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

Percentage Accurate: 99.3% → 99.3%

Time: 20.3s

Alternatives: 33

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

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

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)))
    (* (/ (/ 4.0 (+ (sqrt 5.0) 3.0)) 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))) + (((4.0 / (sqrt(5.0) + 3.0)) / 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))) + (((4.0d0 / (sqrt(5.0d0) + 3.0d0)) / 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))) + (((4.0 / (Math.sqrt(5.0) + 3.0)) / 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))) + (((4.0 / (math.sqrt(5.0) + 3.0)) / 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(4.0 / Float64(sqrt(5.0) + 3.0)) / 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))) + (((4.0 / (sqrt(5.0) + 3.0)) / 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[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * 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

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sin.f64 99.4

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

10. Applied rewrites 99.4%

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

12. Applied rewrites 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sin.f64 99.4

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

10. Applied rewrites 99.4%

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

12. Applied rewrites 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (- (sin x) (* 0.0625 (sin y))) (sqrt 2.0))
     (- (sin y) (* 0.0625 (sin x))))
    (- (cos x) (cos y))))
  (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 + ((((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (sin(y) - (0.0625 * sin(x)))) * (cos(x) - cos(y)))) / 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(Float64(2.0 + Float64(Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)) * Float64(sin(y) - Float64(0.0625 * sin(x)))) * Float64(cos(x) - cos(y)))) / 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[(2.0 + N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sqrt.f64 61.6

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

5. Applied rewrites 61.6%

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

6. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 61.6%

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

9. Taylor expanded in x around inf

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

    1. associate-*r* N/A

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

    2. lower-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower--.f64 N/A

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

    6. lower-sin.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-sin.f64 N/A

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

    9. lower-sqrt.f64 N/A

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

    10. lower--.f64 N/A

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

    11. lower-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-sin.f64 99.4

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

11. Applied rewrites 99.4%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (* (sqrt 2.0) (- (cos x) (cos y)))
    (* (- (sin y) (* 0.0625 (sin x))) (- (sin x) (* 0.0625 (sin y))))))
  (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 + ((sqrt(2.0) * (cos(x) - cos(y))) * ((sin(y) - (0.0625 * sin(x))) * (sin(x) - (0.0625 * sin(y)))))) / 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(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))) * Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) - Float64(0.0625 * sin(y)))))) / 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[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sqrt.f64 61.6

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

5. Applied rewrites 61.6%

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

6. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 61.6%

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

9. Taylor expanded in x around inf

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

    1. associate-*r* N/A

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

    2. lower-*.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-sqrt.f64 N/A

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

    5. lower--.f64 N/A

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

    6. lower-cos.f64 N/A

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

    7. lower-cos.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower--.f64 N/A

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

    11. lower-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-sin.f64 N/A

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

    14. lower--.f64 N/A

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

    15. lower-sin.f64 N/A

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

    16. lower-*.f64 N/A

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

    17. lower-sin.f64 99.3

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

11. Applied rewrites 99.3%

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

Alternative 6: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)\\ \mathbf{if}\;y \leq -0.082 \lor \neg \left(y \leq 0.027\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right) \cdot t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (*
          3.0
          (fma
           (/ (cos y) (+ 3.0 (sqrt 5.0)))
           2.0
           (fma (* 0.5 (- (sqrt 5.0) 1.0)) (cos x) 1.0)))))
   (if (or (<= y -0.082) (not (<= y 0.027)))
     (/
      (+ 2.0 (* (* (* (sin y) (sqrt 2.0)) (- (sin x) (* 0.0625 (sin y)))) t_0))
      t_1)
     (/
      (+
       2.0
       (*
        (*
         (* (- (sin y) (* 0.0625 (sin x))) (sqrt 2.0))
         (fma -0.0625 y (sin x)))
        t_0))
      t_1))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 * fma((cos(y) / (3.0 + sqrt(5.0))), 2.0, fma((0.5 * (sqrt(5.0) - 1.0)), cos(x), 1.0));
	double tmp;
	if ((y <= -0.082) || !(y <= 0.027)) {
		tmp = (2.0 + (((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y)))) * t_0)) / t_1;
	} else {
		tmp = (2.0 + ((((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * fma(-0.0625, y, sin(x))) * t_0)) / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 * fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 2.0, fma(Float64(0.5 * Float64(sqrt(5.0) - 1.0)), cos(x), 1.0)))
	tmp = 0.0
	if ((y <= -0.082) || !(y <= 0.027))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y)))) * t_0)) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * sqrt(2.0)) * fma(-0.0625, y, sin(x))) * t_0)) / 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[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.082], N[Not[LessEqual[y, 0.027]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0820000000000000034 or 0.0269999999999999997 < 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 99.2

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

   10. Applied rewrites 99.2%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 64.0%

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

   if -0.0820000000000000034 < y < 0.0269999999999999997

   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

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 99.7

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 99.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.082 \lor \neg \left(y \leq 0.027\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.082 \lor \neg \left(y \leq 0.027\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot t\_0, \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- (cos x) (cos y))))
   (if (or (<= y -0.082) (not (<= y 0.027)))
     (/
      (+ 2.0 (* (* (* (sin y) (sqrt 2.0)) (- (sin x) (* 0.0625 (sin y)))) t_1))
      (*
       3.0
       (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 2.0 (fma (* 0.5 t_0) (cos x) 1.0))))
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (fma -0.0625 y (sin x))) (- (sin y) (/ (sin x) 16.0)))
        t_1))
      (fma 1.5 (fma t_0 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double tmp;
	if ((y <= -0.082) || !(y <= 0.027)) {
		tmp = (2.0 + (((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y)))) * t_1)) / (3.0 * fma((cos(y) / (3.0 + sqrt(5.0))), 2.0, fma((0.5 * t_0), cos(x), 1.0)));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * fma(-0.0625, y, sin(x))) * (sin(y) - (sin(x) / 16.0))) * t_1)) / fma(1.5, fma(t_0, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((y <= -0.082) || !(y <= 0.027))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y)))) * t_1)) / Float64(3.0 * fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 2.0, fma(Float64(0.5 * t_0), cos(x), 1.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(-0.0625, y, sin(x))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / fma(1.5, fma(t_0, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	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[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.082], N[Not[LessEqual[y, 0.027]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(t$95$0 * 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. Split input into 2 regimes

2. if y < -0.0820000000000000034 or 0.0269999999999999997 < 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 99.2

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

   10. Applied rewrites 99.2%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 64.0%

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

   if -0.0820000000000000034 < y < 0.0269999999999999997

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. +-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-sin.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   11. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.082 \lor \neg \left(y \leq 0.027\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := 2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot t\_3\right) \cdot t\_0\\ t_5 := t\_2 \cdot \cos x\\ \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;\frac{t\_4}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_5\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.035:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot t\_3\right) \cdot t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(1.5, t\_5 - \left(\sqrt{5} - 3\right) \cdot \cos y, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (+ 2.0 (* (* (* (sin x) (sqrt 2.0)) t_3) t_0)))
        (t_5 (* t_2 (cos x))))
   (if (<= x -0.021)
     (/ t_4 (* 3.0 (fma 0.5 (fma t_1 (cos y) t_5) 1.0)))
     (if (<= x 0.035)
       (/
        (+ 2.0 (* (* (* (sqrt 2.0) (fma -0.0625 (sin y) x)) t_3) t_0))
        (fma 1.5 (fma t_2 (cos x) (* t_1 (cos y))) 3.0))
       (/ t_4 (fma 1.5 (- t_5 (* (- (sqrt 5.0) 3.0) (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = 2.0 + (((sin(x) * sqrt(2.0)) * t_3) * t_0);
	double t_5 = t_2 * cos(x);
	double tmp;
	if (x <= -0.021) {
		tmp = t_4 / (3.0 * fma(0.5, fma(t_1, cos(y), t_5), 1.0));
	} else if (x <= 0.035) {
		tmp = (2.0 + (((sqrt(2.0) * fma(-0.0625, sin(y), x)) * t_3) * t_0)) / fma(1.5, fma(t_2, cos(x), (t_1 * cos(y))), 3.0);
	} else {
		tmp = t_4 / fma(1.5, (t_5 - ((sqrt(5.0) - 3.0) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * t_3) * t_0))
	t_5 = Float64(t_2 * cos(x))
	tmp = 0.0
	if (x <= -0.021)
		tmp = Float64(t_4 / Float64(3.0 * fma(0.5, fma(t_1, cos(y), t_5), 1.0)));
	elseif (x <= 0.035)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(-0.0625, sin(y), x)) * t_3) * t_0)) / fma(1.5, fma(t_2, cos(x), Float64(t_1 * cos(y))), 3.0));
	else
		tmp = Float64(t_4 / fma(1.5, Float64(t_5 - Float64(Float64(sqrt(5.0) - 3.0) * cos(y))), 3.0));
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.021], N[(t$95$4 / N[(3.0 * N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.035], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision]), $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[(t$95$4 / N[(1.5 * N[(t$95$5 - N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0210000000000000013

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-cos.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if -0.0210000000000000013 < x < 0.035000000000000003

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 62.0%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sin.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin y}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   11. Applied rewrites 99.7%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if 0.035000000000000003 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 61.1%

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

   10. Applied rewrites 61.1%

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

Alternative 9: 81.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := t\_1 \cdot \cos x\\ t_3 := 3 - \sqrt{5}\\ t_4 := 1 - \cos y\\ \mathbf{if}\;x \leq -0.00029:\\ \;\;\;\;\frac{t\_0}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_2\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.0031:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(t\_4 \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_4 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, t\_2 - \left(\sqrt{5} - 3\right) \cdot \cos y, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
           (- (cos x) (cos y)))))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* t_1 (cos x)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (- 1.0 (cos y))))
   (if (<= x -0.00029)
     (/ t_0 (* 3.0 (fma 0.5 (fma t_3 (cos y) t_2) 1.0)))
     (if (<= x 0.0031)
       (*
        0.3333333333333333
        (/
         (fma
          (* (sqrt 2.0) x)
          (* (* t_4 1.00390625) (sin y))
          (fma (* -0.0625 (pow (sin y) 2.0)) (* t_4 (sqrt 2.0)) 2.0))
         (fma 0.5 (fma t_3 (cos y) t_1) 1.0)))
       (/ t_0 (fma 1.5 (- t_2 (* (- (sqrt 5.0) 3.0) (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = t_1 * cos(x);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = 1.0 - cos(y);
	double tmp;
	if (x <= -0.00029) {
		tmp = t_0 / (3.0 * fma(0.5, fma(t_3, cos(y), t_2), 1.0));
	} else if (x <= 0.0031) {
		tmp = 0.3333333333333333 * (fma((sqrt(2.0) * x), ((t_4 * 1.00390625) * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_4 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_3, cos(y), t_1), 1.0));
	} else {
		tmp = t_0 / fma(1.5, (t_2 - ((sqrt(5.0) - 3.0) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y))))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(t_1 * cos(x))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (x <= -0.00029)
		tmp = Float64(t_0 / Float64(3.0 * fma(0.5, fma(t_3, cos(y), t_2), 1.0)));
	elseif (x <= 0.0031)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * x), Float64(Float64(t_4 * 1.00390625) * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_4 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_3, cos(y), t_1), 1.0)));
	else
		tmp = Float64(t_0 / fma(1.5, Float64(t_2 - Float64(Float64(sqrt(5.0) - 3.0) * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[(N[(N[Sin[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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00029], N[(t$95$0 / N[(3.0 * N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0031], N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(N[(t$95$4 * 1.00390625), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(t$95$2 - N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e-4

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-cos.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if -2.9e-4 < x < 0.00309999999999999989

   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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} + \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

   if 0.00309999999999999989 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 61.1%

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

   10. Applied rewrites 61.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00029:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.0031:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right) \cdot \cos y, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 3 - \sqrt{5}\\ t_3 := 1 - \cos y\\ t_4 := \sqrt{5} - 1\\ t_5 := \sin x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.00029:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_5 \cdot t\_0, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos y, \cos x \cdot t\_4\right), 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 0.0031:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(t\_3 \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_3 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_4\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_5 \cdot t\_1\right) \cdot t\_0}{\mathsf{fma}\left(1.5, t\_4 \cdot \cos x - \left(\sqrt{5} - 3\right) \cdot \cos y, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (- 1.0 (cos y)))
        (t_4 (- (sqrt 5.0) 1.0))
        (t_5 (* (sin x) (sqrt 2.0))))
   (if (<= x -0.00029)
     (/
      (fma t_1 (* t_5 t_0) 2.0)
      (* 3.0 (fma (fma t_2 (cos y) (* (cos x) t_4)) 0.5 1.0)))
     (if (<= x 0.0031)
       (*
        0.3333333333333333
        (/
         (fma
          (* (sqrt 2.0) x)
          (* (* t_3 1.00390625) (sin y))
          (fma (* -0.0625 (pow (sin y) 2.0)) (* t_3 (sqrt 2.0)) 2.0))
         (fma 0.5 (fma t_2 (cos y) t_4) 1.0)))
       (/
        (+ 2.0 (* (* t_5 t_1) t_0))
        (fma 1.5 (- (* t_4 (cos x)) (* (- (sqrt 5.0) 3.0) (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 1.0 - cos(y);
	double t_4 = sqrt(5.0) - 1.0;
	double t_5 = sin(x) * sqrt(2.0);
	double tmp;
	if (x <= -0.00029) {
		tmp = fma(t_1, (t_5 * t_0), 2.0) / (3.0 * fma(fma(t_2, cos(y), (cos(x) * t_4)), 0.5, 1.0));
	} else if (x <= 0.0031) {
		tmp = 0.3333333333333333 * (fma((sqrt(2.0) * x), ((t_3 * 1.00390625) * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_3 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_2, cos(y), t_4), 1.0));
	} else {
		tmp = (2.0 + ((t_5 * t_1) * t_0)) / fma(1.5, ((t_4 * cos(x)) - ((sqrt(5.0) - 3.0) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(1.0 - cos(y))
	t_4 = Float64(sqrt(5.0) - 1.0)
	t_5 = Float64(sin(x) * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.00029)
		tmp = Float64(fma(t_1, Float64(t_5 * t_0), 2.0) / Float64(3.0 * fma(fma(t_2, cos(y), Float64(cos(x) * t_4)), 0.5, 1.0)));
	elseif (x <= 0.0031)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * x), Float64(Float64(t_3 * 1.00390625) * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_3 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_2, cos(y), t_4), 1.0)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_5 * t_1) * t_0)) / fma(1.5, Float64(Float64(t_4 * cos(x)) - Float64(Float64(sqrt(5.0) - 3.0) * cos(y))), 3.0));
	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[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00029], N[(N[(t$95$1 * N[(t$95$5 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0031], N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(N[(t$95$3 * 1.00390625), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$4), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$5 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[(t$95$4 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e-4

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.3

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

   5. Applied rewrites 61.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

   7. Applied rewrites 61.3%

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

   8. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 61.3%

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

   if -2.9e-4 < x < 0.00309999999999999989

   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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} + \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

   if 0.00309999999999999989 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 61.1%

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

   10. Applied rewrites 61.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 0.0031:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right) \cdot \cos y, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 - \cos y\\ t_3 := \sqrt{5} - 1\\ t_4 := t\_3 \cdot \cos x\\ \mathbf{if}\;x \leq -0.00029:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, t\_4\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0031:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(t\_2 \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_2 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_3\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, t\_4 - \left(\sqrt{5} - 3\right) \cdot \cos y, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
           (- (cos x) (cos y)))))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- 1.0 (cos y)))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4 (* t_3 (cos x))))
   (if (<= x -0.00029)
     (/ t_0 (fma 1.5 (fma t_1 (cos y) t_4) 3.0))
     (if (<= x 0.0031)
       (*
        0.3333333333333333
        (/
         (fma
          (* (sqrt 2.0) x)
          (* (* t_2 1.00390625) (sin y))
          (fma (* -0.0625 (pow (sin y) 2.0)) (* t_2 (sqrt 2.0)) 2.0))
         (fma 0.5 (fma t_1 (cos y) t_3) 1.0)))
       (/ t_0 (fma 1.5 (- t_4 (* (- (sqrt 5.0) 3.0) (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 1.0 - cos(y);
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = t_3 * cos(x);
	double tmp;
	if (x <= -0.00029) {
		tmp = t_0 / fma(1.5, fma(t_1, cos(y), t_4), 3.0);
	} else if (x <= 0.0031) {
		tmp = 0.3333333333333333 * (fma((sqrt(2.0) * x), ((t_2 * 1.00390625) * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_2 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_1, cos(y), t_3), 1.0));
	} else {
		tmp = t_0 / fma(1.5, (t_4 - ((sqrt(5.0) - 3.0) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y))))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(1.0 - cos(y))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = Float64(t_3 * cos(x))
	tmp = 0.0
	if (x <= -0.00029)
		tmp = Float64(t_0 / fma(1.5, fma(t_1, cos(y), t_4), 3.0));
	elseif (x <= 0.0031)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * x), Float64(Float64(t_2 * 1.00390625) * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_2 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_1, cos(y), t_3), 1.0)));
	else
		tmp = Float64(t_0 / fma(1.5, Float64(t_4 - Float64(Float64(sqrt(5.0) - 3.0) * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[(N[(N[Sin[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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00029], N[(t$95$0 / N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$4), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0031], N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(N[(t$95$2 * 1.00390625), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(t$95$4 - N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e-4

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 61.3%

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

   10. Applied rewrites 61.3%

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

   if -2.9e-4 < x < 0.00309999999999999989

   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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} + \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

   if 0.00309999999999999989 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 61.1%

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

   10. Applied rewrites 61.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00029:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0031:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right) \cdot \cos y, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 1 - \cos y\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00029 \lor \neg \left(x \leq 0.0031\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos y, t\_0 \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(t\_1 \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_1 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 1.0 (cos y)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.00029) (not (<= x 0.0031)))
     (/
      (+
       2.0
       (*
        (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
        (- (cos x) (cos y))))
      (fma 1.5 (fma t_2 (cos y) (* t_0 (cos x))) 3.0))
     (*
      0.3333333333333333
      (/
       (fma
        (* (sqrt 2.0) x)
        (* (* t_1 1.00390625) (sin y))
        (fma (* -0.0625 (pow (sin y) 2.0)) (* t_1 (sqrt 2.0)) 2.0))
       (fma 0.5 (fma t_2 (cos y) t_0) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 1.0 - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.00029) || !(x <= 0.0031)) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / fma(1.5, fma(t_2, cos(y), (t_0 * cos(x))), 3.0);
	} else {
		tmp = 0.3333333333333333 * (fma((sqrt(2.0) * x), ((t_1 * 1.00390625) * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_1 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_2, cos(y), t_0), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(1.0 - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.00029) || !(x <= 0.0031))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / fma(1.5, fma(t_2, cos(y), Float64(t_0 * cos(x))), 3.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * x), Float64(Float64(t_1 * 1.00390625) * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_1 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)));
	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[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00029], N[Not[LessEqual[x, 0.0031]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(N[Sin[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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(N[(t$95$1 * 1.00390625), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e-4 or 0.00309999999999999989 < x

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 61.2%

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

   10. Applied rewrites 61.2%

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

   if -2.9e-4 < x < 0.00309999999999999989

   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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} + \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00029 \lor \neg \left(x \leq 0.0031\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 1 - \cos y\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00029 \lor \neg \left(x \leq 0.0031\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos y, \cos x \cdot t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(t\_1 \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_1 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 1.0 (cos y)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.00029) (not (<= x 0.0031)))
     (/
      (fma
       (- (sin y) (/ (sin x) 16.0))
       (* (* (sin x) (sqrt 2.0)) (- (cos x) (cos y)))
       2.0)
      (fma 1.5 (fma t_2 (cos y) (* (cos x) t_0)) 3.0))
     (*
      0.3333333333333333
      (/
       (fma
        (* (sqrt 2.0) x)
        (* (* t_1 1.00390625) (sin y))
        (fma (* -0.0625 (pow (sin y) 2.0)) (* t_1 (sqrt 2.0)) 2.0))
       (fma 0.5 (fma t_2 (cos y) t_0) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 1.0 - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.00029) || !(x <= 0.0031)) {
		tmp = fma((sin(y) - (sin(x) / 16.0)), ((sin(x) * sqrt(2.0)) * (cos(x) - cos(y))), 2.0) / fma(1.5, fma(t_2, cos(y), (cos(x) * t_0)), 3.0);
	} else {
		tmp = 0.3333333333333333 * (fma((sqrt(2.0) * x), ((t_1 * 1.00390625) * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_1 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_2, cos(y), t_0), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(1.0 - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.00029) || !(x <= 0.0031))
		tmp = Float64(fma(Float64(sin(y) - Float64(sin(x) / 16.0)), Float64(Float64(sin(x) * sqrt(2.0)) * Float64(cos(x) - cos(y))), 2.0) / fma(1.5, fma(t_2, cos(y), Float64(cos(x) * t_0)), 3.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * x), Float64(Float64(t_1 * 1.00390625) * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_1 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)));
	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[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00029], N[Not[LessEqual[x, 0.0031]], $MachinePrecision]], N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(N[(t$95$1 * 1.00390625), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e-4 or 0.00309999999999999989 < x

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.2

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

   5. Applied rewrites 61.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

   7. Applied rewrites 61.2%

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

   8. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 61.2%

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

   if -2.9e-4 < x < 0.00309999999999999989

   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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} + \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00029 \lor \neg \left(x \leq 0.0031\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 1 - \cos y\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.00029 \lor \neg \left(x \leq 0.0031\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(t\_1 \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_1 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- 1.0 (cos y)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (or (<= x -0.00029) (not (<= x 0.0031)))
     (/
      (fma
       (- (cos x) (cos y))
       (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))
       2.0)
      (fma 1.5 (fma t_2 (cos x) (* t_0 (cos y))) 3.0))
     (*
      0.3333333333333333
      (/
       (fma
        (* (sqrt 2.0) x)
        (* (* t_1 1.00390625) (sin y))
        (fma (* -0.0625 (pow (sin y) 2.0)) (* t_1 (sqrt 2.0)) 2.0))
       (fma 0.5 (fma t_0 (cos y) t_2) 1.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 1.0 - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if ((x <= -0.00029) || !(x <= 0.0031)) {
		tmp = fma((cos(x) - cos(y)), ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(t_2, cos(x), (t_0 * cos(y))), 3.0);
	} else {
		tmp = 0.3333333333333333 * (fma((sqrt(2.0) * x), ((t_1 * 1.00390625) * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_1 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_0, cos(y), t_2), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(1.0 - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((x <= -0.00029) || !(x <= 0.0031))
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(t_2, cos(x), Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * x), Float64(Float64(t_1 * 1.00390625) * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_1 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_0, cos(y), t_2), 1.0)));
	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[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.00029], N[Not[LessEqual[x, 0.0031]], $MachinePrecision]], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(N[(t$95$1 * 1.00390625), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e-4 or 0.00309999999999999989 < x

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 61.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 61.2

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

   10. Applied rewrites 61.2%

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

   if -2.9e-4 < x < 0.00309999999999999989

   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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} + \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00029 \lor \neg \left(x \leq 0.0031\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot x, \left(\left(1 - \cos y\right) \cdot 1.00390625\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{-5} \lor \neg \left(y \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot y, \left(t\_0 \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, t\_0 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, \frac{2}{3 + \sqrt{5}} + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0)) (t_1 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -2.6e-5) (not (<= y 7.5e-5)))
     (/
      (+
       2.0
       (* (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0)) (- (cos x) (cos y))))
      (fma 1.5 (fma t_1 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))
     (*
      0.3333333333333333
      (/
       (fma
        (* (sqrt 2.0) y)
        (* (* t_0 1.00390625) (sin x))
        (fma (* (pow (sin x) 2.0) -0.0625) (* t_0 (sqrt 2.0)) 2.0))
       (fma (* 0.5 (cos x)) t_1 (+ (/ 2.0 (+ 3.0 (sqrt 5.0))) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -2.6e-5) || !(y <= 7.5e-5)) {
		tmp = (2.0 + (((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)) * (cos(x) - cos(y)))) / fma(1.5, fma(t_1, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	} else {
		tmp = 0.3333333333333333 * (fma((sqrt(2.0) * y), ((t_0 * 1.00390625) * sin(x)), fma((pow(sin(x), 2.0) * -0.0625), (t_0 * sqrt(2.0)), 2.0)) / fma((0.5 * cos(x)), t_1, ((2.0 / (3.0 + sqrt(5.0))) + 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -2.6e-5) || !(y <= 7.5e-5))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / fma(1.5, fma(t_1, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * y), Float64(Float64(t_0 * 1.00390625) * sin(x)), fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(t_0 * sqrt(2.0)), 2.0)) / fma(Float64(0.5 * cos(x)), t_1, Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) + 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -2.6e-5], N[Not[LessEqual[y, 7.5e-5]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * y), $MachinePrecision] * N[(N[(t$95$0 * 1.00390625), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.59999999999999984e-5 or 7.49999999999999934e-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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 29.3%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-sqrt.f64 60.8

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

   11. Applied rewrites 60.8%

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

   if -2.59999999999999984e-5 < y < 7.49999999999999934e-5

   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

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{y \cdot \left(\sqrt{2} \cdot \left(\left(\sin x + \frac{1}{256} \cdot \sin x\right) \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} + \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   10. Applied rewrites 99.4%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot y, \left(\left(\cos x - 1\right) \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{3 + \sqrt{5}} + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-5} \lor \neg \left(y \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot y, \left(\left(\cos x - 1\right) \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{3 + \sqrt{5}} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{-5} \lor \neg \left(y \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot y, \left(t\_1 \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, t\_1 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_0\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -2.6e-5) (not (<= y 7.5e-5)))
     (/
      (+
       2.0
       (* (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0)) (- (cos x) (cos y))))
      (fma 1.5 (fma t_2 (cos x) (* t_0 (cos y))) 3.0))
     (*
      0.3333333333333333
      (/
       (fma
        (* (sqrt 2.0) y)
        (* (* t_1 1.00390625) (sin x))
        (fma (* -0.0625 (pow (sin x) 2.0)) (* t_1 (sqrt 2.0)) 2.0))
       (fma 0.5 (fma t_2 (cos x) t_0) 1.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -2.6e-5) || !(y <= 7.5e-5)) {
		tmp = (2.0 + (((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)) * (cos(x) - cos(y)))) / fma(1.5, fma(t_2, cos(x), (t_0 * cos(y))), 3.0);
	} else {
		tmp = 0.3333333333333333 * (fma((sqrt(2.0) * y), ((t_1 * 1.00390625) * sin(x)), fma((-0.0625 * pow(sin(x), 2.0)), (t_1 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_2, cos(x), t_0), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -2.6e-5) || !(y <= 7.5e-5))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / fma(1.5, fma(t_2, cos(x), Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * y), Float64(Float64(t_1 * 1.00390625) * sin(x)), fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(t_1 * sqrt(2.0)), 2.0)) / fma(0.5, fma(t_2, cos(x), t_0), 1.0)));
	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[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -2.6e-5], N[Not[LessEqual[y, 7.5e-5]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * y), $MachinePrecision] * N[(N[(t$95$1 * 1.00390625), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.59999999999999984e-5 or 7.49999999999999934e-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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 29.3%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-sqrt.f64 60.8

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

   11. Applied rewrites 60.8%

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

   if -2.59999999999999984e-5 < y < 7.49999999999999934e-5

   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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{y \cdot \left(\sqrt{2} \cdot \left(\left(\sin x + \frac{1}{256} \cdot \sin x\right) \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} + \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Applied rewrites 99.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot y, \left(\left(\cos x - 1\right) \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-5} \lor \neg \left(y \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot y, \left(\left(\cos x - 1\right) \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-5} \lor \neg \left(y \leq 2900000\right):\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right) \cdot t\_1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sin x \cdot \sqrt{2}\right) \cdot t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -1.15e-5) (not (<= y 2900000.0)))
     (/
      (+ 2.0 (* (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0)) t_1))
      (fma 1.5 (fma t_2 (cos x) (* t_0 (cos y))) 3.0))
     (/
      (fma (- (sin y) (/ (sin x) 16.0)) (* (* (sin x) (sqrt 2.0)) t_1) 2.0)
      (fma 1.5 (fma (cos x) t_2 t_0) 3.0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -1.15e-5) || !(y <= 2900000.0)) {
		tmp = (2.0 + (((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)) * t_1)) / fma(1.5, fma(t_2, cos(x), (t_0 * cos(y))), 3.0);
	} else {
		tmp = fma((sin(y) - (sin(x) / 16.0)), ((sin(x) * sqrt(2.0)) * t_1), 2.0) / fma(1.5, fma(cos(x), t_2, t_0), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -1.15e-5) || !(y <= 2900000.0))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)) * t_1)) / fma(1.5, fma(t_2, cos(x), Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(fma(Float64(sin(y) - Float64(sin(x) / 16.0)), Float64(Float64(sin(x) * sqrt(2.0)) * t_1), 2.0) / fma(1.5, fma(cos(x), t_2, t_0), 3.0));
	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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -1.15e-5], N[Not[LessEqual[y, 2900000.0]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15e-5 or 2.9e6 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 29.3%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-sqrt.f64 61.4

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

   11. Applied rewrites 61.4%

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

   if -1.15e-5 < y < 2.9e6

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 97.6

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

   5. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-+r- N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

   10. Applied rewrites 97.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-5} \lor \neg \left(y \leq 2900000\right):\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.00065 \lor \neg \left(y \leq 2900000\right):\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -0.00065) (not (<= y 2900000.0)))
     (/
      (+
       2.0
       (* (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0)) (- (cos x) (cos y))))
      (fma 1.5 (fma t_0 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))
     (/
      (fma (* (pow (sin x) 2.0) -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
      (*
       3.0
       (+
        (+ 1.0 (* (/ t_0 2.0) (cos x)))
        (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -0.00065) || !(y <= 2900000.0)) {
		tmp = (2.0 + (((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)) * (cos(x) - cos(y)))) / fma(1.5, fma(t_0, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	} else {
		tmp = fma((pow(sin(x), 2.0) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -0.00065) || !(y <= 2900000.0))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / fma(1.5, fma(t_0, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	else
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.00065], N[Not[LessEqual[y, 2900000.0]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4999999999999997e-4 or 2.9e6 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 29.3%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-sqrt.f64 61.4

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

   11. Applied rewrites 61.4%

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

   if -6.4999999999999997e-4 < y < 2.9e6

   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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 97.2

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

   5. Applied rewrites 97.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f64 97.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 97.3

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

   7. Applied rewrites 97.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00065 \lor \neg \left(y \leq 2900000\right):\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -2 \cdot 10^{-5} \lor \neg \left(x \leq 170000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_0, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)))
   (if (or (<= x -2e-5) (not (<= x 170000.0)))
     (/
      (fma (* (pow (sin x) 2.0) -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
      (*
       3.0
       (+
        (+ 1.0 (* (/ t_0 2.0) (cos x)))
        (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma (fma 0.5 t_0 1.0) 3.0 (* 1.5 (* (- 3.0 (sqrt 5.0)) (cos y))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double tmp;
	if ((x <= -2e-5) || !(x <= 170000.0)) {
		tmp = fma((pow(sin(x), 2.0) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, (1.5 * ((3.0 - sqrt(5.0)) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((x <= -2e-5) || !(x <= 170000.0))
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, Float64(1.5 * Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -2e-5], N[Not[LessEqual[x, 170000.0]], $MachinePrecision]], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000016e-5 or 1.7e5 < x

   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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 57.9

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 57.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{\sqrt{5} + 3}}{2} \cdot \cos y\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{\sqrt{5} + 3}}{2} \cdot \cos y\right)} \]

      7. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \color{blue}{5}}{\sqrt{5} + 3}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - 5}{\sqrt{5} + 3}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 3}}{2} \cdot \cos y\right)} \]

      10. lift-/.f64 58.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 58.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 58.0%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   if -2.00000000000000016e-5 < x < 1.7e5

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 61.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

   7. Applied rewrites 61.6%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right) \cdot 3 - \left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right) \cdot 3}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\mathsf{neg}\left(\frac{-3}{2}\right)\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 98.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-5} \lor \neg \left(x \leq 170000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := \sqrt{5} - 1\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(t\_0 \cdot -0.0625\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot t\_2, \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 170000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_2, 1\right), 3, 1.5 \cdot \left(t\_3 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, t\_1 \cdot t\_0, 0.6666666666666666\right)}{\frac{\mathsf{fma}\left(t\_3, \cos y, t\_2 \cdot \cos x\right)}{2} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -2e-5)
     (/
      (+ 2.0 (* (* t_0 -0.0625) t_1))
      (*
       3.0
       (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 2.0 (fma (* 0.5 t_2) (cos x) 1.0))))
     (if (<= x 170000.0)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (fma 0.5 t_2 1.0) 3.0 (* 1.5 (* t_3 (cos y)))))
       (/
        (fma -0.020833333333333332 (* t_1 t_0) 0.6666666666666666)
        (+ (/ (fma t_3 (cos y) (* t_2 (cos x))) 2.0) 1.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -2e-5) {
		tmp = (2.0 + ((t_0 * -0.0625) * t_1)) / (3.0 * fma((cos(y) / (3.0 + sqrt(5.0))), 2.0, fma((0.5 * t_2), cos(x), 1.0)));
	} else if (x <= 170000.0) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_2, 1.0), 3.0, (1.5 * (t_3 * cos(y))));
	} else {
		tmp = fma(-0.020833333333333332, (t_1 * t_0), 0.6666666666666666) / ((fma(t_3, cos(y), (t_2 * cos(x))) / 2.0) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -2e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_0 * -0.0625) * t_1)) / Float64(3.0 * fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 2.0, fma(Float64(0.5 * t_2), cos(x), 1.0))));
	elseif (x <= 170000.0)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_2, 1.0), 3.0, Float64(1.5 * Float64(t_3 * cos(y)))));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(t_1 * t_0), 0.6666666666666666) / Float64(Float64(fma(t_3, cos(y), Float64(t_2 * cos(x))) / 2.0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-5], N[(N[(2.0 + N[(N[(t$95$0 * -0.0625), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 170000.0], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(t$95$1 * t$95$0), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[(t$95$3 * N[Cos[y], $MachinePrecision] + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000016e-5

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      5. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

      12. lower-+.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} \cdot 2 + \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{\cos y}}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \color{blue}{\sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \frac{1}{2} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)} + 1\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x} + 1\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)}\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left({\sin x}^{2} \cdot \frac{-1}{16}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left({\sin x}^{2} \cdot \frac{-1}{16}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \left(\color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{2 + \left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \left(\left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 57.4

          \[\leadsto \frac{2 + \left({\sin x}^{2} \cdot -0.0625\right) \cdot \left(\left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

   10. Applied rewrites 57.4%

       \[\leadsto \frac{2 + \color{blue}{\left({\sin x}^{2} \cdot -0.0625\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)} \]

   if -2.00000000000000016e-5 < x < 1.7e5

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 61.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

   7. Applied rewrites 61.6%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right) \cdot 3 - \left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right) \cdot 3}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\mathsf{neg}\left(\frac{-3}{2}\right)\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 98.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]

   if 1.7e5 < 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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. flip-+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}^{2} - {\left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}^{2}}{1 + \frac{\cos x \cdot \left(\sqrt{5} - 1\right) - \cos y \cdot \left(3 - \sqrt{5}\right)}{2}}}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right), 2\right)}{3}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2\right)}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right) + \frac{1}{3} \cdot 2}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot \frac{-1}{16}\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)} + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{48}} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{48}\right), {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{48}}, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \color{blue}{{\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      16. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\color{blue}{\sin x}}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      17. metadata-eval 58.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \color{blue}{0.6666666666666666}\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_3 := \sqrt{5} - 1\\ t_4 := \mathsf{fma}\left(t\_0, \cos y, t\_3 \cdot \cos x\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot -0.0625, t\_2, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, t\_4, 1\right)}\\ \mathbf{elif}\;x \leq 170000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_3, 1\right), 3, 1.5 \cdot \left(t\_0 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, t\_2 \cdot t\_1, 0.6666666666666666\right)}{\frac{t\_4}{2} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4 (fma t_0 (cos y) (* t_3 (cos x)))))
   (if (<= x -2e-5)
     (/ (fma (* t_1 -0.0625) t_2 2.0) (* 3.0 (fma 0.5 t_4 1.0)))
     (if (<= x 170000.0)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (fma 0.5 t_3 1.0) 3.0 (* 1.5 (* t_0 (cos y)))))
       (/
        (fma -0.020833333333333332 (* t_2 t_1) 0.6666666666666666)
        (+ (/ t_4 2.0) 1.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = (cos(x) - 1.0) * sqrt(2.0);
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = fma(t_0, cos(y), (t_3 * cos(x)));
	double tmp;
	if (x <= -2e-5) {
		tmp = fma((t_1 * -0.0625), t_2, 2.0) / (3.0 * fma(0.5, t_4, 1.0));
	} else if (x <= 170000.0) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_3, 1.0), 3.0, (1.5 * (t_0 * cos(y))));
	} else {
		tmp = fma(-0.020833333333333332, (t_2 * t_1), 0.6666666666666666) / ((t_4 / 2.0) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = fma(t_0, cos(y), Float64(t_3 * cos(x)))
	tmp = 0.0
	if (x <= -2e-5)
		tmp = Float64(fma(Float64(t_1 * -0.0625), t_2, 2.0) / Float64(3.0 * fma(0.5, t_4, 1.0)));
	elseif (x <= 170000.0)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_3, 1.0), 3.0, Float64(1.5 * Float64(t_0 * cos(y)))));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(t_2 * t_1), 0.6666666666666666) / Float64(Float64(t_4 / 2.0) + 1.0));
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-5], N[(N[(N[(t$95$1 * -0.0625), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 170000.0], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$3 + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(t$95$2 * t$95$1), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$4 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000016e-5

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 57.4

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right), 1\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \cos x \cdot \left(\sqrt{5} - 1\right), 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right)}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right)} \cdot \cos x\right), 1\right)} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\color{blue}{\sqrt{5}} - 1\right) \cdot \cos x\right), 1\right)} \]

      14. lower-cos.f64 57.4

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \color{blue}{\cos x}\right), 1\right)} \]

   8. Applied rewrites 57.4%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}} \]

   if -2.00000000000000016e-5 < x < 1.7e5

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 61.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

   7. Applied rewrites 61.6%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right) \cdot 3 - \left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right) \cdot 3}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\mathsf{neg}\left(\frac{-3}{2}\right)\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 98.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]

   if 1.7e5 < 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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. flip-+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}^{2} - {\left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}^{2}}{1 + \frac{\cos x \cdot \left(\sqrt{5} - 1\right) - \cos y \cdot \left(3 - \sqrt{5}\right)}{2}}}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right), 2\right)}{3}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2\right)}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right) + \frac{1}{3} \cdot 2}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot \frac{-1}{16}\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)} + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{48}} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{48}\right), {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{48}}, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \color{blue}{{\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      16. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\color{blue}{\sin x}}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

      17. metadata-eval 58.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \color{blue}{0.6666666666666666}\right)}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-5} \lor \neg \left(x \leq 170000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0 \cdot \cos x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_0, 1\right), 3, 1.5 \cdot \left(t\_1 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -2e-5) (not (<= x 170000.0)))
     (/
      (fma (* (pow (sin x) 2.0) -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
      (* 3.0 (fma 0.5 (fma t_1 (cos y) (* t_0 (cos x))) 1.0)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma (fma 0.5 t_0 1.0) 3.0 (* 1.5 (* t_1 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -2e-5) || !(x <= 170000.0)) {
		tmp = fma((pow(sin(x), 2.0) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (3.0 * fma(0.5, fma(t_1, cos(y), (t_0 * cos(x))), 1.0));
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, (1.5 * (t_1 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -2e-5) || !(x <= 170000.0))
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(0.5, fma(t_1, cos(y), Float64(t_0 * cos(x))), 1.0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, Float64(1.5 * Float64(t_1 * cos(y)))));
	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]}, If[Or[LessEqual[x, -2e-5], N[Not[LessEqual[x, 170000.0]], $MachinePrecision]], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000016e-5 or 1.7e5 < x

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 57.9

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 57.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right), 1\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \cos x \cdot \left(\sqrt{5} - 1\right), 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right)}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right)} \cdot \cos x\right), 1\right)} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\color{blue}{\sqrt{5}} - 1\right) \cdot \cos x\right), 1\right)} \]

      14. lower-cos.f64 57.9

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \color{blue}{\cos x}\right), 1\right)} \]

   8. Applied rewrites 57.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}} \]

   if -2.00000000000000016e-5 < x < 1.7e5

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 61.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

   7. Applied rewrites 61.6%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right) \cdot 3 - \left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right) \cdot 3}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\mathsf{neg}\left(\frac{-3}{2}\right)\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 98.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-5} \lor \neg \left(x \leq 170000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{-6} \lor \neg \left(y \leq 380000\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, \frac{2}{3 + \sqrt{5}} + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -6.8e-6) (not (<= y 380000.0)))
     (/
      (+ 2.0 (* (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0))))
      (fma 1.5 (fma t_0 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))
     (/
      (fma
       -0.020833333333333332
       (* (* (- (cos x) 1.0) (sqrt 2.0)) (pow (sin x) 2.0))
       0.6666666666666666)
      (fma (* 0.5 (cos x)) t_0 (+ (/ 2.0 (+ 3.0 (sqrt 5.0))) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -6.8e-6) || !(y <= 380000.0)) {
		tmp = (2.0 + ((-0.0625 * pow(sin(y), 2.0)) * ((1.0 - cos(y)) * sqrt(2.0)))) / fma(1.5, fma(t_0, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	} else {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * pow(sin(x), 2.0)), 0.6666666666666666) / fma((0.5 * cos(x)), t_0, ((2.0 / (3.0 + sqrt(5.0))) + 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -6.8e-6) || !(y <= 380000.0))
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0)))) / fma(1.5, fma(t_0, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * (sin(x) ^ 2.0)), 0.6666666666666666) / fma(Float64(0.5 * cos(x)), t_0, Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) + 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -6.8e-6], N[Not[LessEqual[y, 380000.0]], $MachinePrecision]], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.80000000000000012e-6 or 3.8e5 < 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

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin x} \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 29.3

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 29.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 29.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

       4. lower-pow.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

       5. lower-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

       8. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

       9. lower-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

       10. lower-sqrt.f64 61.0

           \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   11. Applied rewrites 61.0%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -6.80000000000000012e-6 < y < 3.8e5

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      5. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

      12. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} \cdot 2 + \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{\cos y}}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \color{blue}{\sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \frac{1}{2} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)} + 1\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x} + 1\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)}\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   10. Applied rewrites 97.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{3 + \sqrt{5}} + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-6} \lor \neg \left(y \leq 380000\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{3 + \sqrt{5}} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ \mathbf{if}\;x \leq -2 \cdot 10^{-5} \lor \neg \left(x \leq 170000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_0, 1\right), 3, 1.5 \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (* (- 3.0 (sqrt 5.0)) (cos y))))
   (if (or (<= x -2e-5) (not (<= x 170000.0)))
     (/
      (fma (* (pow (sin x) 2.0) -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
      (fma 1.5 (fma t_0 (cos x) t_1) 3.0))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma (fma 0.5 t_0 1.0) 3.0 (* 1.5 t_1))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (3.0 - sqrt(5.0)) * cos(y);
	double tmp;
	if ((x <= -2e-5) || !(x <= 170000.0)) {
		tmp = fma((pow(sin(x), 2.0) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, (1.5 * t_1));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	tmp = 0.0
	if ((x <= -2e-5) || !(x <= 170000.0))
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, Float64(1.5 * t_1)));
	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[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -2e-5], N[Not[LessEqual[x, 170000.0]], $MachinePrecision]], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000016e-5 or 1.7e5 < x

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 57.9

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 57.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 57.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   if -2.00000000000000016e-5 < x < 1.7e5

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 61.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

   7. Applied rewrites 61.6%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right) \cdot 3 - \left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right) \cdot 3}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\mathsf{neg}\left(\frac{-3}{2}\right)\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 98.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-5} \lor \neg \left(x \leq 170000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-5} \lor \neg \left(x \leq 0.000155\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, \frac{2}{3 + \sqrt{5}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_0, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)))
   (if (or (<= x -5.8e-5) (not (<= x 0.000155)))
     (/
      (fma
       -0.020833333333333332
       (* (* (- (cos x) 1.0) (sqrt 2.0)) (pow (sin x) 2.0))
       0.6666666666666666)
      (fma (* 0.5 (cos x)) t_0 (+ (/ 2.0 (+ 3.0 (sqrt 5.0))) 1.0)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma (fma 0.5 t_0 1.0) 3.0 (* 1.5 (* (- 3.0 (sqrt 5.0)) (cos y))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double tmp;
	if ((x <= -5.8e-5) || !(x <= 0.000155)) {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * pow(sin(x), 2.0)), 0.6666666666666666) / fma((0.5 * cos(x)), t_0, ((2.0 / (3.0 + sqrt(5.0))) + 1.0));
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, (1.5 * ((3.0 - sqrt(5.0)) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((x <= -5.8e-5) || !(x <= 0.000155))
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * (sin(x) ^ 2.0)), 0.6666666666666666) / fma(Float64(0.5 * cos(x)), t_0, Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) + 1.0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, Float64(1.5 * Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -5.8e-5], N[Not[LessEqual[x, 0.000155]], $MachinePrecision]], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e-5 or 1.55e-4 < x

   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. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      5. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

      12. lower-+.f64 99.2

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} \cdot 2 + \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{\cos y}}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \color{blue}{\sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \frac{1}{2} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)} + 1\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x} + 1\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(0.5 \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   10. Applied rewrites 56.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{3 + \sqrt{5}} + 1\right)}} \]

   if -5.8e-5 < x < 1.55e-4

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 62.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

   7. Applied rewrites 62.0%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right) \cdot 3 - \left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right) \cdot 3}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\mathsf{neg}\left(\frac{-3}{2}\right)\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-5} \lor \neg \left(x \leq 0.000155\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{3 + \sqrt{5}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ t_3 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot -0.0625, t\_3, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_0, 1\right), 3, 1.5 \cdot \left(t\_1 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, t\_3 \cdot t\_2, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, t\_1\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (pow (sin x) 2.0))
        (t_3 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -5.8e-5)
     (/ (fma (* t_2 -0.0625) t_3 2.0) (fma 1.5 (fma t_0 (cos x) t_1) 3.0))
     (if (<= x 0.000155)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (fma 0.5 t_0 1.0) 3.0 (* 1.5 (* t_1 (cos y)))))
       (/
        (fma -0.020833333333333332 (* t_3 t_2) 0.6666666666666666)
        (fma (fma (cos x) t_0 t_1) 0.5 1.0))))))

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 = pow(sin(x), 2.0);
	double t_3 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -5.8e-5) {
		tmp = fma((t_2 * -0.0625), t_3, 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	} else if (x <= 0.000155) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, (1.5 * (t_1 * cos(y))));
	} else {
		tmp = fma(-0.020833333333333332, (t_3 * t_2), 0.6666666666666666) / fma(fma(cos(x), t_0, t_1), 0.5, 1.0);
	}
	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 = sin(x) ^ 2.0
	t_3 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -5.8e-5)
		tmp = Float64(fma(Float64(t_2 * -0.0625), t_3, 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0));
	elseif (x <= 0.000155)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, Float64(1.5 * Float64(t_1 * cos(y)))));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(t_3 * t_2), 0.6666666666666666) / fma(fma(cos(x), t_0, t_1), 0.5, 1.0));
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-5], N[(N[(N[(t$95$2 * -0.0625), $MachinePrecision] * t$95$3 + 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, 0.000155], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(t$95$3 * t$95$2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8e-5

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 57.4

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x} + \left(3 - \sqrt{5}\right), 3\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right)}, 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\sqrt{5} - 1}, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\sqrt{5}} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{\cos x}, 3 - \sqrt{5}\right), 3\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      14. lower-sqrt.f64 56.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if -5.8e-5 < x < 1.55e-4

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 62.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right) \cdot 3}} \]

   7. Applied rewrites 62.0%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right) \cdot 3 - \left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right) \cdot 3}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \frac{-3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\mathsf{neg}\left(\frac{-3}{2}\right)\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, 1.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]

   if 1.55e-4 < 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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. flip-+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}^{2} - {\left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}^{2}}{1 + \frac{\cos x \cdot \left(\sqrt{5} - 1\right) - \cos y \cdot \left(3 - \sqrt{5}\right)}{2}}}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right), 2\right)}{3}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right)} - \sqrt{5}\right)} \]

      3. associate-+r- N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   9. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ t_3 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot -0.0625, t\_3, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, t\_3 \cdot t\_2, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, t\_1\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (pow (sin x) 2.0))
        (t_3 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -5.8e-5)
     (/ (fma (* t_2 -0.0625) t_3 2.0) (fma 1.5 (fma t_0 (cos x) t_1) 3.0))
     (if (<= x 0.000155)
       (*
        (/
         (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
        0.3333333333333333)
       (/
        (fma -0.020833333333333332 (* t_3 t_2) 0.6666666666666666)
        (fma (fma (cos x) t_0 t_1) 0.5 1.0))))))

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 = pow(sin(x), 2.0);
	double t_3 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -5.8e-5) {
		tmp = fma((t_2 * -0.0625), t_3, 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	} else if (x <= 0.000155) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma(-0.020833333333333332, (t_3 * t_2), 0.6666666666666666) / fma(fma(cos(x), t_0, t_1), 0.5, 1.0);
	}
	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 = sin(x) ^ 2.0
	t_3 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -5.8e-5)
		tmp = Float64(fma(Float64(t_2 * -0.0625), t_3, 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0));
	elseif (x <= 0.000155)
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(t_3 * t_2), 0.6666666666666666) / fma(fma(cos(x), t_0, t_1), 0.5, 1.0));
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-5], N[(N[(N[(t$95$2 * -0.0625), $MachinePrecision] * t$95$3 + 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, 0.000155], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(t$95$3 * t$95$2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8e-5

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 57.4

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x} + \left(3 - \sqrt{5}\right), 3\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right)}, 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\sqrt{5} - 1}, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\sqrt{5}} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{\cos x}, 3 - \sqrt{5}\right), 3\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      14. lower-sqrt.f64 56.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if -5.8e-5 < x < 1.55e-4

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 62.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]

   if 1.55e-4 < 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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. flip-+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}^{2} - {\left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}^{2}}{1 + \frac{\cos x \cdot \left(\sqrt{5} - 1\right) - \cos y \cdot \left(3 - \sqrt{5}\right)}{2}}}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right), 2\right)}{3}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right)} - \sqrt{5}\right)} \]

      3. associate-+r- N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   9. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-5} \lor \neg \left(x \leq 0.000155\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, t\_1\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -5.8e-5) (not (<= x 0.000155)))
     (/
      (fma
       -0.020833333333333332
       (* (* (- (cos x) 1.0) (sqrt 2.0)) (pow (sin x) 2.0))
       0.6666666666666666)
      (fma (fma (cos x) t_0 t_1) 0.5 1.0))
     (/
      (fma
       -0.020833333333333332
       (* (* (- 1.0 (cos y)) (sqrt 2.0)) (pow (sin y) 2.0))
       0.6666666666666666)
      (fma (fma t_1 (cos y) t_0) 0.5 1.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -5.8e-5) || !(x <= 0.000155)) {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * pow(sin(x), 2.0)), 0.6666666666666666) / fma(fma(cos(x), t_0, t_1), 0.5, 1.0);
	} else {
		tmp = fma(-0.020833333333333332, (((1.0 - cos(y)) * sqrt(2.0)) * pow(sin(y), 2.0)), 0.6666666666666666) / fma(fma(t_1, cos(y), t_0), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -5.8e-5) || !(x <= 0.000155))
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * (sin(x) ^ 2.0)), 0.6666666666666666) / fma(fma(cos(x), t_0, t_1), 0.5, 1.0));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * (sin(y) ^ 2.0)), 0.6666666666666666) / fma(fma(t_1, cos(y), t_0), 0.5, 1.0));
	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]}, If[Or[LessEqual[x, -5.8e-5], N[Not[LessEqual[x, 0.000155]], $MachinePrecision]], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e-5 or 1.55e-4 < x

   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. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. flip-+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}^{2} - {\left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}^{2}}{1 + \frac{\cos x \cdot \left(\sqrt{5} - 1\right) - \cos y \cdot \left(3 - \sqrt{5}\right)}{2}}}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right), 2\right)}{3}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right)} - \sqrt{5}\right)} \]

      3. associate-+r- N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   9. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   if -5.8e-5 < x < 1.55e-4

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. flip-+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}^{2} - {\left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}^{2}}{1 + \frac{\cos x \cdot \left(\sqrt{5} - 1\right) - \cos y \cdot \left(3 - \sqrt{5}\right)}{2}}}} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right), 2\right)}{3}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]

   9. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-5} \lor \neg \left(x \leq 0.000155\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ t_3 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot -0.0625, t\_3, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_0\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, t\_3 \cdot t\_2, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, t\_1\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (pow (sin x) 2.0))
        (t_3 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -5.8e-5)
     (/ (fma (* t_2 -0.0625) t_3 2.0) (fma 1.5 (fma t_0 (cos x) t_1) 3.0))
     (if (<= x 0.000155)
       (/
        (fma
         -0.020833333333333332
         (* (* (- 1.0 (cos y)) (sqrt 2.0)) (pow (sin y) 2.0))
         0.6666666666666666)
        (fma (fma t_1 (cos y) t_0) 0.5 1.0))
       (/
        (fma -0.020833333333333332 (* t_3 t_2) 0.6666666666666666)
        (fma (fma (cos x) t_0 t_1) 0.5 1.0))))))

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 = pow(sin(x), 2.0);
	double t_3 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -5.8e-5) {
		tmp = fma((t_2 * -0.0625), t_3, 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	} else if (x <= 0.000155) {
		tmp = fma(-0.020833333333333332, (((1.0 - cos(y)) * sqrt(2.0)) * pow(sin(y), 2.0)), 0.6666666666666666) / fma(fma(t_1, cos(y), t_0), 0.5, 1.0);
	} else {
		tmp = fma(-0.020833333333333332, (t_3 * t_2), 0.6666666666666666) / fma(fma(cos(x), t_0, t_1), 0.5, 1.0);
	}
	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 = sin(x) ^ 2.0
	t_3 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -5.8e-5)
		tmp = Float64(fma(Float64(t_2 * -0.0625), t_3, 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0));
	elseif (x <= 0.000155)
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * (sin(y) ^ 2.0)), 0.6666666666666666) / fma(fma(t_1, cos(y), t_0), 0.5, 1.0));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(t_3 * t_2), 0.6666666666666666) / fma(fma(cos(x), t_0, t_1), 0.5, 1.0));
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-5], N[(N[(N[(t$95$2 * -0.0625), $MachinePrecision] * t$95$3 + 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, 0.000155], N[(N[(-0.020833333333333332 * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(t$95$3 * t$95$2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8e-5

   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. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 57.4

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 57.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x} + \left(3 - \sqrt{5}\right), 3\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right)}, 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\sqrt{5} - 1}, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\sqrt{5}} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{\cos x}, 3 - \sqrt{5}\right), 3\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      14. lower-sqrt.f64 56.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if -5.8e-5 < x < 1.55e-4

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. flip-+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}^{2} - {\left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}^{2}}{1 + \frac{\cos x \cdot \left(\sqrt{5} - 1\right) - \cos y \cdot \left(3 - \sqrt{5}\right)}{2}}}} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right), 2\right)}{3}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]

   9. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)}} \]

   if 1.55e-4 < 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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. flip-+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}^{2} - {\left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}^{2}}{1 + \frac{\cos x \cdot \left(\sqrt{5} - 1\right) - \cos y \cdot \left(3 - \sqrt{5}\right)}{2}}}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right), 2\right)}{3}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right)} - \sqrt{5}\right)} \]

      3. associate-+r- N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   9. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 59.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   -0.020833333333333332
   (* (* (- 1.0 (cos y)) (sqrt 2.0)) (pow (sin y) 2.0))
   0.6666666666666666)
  (fma (fma (- 3.0 (sqrt 5.0)) (cos y) (- (sqrt 5.0) 1.0)) 0.5 1.0)))

C

double code(double x, double y) {
	return fma(-0.020833333333333332, (((1.0 - cos(y)) * sqrt(2.0)) * pow(sin(y), 2.0)), 0.6666666666666666) / fma(fma((3.0 - sqrt(5.0)), cos(y), (sqrt(5.0) - 1.0)), 0.5, 1.0);
}

Julia

function code(x, y)
	return Float64(fma(-0.020833333333333332, Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * (sin(y) ^ 2.0)), 0.6666666666666666) / fma(fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(sqrt(5.0) - 1.0)), 0.5, 1.0))
end

Wolfram

code[x_, y_] := N[(N[(-0.020833333333333332 * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. flip-+ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \frac{3 - \sqrt{5}}{2} \cdot \cos y}}} \]

4. Applied rewrites 98.9%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}^{2} - {\left(\frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}^{2}}{1 + \frac{\cos x \cdot \left(\sqrt{5} - 1\right) - \cos y \cdot \left(3 - \sqrt{5}\right)}{2}}}} \]

6. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right), 2\right)}{3}}{\frac{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)}{2} + 1}} \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]

9. Applied rewrites 59.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)}} \]
10. Add Preprocessing

Alternative 31: 45.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 3, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   (* 0.5 (fma (- 3.0 (sqrt 5.0)) (cos y) (* (cos x) (- (sqrt 5.0) 1.0))))
   3.0
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma((0.5 * fma((3.0 - sqrt(5.0)), cos(y), (cos(x) * (sqrt(5.0) - 1.0)))), 3.0, 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(Float64(0.5 * fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(cos(x) * Float64(sqrt(5.0) - 1.0)))), 3.0, 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 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. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. lower-sqrt.f64 59.8

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 59.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 3\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 3\right)} \]

   13. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

   14. lower-sqrt.f64 41.4

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

8. Applied rewrites 41.4%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 41.4%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

12. Taylor expanded in x 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-rgt-in N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3 + 1 \cdot 3}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{2}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3 + \color{blue}{3}} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right), 3, 3\right)}} \]

14. Applied rewrites 43.9%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 3, 3\right)}} \]
15. Add Preprocessing

Alternative 32: 42.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (* (fma 0.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 1.0) 3.0)))

C

double code(double x, double y) {
	return 2.0 / (fma(0.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 1.0) * 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(fma(0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 1.0) * 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. lower-sqrt.f64 59.8

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 59.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 3\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 3\right)} \]

   13. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

   14. lower-sqrt.f64 41.4

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

8. Applied rewrites 41.4%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 41.4%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

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

       \[\leadsto \frac{2}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3}} \]

    3. +-commutative N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)} \cdot 3} \]

    4. distribute-lft-out N/A

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 1\right) \cdot 3} \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 1\right)} \cdot 3} \]

    6. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 1\right) \cdot 3} \]

    7. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right) \cdot 3} \]

    8. lower--.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 1\right) \cdot 3} \]

    9. lower-sqrt.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 1\right) \cdot 3} \]

    10. lower--.f64 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 1\right) \cdot 3} \]

    11. lower-sqrt.f64 41.6

        \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 1\right) \cdot 3} \]

14. Applied rewrites 41.6%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right) \cdot 3}} \]
15. Add Preprocessing

Alternative 33: 42.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (- 3.0 (sqrt 5.0)) (cos y) (- (sqrt 5.0) 1.0)) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma((3.0 - sqrt(5.0)), cos(y), (sqrt(5.0) - 1.0)), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(sqrt(5.0) - 1.0)), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $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. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. lower-sqrt.f64 59.8

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 59.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 3\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 3\right)} \]

   13. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

   14. lower-sqrt.f64 41.4

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

8. Applied rewrites 41.4%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 41.4%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(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))))))