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{96}, y \cdot y, \frac{-1}{16}\right), y, \sqrt{2} \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{24}, y \cdot y, \frac{1}{2}\right) \cdot y, y, \color{blue}{\cos x - 1}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       13. lower-cos.f64 99.2

           \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right), y, \sqrt{2} \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, y \cdot y, 0.5\right) \cdot y, y, \color{blue}{\cos x} - 1\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   12. Applied rewrites 99.2%

       \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right), y, \sqrt{2} \cdot \sin x\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, y \cdot y, 0.5\right) \cdot y, y, \cos x - 1\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if 5.8e-5 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 28.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 28.0

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sqrt.f64 60.8

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

   10. Applied rewrites 60.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.48:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, y \cdot y, 0.5\right) \cdot y, y, \cos x - 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right) \cdot \sqrt{2}, y, \sqrt{2} \cdot \sin x\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2} \cdot -0.0625\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left(t\_0 \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(t\_1 \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - 1\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{t\_2}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (pow (sin y) 2.0) -0.0625))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -2.8e+33)
     (/
      (+ (* (* t_0 (sqrt 2.0)) (- (cos x) (cos y))) 2.0)
      (fma
       (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
       3.0
       (* (* t_1 0.5) (* 3.0 (cos y)))))
     (if (<= y 7.5e-6)
       (/
        (fma
         (sqrt 2.0)
         (*
          (- (cos x) 1.0)
          (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x))))
         2.0)
        (fma 1.5 (fma t_2 (cos x) t_1) 3.0))
       (/
        (fma t_0 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (*
         (+
          (* (/ 2.0 (+ 3.0 (sqrt 5.0))) (cos y))
          (+ (* (/ t_2 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0) * -0.0625;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -2.8e+33) {
		tmp = (((t_0 * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((t_1 * 0.5) * (3.0 * cos(y))));
	} else if (y <= 7.5e-6) {
		tmp = fma(sqrt(2.0), ((cos(x) - 1.0) * (fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x)))), 2.0) / fma(1.5, fma(t_2, cos(x), t_1), 3.0);
	} else {
		tmp = fma(t_0, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + (((t_2 / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64((sin(y) ^ 2.0) * -0.0625)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -2.8e+33)
		tmp = Float64(Float64(Float64(Float64(t_0 * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(t_1 * 0.5) * Float64(3.0 * cos(y)))));
	elseif (y <= 7.5e-6)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(cos(x) - 1.0) * Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x)))), 2.0) / fma(1.5, fma(t_2, cos(x), t_1), 3.0));
	else
		tmp = Float64(fma(t_0, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(Float64(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * cos(y)) + Float64(Float64(Float64(t_2 / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -2.8e+33], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(t$95$1 * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e-6], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8000000000000001e33

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 58.9

         \[\leadsto \frac{2 + \left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 58.9%

      \[\leadsto \frac{2 + \color{blue}{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -2.8000000000000001e33 < y < 7.50000000000000019e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) \cdot \left(\cos x - 1\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 97.4%

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

   if 7.50000000000000019e-6 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 28.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 28.0

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sqrt.f64 60.8

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

   10. Applied rewrites 60.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - 1\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(t\_1 \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - 1\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{t\_2}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (pow (sin y) 2.0) -0.0625)
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -2.8e+33)
     (/
      t_0
      (fma
       (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
       3.0
       (* (* t_1 0.5) (* 3.0 (cos y)))))
     (if (<= y 7.5e-6)
       (/
        (fma
         (sqrt 2.0)
         (*
          (- (cos x) 1.0)
          (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x))))
         2.0)
        (fma 1.5 (fma t_2 (cos x) t_1) 3.0))
       (/
        t_0
        (*
         (+
          (* (/ 2.0 (+ 3.0 (sqrt 5.0))) (cos y))
          (+ (* (/ t_2 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -2.8e+33) {
		tmp = t_0 / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((t_1 * 0.5) * (3.0 * cos(y))));
	} else if (y <= 7.5e-6) {
		tmp = fma(sqrt(2.0), ((cos(x) - 1.0) * (fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x)))), 2.0) / fma(1.5, fma(t_2, cos(x), t_1), 3.0);
	} else {
		tmp = t_0 / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + (((t_2 / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -2.8e+33)
		tmp = Float64(t_0 / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(t_1 * 0.5) * Float64(3.0 * cos(y)))));
	elseif (y <= 7.5e-6)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(cos(x) - 1.0) * Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x)))), 2.0) / fma(1.5, fma(t_2, cos(x), t_1), 3.0));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * cos(y)) + Float64(Float64(Float64(t_2 / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -2.8e+33], N[(t$95$0 / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(t$95$1 * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e-6], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8000000000000001e33

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 58.7

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 58.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -2.8000000000000001e33 < y < 7.50000000000000019e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) \cdot \left(\cos x - 1\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 97.4%

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

   if 7.50000000000000019e-6 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 28.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 28.0

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sqrt.f64 60.8

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

   10. Applied rewrites 60.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - 1\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot t\_1, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(t\_1, t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.0625 (cos x) 0.0625)) (t_1 (pow (sin x) 2.0)))
   (if (<= x -0.0102)
     (/
      (fma (* t_0 t_1) (sqrt 2.0) 2.0)
      (fma
       (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
       3.0
       (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))
     (if (<= x 0.018)
       (/
        (+
         (*
          (*
           (* (fma (sin y) -0.0625 x) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0)))
          (fma
           (* (fma 0.041666666666666664 (* x x) -0.5) x)
           x
           (- 1.0 (cos y))))
         2.0)
        (*
         (+
          (fma
           (- (sqrt 5.0) 1.0)
           (fma (* -0.25 x) x 0.5)
           (* (fma -0.5 (sqrt 5.0) 1.5) (cos y)))
          1.0)
         3.0))
       (/
        (* 0.3333333333333333 (fma t_1 (* t_0 (sqrt 2.0)) 2.0))
        (fma
         (* 0.5 (cos y))
         (- 3.0 (sqrt 5.0))
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, cos(x), 0.0625);
	double t_1 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -0.0102) {
		tmp = fma((t_0 * t_1), sqrt(2.0), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	} else if (x <= 0.018) {
		tmp = ((((fma(sin(y), -0.0625, x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * fma((fma(0.041666666666666664, (x * x), -0.5) * x), x, (1.0 - cos(y)))) + 2.0) / ((fma((sqrt(5.0) - 1.0), fma((-0.25 * x), x, 0.5), (fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0);
	} else {
		tmp = (0.3333333333333333 * fma(t_1, (t_0 * sqrt(2.0)), 2.0)) / fma((0.5 * cos(y)), (3.0 - sqrt(5.0)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, cos(x), 0.0625)
	t_1 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -0.0102)
		tmp = Float64(fma(Float64(t_0 * t_1), sqrt(2.0), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))));
	elseif (x <= 0.018)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(sin(y), -0.0625, x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * fma(Float64(fma(0.041666666666666664, Float64(x * x), -0.5) * x), x, Float64(1.0 - cos(y)))) + 2.0) / Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), fma(Float64(-0.25 * x), x, 0.5), Float64(fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0));
	else
		tmp = Float64(Float64(0.3333333333333333 * fma(t_1, Float64(t_0 * sqrt(2.0)), 2.0)) / fma(Float64(0.5 * cos(y)), Float64(3.0 - sqrt(5.0)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -0.0102], N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.018], N[(N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.25 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$1 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.010200000000000001

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y + \left(\mathsf{neg}\left(\frac{\sin x}{16}\right)\right)\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\color{blue}{\frac{\sin x}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \frac{1}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\sin x \cdot \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\sin y + \color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\sin y + \sin x \cdot \color{blue}{\frac{-1}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\left(3 \cdot \cos y\right) \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\color{blue}{\left(3 \cdot \cos y\right)} \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\color{blue}{\left(\cos y \cdot 3\right)} \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\cos y \cdot \left(3 \cdot \frac{1}{2}\right)\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\cos y \cdot \color{blue}{\frac{3}{2}}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\frac{3}{2} \cdot \cos y\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\frac{3}{2} \cdot \cos y\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      11. flip-- N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\frac{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{\sqrt{5} + 3}}\right)} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right) \cdot \sqrt{2}} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 56.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   if -0.010200000000000001 < x < 0.0179999999999999986

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}} + x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \sin y + x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \sin y + x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{16} \cdot \sin y + x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\sin y \cdot \frac{-1}{16}} + x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      7. lower-sin.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\color{blue}{\sin y}, -0.0625, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       2. associate--l+ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot x} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x, x, 1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x}, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       8. sub-neg N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       9. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}\right) \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right)} \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       11. unpow2 N/A

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       13. lower--.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right) \cdot x, x, \color{blue}{1 - \cos y}\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       14. lower-cos.f64 99.4

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \color{blue}{\cos y}\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   11. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   if 0.0179999999999999986 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 61.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 61.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot t\_1, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(t\_1, t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.0625 (cos x) 0.0625)) (t_1 (pow (sin x) 2.0)))
   (if (<= x -0.0102)
     (/
      (fma (* t_0 t_1) (sqrt 2.0) 2.0)
      (fma
       (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
       3.0
       (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))
     (if (<= x 0.018)
       (/
        (+
         (*
          (*
           (* (fma (sin y) -0.0625 x) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0)))
          (fma (* x x) -0.5 (- 1.0 (cos y))))
         2.0)
        (*
         (+
          (fma
           (- (sqrt 5.0) 1.0)
           (fma (* -0.25 x) x 0.5)
           (* (fma -0.5 (sqrt 5.0) 1.5) (cos y)))
          1.0)
         3.0))
       (/
        (* 0.3333333333333333 (fma t_1 (* t_0 (sqrt 2.0)) 2.0))
        (fma
         (* 0.5 (cos y))
         (- 3.0 (sqrt 5.0))
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, cos(x), 0.0625);
	double t_1 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -0.0102) {
		tmp = fma((t_0 * t_1), sqrt(2.0), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	} else if (x <= 0.018) {
		tmp = ((((fma(sin(y), -0.0625, x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * fma((x * x), -0.5, (1.0 - cos(y)))) + 2.0) / ((fma((sqrt(5.0) - 1.0), fma((-0.25 * x), x, 0.5), (fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0);
	} else {
		tmp = (0.3333333333333333 * fma(t_1, (t_0 * sqrt(2.0)), 2.0)) / fma((0.5 * cos(y)), (3.0 - sqrt(5.0)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, cos(x), 0.0625)
	t_1 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -0.0102)
		tmp = Float64(fma(Float64(t_0 * t_1), sqrt(2.0), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))));
	elseif (x <= 0.018)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(sin(y), -0.0625, x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * fma(Float64(x * x), -0.5, Float64(1.0 - cos(y)))) + 2.0) / Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), fma(Float64(-0.25 * x), x, 0.5), Float64(fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0));
	else
		tmp = Float64(Float64(0.3333333333333333 * fma(t_1, Float64(t_0 * sqrt(2.0)), 2.0)) / fma(Float64(0.5 * cos(y)), Float64(3.0 - sqrt(5.0)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -0.0102], N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.018], N[(N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.25 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$1 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.010200000000000001

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y + \left(\mathsf{neg}\left(\frac{\sin x}{16}\right)\right)\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\color{blue}{\frac{\sin x}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \frac{1}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\sin x \cdot \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\sin y + \color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\sin y + \sin x \cdot \color{blue}{\frac{-1}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\left(3 \cdot \cos y\right) \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\color{blue}{\left(3 \cdot \cos y\right)} \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\color{blue}{\left(\cos y \cdot 3\right)} \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\cos y \cdot \left(3 \cdot \frac{1}{2}\right)\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\cos y \cdot \color{blue}{\frac{3}{2}}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\frac{3}{2} \cdot \cos y\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\frac{3}{2} \cdot \cos y\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      11. flip-- N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\frac{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{\sqrt{5} + 3}}\right)} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right) \cdot \sqrt{2}} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 56.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   if -0.010200000000000001 < x < 0.0179999999999999986

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}} + x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \sin y + x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \sin y + x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{16} \cdot \sin y + x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\sin y \cdot \frac{-1}{16}} + x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      7. lower-sin.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\color{blue}{\sin y}, -0.0625, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       2. associate--l+ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + \left(1 - \cos y\right)\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + \left(1 - \cos y\right)\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1 - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, \color{blue}{1 - \cos y}\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       8. lower-cos.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \color{blue}{\cos y}\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   11. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   if 0.0179999999999999986 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 61.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 61.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.009:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot t\_1, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\left(1 - \cos y\right) \cdot \left(\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(t\_1, t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.0625 (cos x) 0.0625)) (t_1 (pow (sin x) 2.0)))
   (if (<= x -0.009)
     (/
      (fma (* t_0 t_1) (sqrt 2.0) 2.0)
      (fma
       (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
       3.0
       (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))
     (if (<= x 0.018)
       (/
        (+
         (*
          (- 1.0 (cos y))
          (*
           (* (fma (sin y) -0.0625 x) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0))))
         2.0)
        (*
         (+
          (fma
           (- (sqrt 5.0) 1.0)
           (fma (* -0.25 x) x 0.5)
           (* (fma -0.5 (sqrt 5.0) 1.5) (cos y)))
          1.0)
         3.0))
       (/
        (* 0.3333333333333333 (fma t_1 (* t_0 (sqrt 2.0)) 2.0))
        (fma
         (* 0.5 (cos y))
         (- 3.0 (sqrt 5.0))
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, cos(x), 0.0625);
	double t_1 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -0.009) {
		tmp = fma((t_0 * t_1), sqrt(2.0), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	} else if (x <= 0.018) {
		tmp = (((1.0 - cos(y)) * ((fma(sin(y), -0.0625, x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0)))) + 2.0) / ((fma((sqrt(5.0) - 1.0), fma((-0.25 * x), x, 0.5), (fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0);
	} else {
		tmp = (0.3333333333333333 * fma(t_1, (t_0 * sqrt(2.0)), 2.0)) / fma((0.5 * cos(y)), (3.0 - sqrt(5.0)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, cos(x), 0.0625)
	t_1 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -0.009)
		tmp = Float64(fma(Float64(t_0 * t_1), sqrt(2.0), 2.0) / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))));
	elseif (x <= 0.018)
		tmp = Float64(Float64(Float64(Float64(1.0 - cos(y)) * Float64(Float64(fma(sin(y), -0.0625, x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))) + 2.0) / Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), fma(Float64(-0.25 * x), x, 0.5), Float64(fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0));
	else
		tmp = Float64(Float64(0.3333333333333333 * fma(t_1, Float64(t_0 * sqrt(2.0)), 2.0)) / fma(Float64(0.5 * cos(y)), Float64(3.0 - sqrt(5.0)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -0.009], N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.018], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.25 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$1 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00899999999999999932

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y + \left(\mathsf{neg}\left(\frac{\sin x}{16}\right)\right)\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\color{blue}{\frac{\sin x}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \frac{1}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\sin x \cdot \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\sin y + \color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\sin y + \sin x \cdot \color{blue}{\frac{-1}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\left(3 \cdot \cos y\right) \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\color{blue}{\left(3 \cdot \cos y\right)} \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\color{blue}{\left(\cos y \cdot 3\right)} \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\cos y \cdot \left(3 \cdot \frac{1}{2}\right)\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\cos y \cdot \color{blue}{\frac{3}{2}}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\frac{3}{2} \cdot \cos y\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\frac{3}{2} \cdot \cos y\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      11. flip-- N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\frac{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{\sqrt{5} + 3}}\right)} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right) \cdot \sqrt{2}} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 56.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   if -0.00899999999999999932 < x < 0.0179999999999999986

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}} + x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \sin y + x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \sin y + x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{16} \cdot \sin y + x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\sin y \cdot \frac{-1}{16}} + x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

      7. lower-sin.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\color{blue}{\sin y}, -0.0625, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4} \cdot x, x, \frac{1}{2}\right), \cos y \cdot \mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)\right) + 1\right)} \]

       2. lower-cos.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \color{blue}{\cos y}\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   11. Applied rewrites 99.3%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(1 - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \cos y \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)\right) + 1\right)} \]

   if 0.0179999999999999986 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 61.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 61.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.009:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\left(1 - \cos y\right) \cdot \left(\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -39000000:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(t\_1 \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_1, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (pow (sin y) 2.0) -0.0625)
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= y -39000000.0)
     (/
      t_0
      (fma
       (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)
       3.0
       (* (* t_1 0.5) (* 3.0 (cos y)))))
     (if (<= y 5.8e-5)
       (/
        (*
         0.3333333333333333
         (fma
          (pow (sin x) 2.0)
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          2.0))
        (fma (* 0.5 (cos y)) t_1 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
       (/
        t_0
        (*
         (+
          (* (/ 2.0 (+ 3.0 (sqrt 5.0))) (cos y))
          (+ (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -39000000.0) {
		tmp = t_0 / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, ((t_1 * 0.5) * (3.0 * cos(y))));
	} else if (y <= 5.8e-5) {
		tmp = (0.3333333333333333 * fma(pow(sin(x), 2.0), (fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0)) / fma((0.5 * cos(y)), t_1, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	} else {
		tmp = t_0 / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + ((((sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -39000000.0)
		tmp = Float64(t_0 / fma(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0), 3.0, Float64(Float64(t_1 * 0.5) * Float64(3.0 * cos(y)))));
	elseif (y <= 5.8e-5)
		tmp = Float64(Float64(0.3333333333333333 * fma((sin(x) ^ 2.0), Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0)) / fma(Float64(0.5 * cos(y)), t_1, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * cos(y)) + Float64(Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -39000000.0], N[(t$95$0 / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(t$95$1 * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e-5], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9e7

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. lower-sqrt.f64 57.5

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 57.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -3.9e7 < y < 5.8e-5

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 98.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 98.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if 5.8e-5 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 28.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 28.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 60.8

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   10. Applied rewrites 60.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -39000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := t\_2 \cdot 0.5\\ \mathbf{if}\;y \leq -39000000:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, 1\right), 3, t\_3 \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(t\_0 \cdot \cos x + \mathsf{fma}\left(t\_3, \cos y, 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1
         (fma
          (* (pow (sin y) 2.0) -0.0625)
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (* t_2 0.5)))
   (if (<= y -39000000.0)
     (/ t_1 (fma (fma (cos x) t_0 1.0) 3.0 (* t_3 (* 3.0 (cos y)))))
     (if (<= y 5.8e-5)
       (/
        (*
         0.3333333333333333
         (fma
          (pow (sin x) 2.0)
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          2.0))
        (fma (* 0.5 (cos y)) t_2 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
       (/ t_1 (* (+ (* t_0 (cos x)) (fma t_3 (cos y) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = t_2 * 0.5;
	double tmp;
	if (y <= -39000000.0) {
		tmp = t_1 / fma(fma(cos(x), t_0, 1.0), 3.0, (t_3 * (3.0 * cos(y))));
	} else if (y <= 5.8e-5) {
		tmp = (0.3333333333333333 * fma(pow(sin(x), 2.0), (fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0)) / fma((0.5 * cos(y)), t_2, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	} else {
		tmp = t_1 / (((t_0 * cos(x)) + fma(t_3, cos(y), 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(t_2 * 0.5)
	tmp = 0.0
	if (y <= -39000000.0)
		tmp = Float64(t_1 / fma(fma(cos(x), t_0, 1.0), 3.0, Float64(t_3 * Float64(3.0 * cos(y)))));
	elseif (y <= 5.8e-5)
		tmp = Float64(Float64(0.3333333333333333 * fma((sin(x) ^ 2.0), Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0)) / fma(Float64(0.5 * cos(y)), t_2, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	else
		tmp = Float64(t_1 / Float64(Float64(Float64(t_0 * cos(x)) + fma(t_3, cos(y), 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 0.5), $MachinePrecision]}, If[LessEqual[y, -39000000.0], N[(t$95$1 / N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(t$95$3 * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e-5], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9e7

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. lower-sqrt.f64 57.5

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 57.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -3.9e7 < y < 5.8e-5

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 98.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 98.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if 5.8e-5 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 28.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x\right)} \]

      9. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x\right)} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)}\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)}\right)} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      11. lower-sqrt.f64 60.8

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)} \]

   10. Applied rewrites 60.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -39000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x + \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ t_2 := {\sin x}^{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00066:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot t\_2, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(t\_0 \cdot \cos x + \mathsf{fma}\left(t\_3 \cdot 0.5, \cos y, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(t\_2, t\_1 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (fma -0.0625 (cos x) 0.0625))
        (t_2 (pow (sin x) 2.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.00066)
     (/
      (fma (* t_1 t_2) (sqrt 2.0) 2.0)
      (fma (fma (cos x) t_0 1.0) 3.0 (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))
     (if (<= x 0.00165)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* (+ (* t_0 (cos x)) (fma (* t_3 0.5) (cos y) 1.0)) 3.0))
       (/
        (* 0.3333333333333333 (fma t_2 (* t_1 (sqrt 2.0)) 2.0))
        (fma
         (* 0.5 (cos y))
         t_3
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = fma(-0.0625, cos(x), 0.0625);
	double t_2 = pow(sin(x), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.00066) {
		tmp = fma((t_1 * t_2), sqrt(2.0), 2.0) / fma(fma(cos(x), t_0, 1.0), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	} else if (x <= 0.00165) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (((t_0 * cos(x)) + fma((t_3 * 0.5), cos(y), 1.0)) * 3.0);
	} else {
		tmp = (0.3333333333333333 * fma(t_2, (t_1 * sqrt(2.0)), 2.0)) / fma((0.5 * cos(y)), t_3, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = fma(-0.0625, cos(x), 0.0625)
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.00066)
		tmp = Float64(fma(Float64(t_1 * t_2), sqrt(2.0), 2.0) / fma(fma(cos(x), t_0, 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))));
	elseif (x <= 0.00165)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(Float64(Float64(t_0 * cos(x)) + fma(Float64(t_3 * 0.5), cos(y), 1.0)) * 3.0));
	else
		tmp = Float64(Float64(0.3333333333333333 * fma(t_2, Float64(t_1 * sqrt(2.0)), 2.0)) / fma(Float64(0.5 * cos(y)), t_3, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00066], N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00165], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$2 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6e-4

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y + \left(\mathsf{neg}\left(\frac{\sin x}{16}\right)\right)\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\color{blue}{\frac{\sin x}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \frac{1}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sin y + \left(\mathsf{neg}\left(\sin x \cdot \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\sin y + \color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\sin y + \sin x \cdot \color{blue}{\frac{-1}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)} \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\left(3 \cdot \cos y\right) \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\color{blue}{\left(3 \cdot \cos y\right)} \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\color{blue}{\left(\cos y \cdot 3\right)} \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\cos y \cdot \left(3 \cdot \frac{1}{2}\right)\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\cos y \cdot \color{blue}{\frac{3}{2}}\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\frac{3}{2} \cdot \cos y\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\left(\frac{3}{2} \cdot \cos y\right)} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      11. flip-- N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \color{blue}{\frac{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{\sqrt{5} + 3}}\right)} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \color{blue}{\frac{6 \cdot \cos y}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       6. associate-*r* N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right) \cdot \sqrt{2}} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 56.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \]

   if -6.6e-4 < x < 0.00165

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 66.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x\right)} \]

      9. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x\right)} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)}\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)}\right)} \]

   7. Applied rewrites 66.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      11. lower-sqrt.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)} \]

   10. Applied rewrites 99.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)} \]

   if 0.00165 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 61.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 61.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00066:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x + \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x + \mathsf{fma}\left(t\_0 \cdot 0.5, \cos y, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -39000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (pow (sin y) 2.0) -0.0625)
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (*
           (+
            (* (fma (sqrt 5.0) 0.5 -0.5) (cos x))
            (fma (* t_0 0.5) (cos y) 1.0))
           3.0))))
   (if (<= y -39000000.0)
     t_1
     (if (<= y 5.8e-5)
       (/
        (*
         0.3333333333333333
         (fma
          (pow (sin x) 2.0)
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          2.0))
        (fma (* 0.5 (cos y)) t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (((fma(sqrt(5.0), 0.5, -0.5) * cos(x)) + fma((t_0 * 0.5), cos(y), 1.0)) * 3.0);
	double tmp;
	if (y <= -39000000.0) {
		tmp = t_1;
	} else if (y <= 5.8e-5) {
		tmp = (0.3333333333333333 * fma(pow(sin(x), 2.0), (fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0)) / fma((0.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(Float64(Float64(fma(sqrt(5.0), 0.5, -0.5) * cos(x)) + fma(Float64(t_0 * 0.5), cos(y), 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -39000000.0)
		tmp = t_1;
	elseif (y <= 5.8e-5)
		tmp = Float64(Float64(0.3333333333333333 * fma((sin(x) ^ 2.0), Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0)) / fma(Float64(0.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -39000000.0], t$95$1, If[LessEqual[y, 5.8e-5], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9e7 or 5.8e-5 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 26.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x\right)} \]

      9. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x\right)} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)}\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)}\right)} \]

   7. Applied rewrites 26.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right)} \]

      11. lower-sqrt.f64 59.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)} \]

   10. Applied rewrites 59.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)} \]

   if -3.9e7 < y < 5.8e-5

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 98.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 98.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -39000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x + \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x + \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(0.5 \cdot \cos y, t\_1, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{t\_2} \cdot t\_0\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot t\_0}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (pow (sin x) 2.0)
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (fma
          (* 0.5 (cos y))
          t_1
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
   (if (<= x -3.7e-6)
     (* (/ 0.3333333333333333 t_2) t_0)
     (if (<= x 0.00165)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos y) t_1 (- (sqrt 5.0) 1.0)) 3.0))
       (/ (* 0.3333333333333333 t_0) t_2)))))

C

double code(double x, double y) {
	double t_0 = fma(pow(sin(x), 2.0), (fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((0.5 * cos(y)), t_1, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	double tmp;
	if (x <= -3.7e-6) {
		tmp = (0.3333333333333333 / t_2) * t_0;
	} else if (x <= 0.00165) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, (sqrt(5.0) - 1.0)), 3.0);
	} else {
		tmp = (0.3333333333333333 * t_0) / t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma((sin(x) ^ 2.0), Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(Float64(0.5 * cos(y)), t_1, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))
	tmp = 0.0
	if (x <= -3.7e-6)
		tmp = Float64(Float64(0.3333333333333333 / t_2) * t_0);
	elseif (x <= 0.00165)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, Float64(sqrt(5.0) - 1.0)), 3.0));
	else
		tmp = Float64(Float64(0.3333333333333333 * t_0) / t_2);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e-6], N[(N[(0.3333333333333333 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 0.00165], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000002e-6

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 56.4

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 56.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)} \]

   if -3.7000000000000002e-6 < x < 0.00165

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 66.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. Applied rewrites 66.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       11. lower-sqrt.f64 99.1

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   if 0.00165 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 61.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 61.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (/
           0.3333333333333333
           (fma
            (* 0.5 (cos y))
            t_0
            (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
          (fma
           (pow (sin x) 2.0)
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           2.0))))
   (if (<= x -3.7e-6)
     t_1
     (if (<= x 0.00165)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos y) t_0 (- (sqrt 5.0) 1.0)) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (0.3333333333333333 / fma((0.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))) * fma(pow(sin(x), 2.0), (fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0);
	double tmp;
	if (x <= -3.7e-6) {
		tmp = t_1;
	} else if (x <= 0.00165) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_0, (sqrt(5.0) - 1.0)), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(0.3333333333333333 / fma(Float64(0.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))) * fma((sin(x) ^ 2.0), Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), 2.0))
	tmp = 0.0
	if (x <= -3.7e-6)
		tmp = t_1;
	elseif (x <= 0.00165)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_0, Float64(sqrt(5.0) - 1.0)), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e-6], t$95$1, If[LessEqual[x, 0.00165], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000002e-6 or 0.00165 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 58.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 58.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 58.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   9. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\mathsf{fma}\left(\cos y \cdot 0.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)} \]

   if -3.7000000000000002e-6 < x < 0.00165

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 66.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. Applied rewrites 66.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       11. lower-sqrt.f64 99.1

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma t_0 (cos x) (* t_1 (cos y))) 3.0))))
   (if (<= x -3.7e-6)
     t_2
     (if (<= x 0.00165)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(t_0, cos(x), (t_1 * cos(y))), 3.0);
	double tmp;
	if (x <= -3.7e-6) {
		tmp = t_2;
	} else if (x <= 0.00165) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -3.7e-6)
		tmp = t_2;
	elseif (x <= 0.00165)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e-6], t$95$2, If[LessEqual[x, 0.00165], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000002e-6 or 0.00165 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 58.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 58.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]

   if -3.7000000000000002e-6 < x < 0.00165

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 66.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. Applied rewrites 66.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       11. lower-sqrt.f64 99.1

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{2}{3 + \sqrt{5}} + 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_2\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0))
          (pow (sin x) 2.0)
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -2.2e-5)
     (*
      (/
       t_0
       (fma
        (fma (sqrt 5.0) 0.5 -0.5)
        (cos x)
        (+ (/ 2.0 (+ 3.0 (sqrt 5.0))) 1.0)))
      0.3333333333333333)
     (if (<= x 0.00165)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_2) 3.0))
       (* (/ t_0 (fma 0.5 (fma t_2 (cos x) t_1) 1.0)) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -2.2e-5) {
		tmp = (t_0 / fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), ((2.0 / (3.0 + sqrt(5.0))) + 1.0))) * 0.3333333333333333;
	} else if (x <= 0.00165) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_2), 3.0);
	} else {
		tmp = (t_0 / fma(0.5, fma(t_2, cos(x), t_1), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -2.2e-5)
		tmp = Float64(Float64(t_0 / fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) + 1.0))) * 0.3333333333333333);
	elseif (x <= 0.00165)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_2), 3.0));
	else
		tmp = Float64(Float64(t_0 / fma(0.5, fma(t_2, cos(x), t_1), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -2.2e-5], N[(N[(t$95$0 / N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 0.00165], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999999e-5

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 56.4

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   7. Applied rewrites 56.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   10. Applied rewrites 55.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3} + 1\right)} \cdot 0.3333333333333333} \]

   if -2.1999999999999999e-5 < x < 0.00165

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 66.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. Applied rewrites 66.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       11. lower-sqrt.f64 99.1

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   if 0.00165 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. Applied rewrites 21.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   11. Applied rewrites 60.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{2}{3 + \sqrt{5}} + 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (/
           (fma
            (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0))
            (pow (sin x) 2.0)
            2.0)
           (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
          0.3333333333333333)))
   (if (<= x -2.2e-5)
     t_2
     (if (<= x 0.00165)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -2.2e-5) {
		tmp = t_2;
	} else if (x <= 0.00165) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -2.2e-5)
		tmp = t_2;
	elseif (x <= 0.00165)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -2.2e-5], t$95$2, If[LessEqual[x, 0.00165], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1999999999999999e-5 or 0.00165 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 58.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. Applied rewrites 21.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   11. Applied rewrites 57.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -2.1999999999999999e-5 < x < 0.00165

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 66.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. Applied rewrites 66.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       11. lower-sqrt.f64 99.1

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma t_0 (cos x) t_1) 3.0))))
   (if (<= x -2.2e-5)
     t_2
     (if (<= x 0.00165)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	double tmp;
	if (x <= -2.2e-5) {
		tmp = t_2;
	} else if (x <= 0.00165) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0))
	tmp = 0.0
	if (x <= -2.2e-5)
		tmp = t_2;
	elseif (x <= 0.00165)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2e-5], t$95$2, If[LessEqual[x, 0.00165], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1999999999999999e-5 or 0.00165 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 58.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 57.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if -2.1999999999999999e-5 < x < 0.00165

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 66.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. Applied rewrites 66.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       11. lower-sqrt.f64 99.1

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 60.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) (pow (sin x) 2.0) 2.0)
  (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 62.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 60.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]
9. Add Preprocessing

Alternative 31: 45.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   (+
    (* (/ 2.0 (+ 3.0 (sqrt 5.0))) (cos y))
    (+ (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)) 1.0))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + ((((sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 / ((((2.0d0 / (3.0d0 + sqrt(5.0d0))) * cos(y)) + ((((sqrt(5.0d0) - 1.0d0) / 2.0d0) * cos(x)) + 1.0d0)) * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return 2.0 / ((((2.0 / (3.0 + Math.sqrt(5.0))) * Math.cos(y)) + ((((Math.sqrt(5.0) - 1.0) / 2.0) * Math.cos(x)) + 1.0)) * 3.0);
}

Python

def code(x, y):
	return 2.0 / ((((2.0 / (3.0 + math.sqrt(5.0))) * math.cos(y)) + ((((math.sqrt(5.0) - 1.0) / 2.0) * math.cos(x)) + 1.0)) * 3.0)

Julia

function code(x, y)
	return Float64(2.0 / Float64(Float64(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * cos(y)) + Float64(Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + ((((sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0);
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 62.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

   2. div-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   4. flip-- N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   9. rem-square-sqrt N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   15. lower-+.f64 62.5

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

7. Applied rewrites 62.5%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 45.8%

    \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

11. Final simplification 45.8%

    \[\leadsto \frac{2}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3} \]
12. Add Preprocessing

Alternative 32: 45.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   1.5
   (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 62.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

8. Applied rewrites 43.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 43.0%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

12. Taylor expanded in y around inf

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-*r* N/A

       \[\leadsto \frac{2}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

14. Applied rewrites 45.8%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]

15. Final simplification 45.8%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
16. Add Preprocessing

Alternative 33: 43.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 62.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

8. Applied rewrites 43.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 43.0%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-*r* N/A

       \[\leadsto \frac{2}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

14. Applied rewrites 43.6%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]
15. Add Preprocessing

Alternative 34: 42.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (- (sqrt 5.0) 1.0)) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), (sqrt(5.0) - 1.0)), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(sqrt(5.0) - 1.0)), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 62.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

8. Applied rewrites 43.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 43.0%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
12. Add Preprocessing

Alternative 35: 40.9% accurate, 78.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{6} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 2.0 6.0))

C

double code(double x, double y) {
	return 2.0 / 6.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 / 6.0d0
end function

Java

public static double code(double x, double y) {
	return 2.0 / 6.0;
}

Python

def code(x, y):
	return 2.0 / 6.0

Julia

function code(x, y)
	return Float64(2.0 / 6.0)
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 / 6.0;
end

Wolfram

code[x_, y_] := N[(2.0 / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 62.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

8. Applied rewrites 43.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 43.0%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{2}{6} \]
13. Step-by-step derivation

14. Applied rewrites 41.1%

    \[\leadsto \frac{2}{6} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(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))))))