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

Percentage Accurate: 99.3% → 99.4%

Time: 21.4s

Alternatives: 32

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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \end{array} \]

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))))
  (fma
   1.5
   (fma (cos x) (- (sqrt 5.0) 1.0) (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
   3.0)))

C

double code(double x, double y) {
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
}

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $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 99.2%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $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 99.2%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 99.4%

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

Alternative 5: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 1 - \cos y\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.005 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, t\_1 \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_1 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, t\_2\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

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 -0.005) (not (<= x 3.6e-12)))
     (/
      (+
       2.0
       (*
        (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
        (- (cos x) (cos y))))
      (fma 1.5 (fma (cos x) t_2 (* t_0 (cos y))) 3.0))
     (/
      (fma
       (* (* (sqrt 2.0) x) 1.00390625)
       (* t_1 (sin y))
       (fma (* -0.0625 (pow (sin y) 2.0)) (* t_1 (sqrt 2.0)) 2.0))
      (fma
       (fma 0.5 (fma (cos y) t_0 t_2) 1.0)
       3.0
       (* (fma -0.75 (sqrt 5.0) 0.75) (* x x)))))))

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 <= -0.005) || !(x <= 3.6e-12)) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / fma(1.5, fma(cos(x), t_2, (t_0 * cos(y))), 3.0);
	} else {
		tmp = fma(((sqrt(2.0) * x) * 1.00390625), (t_1 * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_1 * sqrt(2.0)), 2.0)) / fma(fma(0.5, fma(cos(y), t_0, t_2), 1.0), 3.0, (fma(-0.75, sqrt(5.0), 0.75) * (x * x)));
	}
	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 <= -0.005) || !(x <= 3.6e-12))
		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(cos(x), t_2, Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(fma(Float64(Float64(sqrt(2.0) * x) * 1.00390625), Float64(t_1 * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_1 * sqrt(2.0)), 2.0)) / fma(fma(0.5, fma(cos(y), t_0, t_2), 1.0), 3.0, Float64(fma(-0.75, sqrt(5.0), 0.75) * Float64(x * x))));
	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, -0.005], N[Not[LessEqual[x, 3.6e-12]], $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[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * 1.00390625), $MachinePrecision] * N[(t$95$1 * 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[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0050000000000000001 or 3.6e-12 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 66.0

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

   8. Applied rewrites 66.0%

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

   if -0.0050000000000000001 < x < 3.6e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       3. associate-+l+ N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{\left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, \left(1 - \cos y\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.005 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, \left(1 - \cos y\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 1 - \cos y\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.005 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), \sqrt{2} \cdot \sin x, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0 \cdot \cos x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, t\_1 \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_1 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, t\_0\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

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 -0.005) (not (<= x 3.6e-12)))
     (/
      (fma
       (* (fma -0.0625 (sin x) (sin y)) (- (cos x) (cos y)))
       (* (sqrt 2.0) (sin x))
       2.0)
      (* 3.0 (fma 0.5 (fma t_2 (cos y) (* t_0 (cos x))) 1.0)))
     (/
      (fma
       (* (* (sqrt 2.0) x) 1.00390625)
       (* t_1 (sin y))
       (fma (* -0.0625 (pow (sin y) 2.0)) (* t_1 (sqrt 2.0)) 2.0))
      (fma
       (fma 0.5 (fma (cos y) t_2 t_0) 1.0)
       3.0
       (* (fma -0.75 (sqrt 5.0) 0.75) (* x x)))))))

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 <= -0.005) || !(x <= 3.6e-12)) {
		tmp = fma((fma(-0.0625, sin(x), sin(y)) * (cos(x) - cos(y))), (sqrt(2.0) * sin(x)), 2.0) / (3.0 * fma(0.5, fma(t_2, cos(y), (t_0 * cos(x))), 1.0));
	} else {
		tmp = fma(((sqrt(2.0) * x) * 1.00390625), (t_1 * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_1 * sqrt(2.0)), 2.0)) / fma(fma(0.5, fma(cos(y), t_2, t_0), 1.0), 3.0, (fma(-0.75, sqrt(5.0), 0.75) * (x * x)));
	}
	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 <= -0.005) || !(x <= 3.6e-12))
		tmp = Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * Float64(cos(x) - cos(y))), Float64(sqrt(2.0) * sin(x)), 2.0) / Float64(3.0 * fma(0.5, fma(t_2, cos(y), Float64(t_0 * cos(x))), 1.0)));
	else
		tmp = Float64(fma(Float64(Float64(sqrt(2.0) * x) * 1.00390625), Float64(t_1 * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_1 * sqrt(2.0)), 2.0)) / fma(fma(0.5, fma(cos(y), t_2, t_0), 1.0), 3.0, Float64(fma(-0.75, sqrt(5.0), 0.75) * Float64(x * x))));
	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, -0.005], N[Not[LessEqual[x, 3.6e-12]], $MachinePrecision]], N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * 1.00390625), $MachinePrecision] * N[(t$95$1 * 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[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0050000000000000001 or 3.6e-12 < x

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 98.9

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.9

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. sub-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 65.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

   12. Applied rewrites 65.9%

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

   if -0.0050000000000000001 < x < 3.6e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       3. associate-+l+ N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{\left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, \left(1 - \cos y\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.005 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), \sqrt{2} \cdot \sin x, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, \left(1 - \cos y\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := 1 - \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \cos x - \cos y\\ t_4 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_5 := 3 - \sqrt{5}\\ t_6 := 3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_5, \cos y, t\_2 \cdot \cos x\right), 1\right)\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_4 \cdot t\_3, t\_0, 2\right)}{t\_6}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, t\_1 \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_1 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_5, t\_2\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, t\_4 \cdot t\_0, 2\right)}{t\_6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sin x)))
        (t_1 (- 1.0 (cos y)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- (cos x) (cos y)))
        (t_4 (fma -0.0625 (sin x) (sin y)))
        (t_5 (- 3.0 (sqrt 5.0)))
        (t_6 (* 3.0 (fma 0.5 (fma t_5 (cos y) (* t_2 (cos x))) 1.0))))
   (if (<= x -0.005)
     (/ (fma (* t_4 t_3) t_0 2.0) t_6)
     (if (<= x 3.6e-12)
       (/
        (fma
         (* (* (sqrt 2.0) x) 1.00390625)
         (* t_1 (sin y))
         (fma (* -0.0625 (pow (sin y) 2.0)) (* t_1 (sqrt 2.0)) 2.0))
        (fma
         (fma 0.5 (fma (cos y) t_5 t_2) 1.0)
         3.0
         (* (fma -0.75 (sqrt 5.0) 0.75) (* x x))))
       (/ (fma t_3 (* t_4 t_0) 2.0) t_6)))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * sin(x);
	double t_1 = 1.0 - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = cos(x) - cos(y);
	double t_4 = fma(-0.0625, sin(x), sin(y));
	double t_5 = 3.0 - sqrt(5.0);
	double t_6 = 3.0 * fma(0.5, fma(t_5, cos(y), (t_2 * cos(x))), 1.0);
	double tmp;
	if (x <= -0.005) {
		tmp = fma((t_4 * t_3), t_0, 2.0) / t_6;
	} else if (x <= 3.6e-12) {
		tmp = fma(((sqrt(2.0) * x) * 1.00390625), (t_1 * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_1 * sqrt(2.0)), 2.0)) / fma(fma(0.5, fma(cos(y), t_5, t_2), 1.0), 3.0, (fma(-0.75, sqrt(5.0), 0.75) * (x * x)));
	} else {
		tmp = fma(t_3, (t_4 * t_0), 2.0) / t_6;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * sin(x))
	t_1 = Float64(1.0 - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(cos(x) - cos(y))
	t_4 = fma(-0.0625, sin(x), sin(y))
	t_5 = Float64(3.0 - sqrt(5.0))
	t_6 = Float64(3.0 * fma(0.5, fma(t_5, cos(y), Float64(t_2 * cos(x))), 1.0))
	tmp = 0.0
	if (x <= -0.005)
		tmp = Float64(fma(Float64(t_4 * t_3), t_0, 2.0) / t_6);
	elseif (x <= 3.6e-12)
		tmp = Float64(fma(Float64(Float64(sqrt(2.0) * x) * 1.00390625), Float64(t_1 * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_1 * sqrt(2.0)), 2.0)) / fma(fma(0.5, fma(cos(y), t_5, t_2), 1.0), 3.0, Float64(fma(-0.75, sqrt(5.0), 0.75) * Float64(x * x))));
	else
		tmp = Float64(fma(t_3, Float64(t_4 * t_0), 2.0) / t_6);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(3.0 * N[(0.5 * N[(t$95$5 * N[Cos[y], $MachinePrecision] + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.005], N[(N[(N[(t$95$4 * t$95$3), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[x, 3.6e-12], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * 1.00390625), $MachinePrecision] * N[(t$95$1 * 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[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$5 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 * N[(t$95$4 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$6), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0050000000000000001

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 98.9

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.7

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. sub-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 63.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

   12. Applied rewrites 63.7%

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

   if -0.0050000000000000001 < x < 3.6e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       3. associate-+l+ N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{\left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, \left(1 - \cos y\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)} \]

   if 3.6e-12 < x

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.0

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 99.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.0

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 68.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. sub-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 68.4%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 68.4

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

   12. Applied rewrites 68.4%

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

Alternative 8: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)\\ t_2 := 1 - \cos y\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot t\_3, 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, t\_2 \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_2 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_3, t\_0\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_0 \cdot \cos x\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (+
          2.0
          (*
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
           (- (cos x) 1.0))))
        (t_2 (- 1.0 (cos y)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.005)
     (/
      t_1
      (*
       3.0
       (+
        (fma (cos y) (* 0.5 t_3) 1.0)
        (* (cos x) (fma (sqrt 5.0) 0.5 -0.5)))))
     (if (<= x 3.6e-12)
       (/
        (fma
         (* (* (sqrt 2.0) x) 1.00390625)
         (* t_2 (sin y))
         (fma (* -0.0625 (pow (sin y) 2.0)) (* t_2 (sqrt 2.0)) 2.0))
        (fma
         (fma 0.5 (fma (cos y) t_3 t_0) 1.0)
         3.0
         (* (fma -0.75 (sqrt 5.0) 0.75) (* x x))))
       (/ t_1 (* 3.0 (fma 0.5 (fma t_3 (cos y) (* t_0 (cos x))) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0));
	double t_2 = 1.0 - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.005) {
		tmp = t_1 / (3.0 * (fma(cos(y), (0.5 * t_3), 1.0) + (cos(x) * fma(sqrt(5.0), 0.5, -0.5))));
	} else if (x <= 3.6e-12) {
		tmp = fma(((sqrt(2.0) * x) * 1.00390625), (t_2 * sin(y)), fma((-0.0625 * pow(sin(y), 2.0)), (t_2 * sqrt(2.0)), 2.0)) / fma(fma(0.5, fma(cos(y), t_3, t_0), 1.0), 3.0, (fma(-0.75, sqrt(5.0), 0.75) * (x * x)));
	} else {
		tmp = t_1 / (3.0 * fma(0.5, fma(t_3, cos(y), (t_0 * cos(x))), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0)))
	t_2 = Float64(1.0 - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.005)
		tmp = Float64(t_1 / Float64(3.0 * Float64(fma(cos(y), Float64(0.5 * t_3), 1.0) + Float64(cos(x) * fma(sqrt(5.0), 0.5, -0.5)))));
	elseif (x <= 3.6e-12)
		tmp = Float64(fma(Float64(Float64(sqrt(2.0) * x) * 1.00390625), Float64(t_2 * sin(y)), fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(t_2 * sqrt(2.0)), 2.0)) / fma(fma(0.5, fma(cos(y), t_3, t_0), 1.0), 3.0, Float64(fma(-0.75, sqrt(5.0), 0.75) * Float64(x * x))));
	else
		tmp = Float64(t_1 / Float64(3.0 * fma(0.5, fma(t_3, cos(y), Float64(t_0 * cos(x))), 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[(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] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.005], N[(t$95$1 / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-12], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * 1.00390625), $MachinePrecision] * N[(t$95$2 * 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[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$3 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0050000000000000001

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 98.9

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.7

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 60.6

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

   10. Applied rewrites 60.6%

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

   if -0.0050000000000000001 < x < 3.6e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       3. associate-+l+ N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{\left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot \left(x \cdot x\right)\right)} \]

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot 1.00390625, \left(1 - \cos y\right) \cdot \sin y, \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)} \]

   if 3.6e-12 < x

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.0

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 99.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.0

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 68.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. sub-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 68.4%

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

   11. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-cos.f64 66.2

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

   13. Applied rewrites 66.2%

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

Alternative 9: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot t\_2, 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{t\_2}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0 \cdot \cos x\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (+
          2.0
          (*
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
           (- (cos x) 1.0))))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.005)
     (/
      t_1
      (*
       3.0
       (+
        (fma (cos y) (* 0.5 t_2) 1.0)
        (* (cos x) (fma (sqrt 5.0) 0.5 -0.5)))))
     (if (<= x 3.6e-12)
       (/
        (fma
         (- 1.0 (cos y))
         (*
          (sqrt 2.0)
          (fma (* 1.00390625 (sin y)) x (* (pow (sin y) 2.0) -0.0625)))
         2.0)
        (* 3.0 (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) (* (/ t_2 2.0) (cos y)))))
       (/ t_1 (* 3.0 (fma 0.5 (fma t_2 (cos y) (* t_0 (cos x))) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0));
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.005) {
		tmp = t_1 / (3.0 * (fma(cos(y), (0.5 * t_2), 1.0) + (cos(x) * fma(sqrt(5.0), 0.5, -0.5))));
	} else if (x <= 3.6e-12) {
		tmp = fma((1.0 - cos(y)), (sqrt(2.0) * fma((1.00390625 * sin(y)), x, (pow(sin(y), 2.0) * -0.0625))), 2.0) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + ((t_2 / 2.0) * cos(y))));
	} else {
		tmp = t_1 / (3.0 * fma(0.5, fma(t_2, cos(y), (t_0 * cos(x))), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0)))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.005)
		tmp = Float64(t_1 / Float64(3.0 * Float64(fma(cos(y), Float64(0.5 * t_2), 1.0) + Float64(cos(x) * fma(sqrt(5.0), 0.5, -0.5)))));
	elseif (x <= 3.6e-12)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(sqrt(2.0) * fma(Float64(1.00390625 * sin(y)), x, Float64((sin(y) ^ 2.0) * -0.0625))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(t_2 / 2.0) * cos(y)))));
	else
		tmp = Float64(t_1 / Float64(3.0 * fma(0.5, fma(t_2, cos(y), Float64(t_0 * cos(x))), 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[(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] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.005], N[(t$95$1 / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-12], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $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[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0050000000000000001

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 98.9

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.7

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 60.6

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

   10. Applied rewrites 60.6%

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

   if -0.0050000000000000001 < x < 3.6e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   if 3.6e-12 < x

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.0

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 99.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.0

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 68.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. sub-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 68.4%

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

   11. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-cos.f64 66.2

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

   13. Applied rewrites 66.2%

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

Alternative 10: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(0.5 \cdot t\_0, \cos y, 1\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -5.2) (not (<= x 3.6e-12)))
     (/
      (+
       2.0
       (*
        (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
        (- (cos x) 1.0)))
      (* 3.0 (fma 0.5 (fma t_0 (cos y) (* (- (sqrt 5.0) 1.0) (cos x))) 1.0)))
     (/
      (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
      (fma
       (fma 0.5 (sqrt 5.0) -0.5)
       (* (cos x) 3.0)
       (* (fma (* 0.5 t_0) (cos y) 1.0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -5.2) || !(x <= 3.6e-12)) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / (3.0 * fma(0.5, fma(t_0, cos(y), ((sqrt(5.0) - 1.0) * cos(x))), 1.0));
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(fma(0.5, sqrt(5.0), -0.5), (cos(x) * 3.0), (fma((0.5 * t_0), cos(y), 1.0) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -5.2) || !(x <= 3.6e-12))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / Float64(3.0 * fma(0.5, fma(t_0, cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 1.0)));
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(fma(0.5, sqrt(5.0), -0.5), Float64(cos(x) * 3.0), Float64(fma(Float64(0.5 * t_0), cos(y), 1.0) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -5.2], N[Not[LessEqual[x, 3.6e-12]], $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] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000018 or 3.6e-12 < x

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 98.9

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 66.1

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

   7. Applied rewrites 66.1%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. sub-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 66.2%

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

   11. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-cos.f64 63.5

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

   13. Applied rewrites 63.5%

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

   if -5.20000000000000018 < x < 3.6e-12

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 98.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 0.5 \cdot t\_1\\ \mathbf{if}\;x \leq -5.2:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(\mathsf{fma}\left(\cos y, t\_2, 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(t\_2, \cos y, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
           (- (cos x) 1.0))))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* 0.5 t_1)))
   (if (<= x -5.2)
     (/
      t_0
      (* 3.0 (+ (fma (cos y) t_2 1.0) (* (cos x) (fma (sqrt 5.0) 0.5 -0.5)))))
     (if (<= x 3.6e-12)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (fma
         (fma 0.5 (sqrt 5.0) -0.5)
         (* (cos x) 3.0)
         (* (fma t_2 (cos y) 1.0) 3.0)))
       (/
        t_0
        (*
         3.0
         (fma 0.5 (fma t_1 (cos y) (* (- (sqrt 5.0) 1.0) (cos x))) 1.0)))))))

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) - 1.0));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 0.5 * t_1;
	double tmp;
	if (x <= -5.2) {
		tmp = t_0 / (3.0 * (fma(cos(y), t_2, 1.0) + (cos(x) * fma(sqrt(5.0), 0.5, -0.5))));
	} else if (x <= 3.6e-12) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(fma(0.5, sqrt(5.0), -0.5), (cos(x) * 3.0), (fma(t_2, cos(y), 1.0) * 3.0));
	} else {
		tmp = t_0 / (3.0 * fma(0.5, fma(t_1, cos(y), ((sqrt(5.0) - 1.0) * cos(x))), 1.0));
	}
	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) - 1.0)))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(0.5 * t_1)
	tmp = 0.0
	if (x <= -5.2)
		tmp = Float64(t_0 / Float64(3.0 * Float64(fma(cos(y), t_2, 1.0) + Float64(cos(x) * fma(sqrt(5.0), 0.5, -0.5)))));
	elseif (x <= 3.6e-12)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(fma(0.5, sqrt(5.0), -0.5), Float64(cos(x) * 3.0), Float64(fma(t_2, cos(y), 1.0) * 3.0)));
	else
		tmp = Float64(t_0 / Float64(3.0 * fma(0.5, fma(t_1, cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 1.0)));
	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] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * t$95$1), $MachinePrecision]}, If[LessEqual[x, -5.2], N[(t$95$0 / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-12], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000018

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 98.9

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 61.1

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

   10. Applied rewrites 61.1%

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

   if -5.20000000000000018 < x < 3.6e-12

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 98.4%

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

   if 3.6e-12 < x

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.0

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 99.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.0

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 68.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. sub-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 68.4%

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

   11. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-cos.f64 66.2

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

   13. Applied rewrites 66.2%

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

Alternative 12: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := {\sin y}^{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -400000000000:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot t\_2\right) \cdot \sqrt{2}\right) \cdot t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos x, t\_3\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, t\_2, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(0.5 \cdot t\_3, \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (pow (sin y) 2.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -400000000000.0)
     (/
      (+ 2.0 (* (* (* -0.0625 t_2) (sqrt 2.0)) t_0))
      (fma 1.5 (fma (cos x) t_1 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y))) 3.0))
     (if (<= y 8.2e-6)
       (/
        (+ 2.0 (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0))
        (fma 1.5 (fma t_1 (cos x) t_3) 3.0))
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) t_2 2.0)
        (*
         3.0
         (fma
          (fma 0.5 (sqrt 5.0) -0.5)
          (cos x)
          (fma (* 0.5 t_3) (cos y) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = pow(sin(y), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -400000000000.0) {
		tmp = (2.0 + (((-0.0625 * t_2) * sqrt(2.0)) * t_0)) / fma(1.5, fma(cos(x), t_1, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	} else if (y <= 8.2e-6) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma(1.5, fma(t_1, cos(x), t_3), 3.0);
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_2, 2.0) / (3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma((0.5 * t_3), cos(y), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = sin(y) ^ 2.0
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -400000000000.0)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * t_2) * sqrt(2.0)) * t_0)) / fma(1.5, fma(cos(x), t_1, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	elseif (y <= 8.2e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(1.5, fma(t_1, cos(x), t_3), 3.0));
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_2, 2.0) / Float64(3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma(Float64(0.5 * t_3), cos(y), 1.0))));
	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[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -400000000000.0], N[(N[(2.0 + N[(N[(N[(-0.0625 * t$95$2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-6], 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] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(0.5 * t$95$3), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4e11

   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 x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 57.6

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 57.6%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   if -4e11 < y < 8.1999999999999994e-6

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.4

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 99.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.4

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

   10. Applied rewrites 97.5%

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

   if 8.1999999999999994e-6 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. div-sub N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-/.f64 N/A

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

   7. Applied rewrites 61.9%

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

Alternative 13: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -400000000000:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot t\_3\right) \cdot \sqrt{2}\right) \cdot t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, t\_2 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos x, t\_2\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, t\_3, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(0.5 \cdot t\_2, \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (pow (sin y) 2.0)))
   (if (<= y -400000000000.0)
     (/
      (+ 2.0 (* (* (* -0.0625 t_3) (sqrt 2.0)) t_0))
      (fma 1.5 (fma (cos x) t_1 (* t_2 (cos y))) 3.0))
     (if (<= y 8.2e-6)
       (/
        (+ 2.0 (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0))
        (fma 1.5 (fma t_1 (cos x) t_2) 3.0))
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) t_3 2.0)
        (*
         3.0
         (fma
          (fma 0.5 (sqrt 5.0) -0.5)
          (cos x)
          (fma (* 0.5 t_2) (cos y) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -400000000000.0) {
		tmp = (2.0 + (((-0.0625 * t_3) * sqrt(2.0)) * t_0)) / fma(1.5, fma(cos(x), t_1, (t_2 * cos(y))), 3.0);
	} else if (y <= 8.2e-6) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma(1.5, fma(t_1, cos(x), t_2), 3.0);
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_3, 2.0) / (3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma((0.5 * t_2), cos(y), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -400000000000.0)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * t_3) * sqrt(2.0)) * t_0)) / fma(1.5, fma(cos(x), t_1, Float64(t_2 * cos(y))), 3.0));
	elseif (y <= 8.2e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(1.5, fma(t_1, cos(x), t_2), 3.0));
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_3, 2.0) / Float64(3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma(Float64(0.5 * t_2), cos(y), 1.0))));
	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[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -400000000000.0], N[(N[(2.0 + N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-6], 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] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4e11

   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 x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 57.5

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

   8. Applied rewrites 57.5%

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

   if -4e11 < y < 8.1999999999999994e-6

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.4

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 99.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.4

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

   10. Applied rewrites 97.5%

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

   if 8.1999999999999994e-6 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. div-sub N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-/.f64 N/A

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

   7. Applied rewrites 61.9%

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

Alternative 14: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot t\_2\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, t\_2, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(0.5 \cdot t\_1, \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (pow (sin y) 2.0)))
   (if (<= y -2.3e-5)
     (/
      (+ 2.0 (* (* (* -0.0625 t_2) (sqrt 2.0)) (- (cos x) (cos y))))
      (fma 1.5 (fma (cos x) t_0 (* t_1 (cos y))) 3.0))
     (if (<= y 5e-6)
       (*
        (/
         (fma
          (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625))
          (sqrt 2.0)
          2.0)
         (fma (fma t_0 (cos x) t_1) 0.5 1.0))
        0.3333333333333333)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) t_2 2.0)
        (*
         3.0
         (fma
          (fma 0.5 (sqrt 5.0) -0.5)
          (cos x)
          (fma (* 0.5 t_1) (cos y) 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(y), 2.0);
	double tmp;
	if (y <= -2.3e-5) {
		tmp = (2.0 + (((-0.0625 * t_2) * sqrt(2.0)) * (cos(x) - cos(y)))) / fma(1.5, fma(cos(x), t_0, (t_1 * cos(y))), 3.0);
	} else if (y <= 5e-6) {
		tmp = (fma((pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_2, 2.0) / (3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma((0.5 * t_1), cos(y), 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(y) ^ 2.0
	tmp = 0.0
	if (y <= -2.3e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * t_2) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / fma(1.5, fma(cos(x), t_0, Float64(t_1 * cos(y))), 3.0));
	elseif (y <= 5e-6)
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_2, 2.0) / Float64(3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma(Float64(0.5 * t_1), cos(y), 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[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -2.3e-5], N[(N[(2.0 + N[(N[(N[(-0.0625 * t$95$2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-6], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e-5

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 55.9

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

   8. Applied rewrites 55.9%

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

   if -2.3e-5 < y < 5.00000000000000041e-6

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.4

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 99.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.4

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 99.1%

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

   if 5.00000000000000041e-6 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. div-sub N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-/.f64 N/A

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

   7. Applied rewrites 61.9%

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

Alternative 15: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(0.5 \cdot t\_2, \cos y, 1\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(t\_0, \cos x \cdot 3, t\_3 \cdot 3\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_2\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, t\_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.5 (sqrt 5.0) -0.5))
        (t_1
         (fma
          (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0))
          (pow (sin y) 2.0)
          2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (* 0.5 t_2) (cos y) 1.0)))
   (if (<= y -2.3e-5)
     (/ t_1 (fma t_0 (* (cos x) 3.0) (* t_3 3.0)))
     (if (<= y 5e-6)
       (*
        (/
         (fma
          (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625))
          (sqrt 2.0)
          2.0)
         (fma (fma (- (sqrt 5.0) 1.0) (cos x) t_2) 0.5 1.0))
        0.3333333333333333)
       (/ t_1 (* 3.0 (fma t_0 (cos x) t_3)))))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, sqrt(5.0), -0.5);
	double t_1 = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma((0.5 * t_2), cos(y), 1.0);
	double tmp;
	if (y <= -2.3e-5) {
		tmp = t_1 / fma(t_0, (cos(x) * 3.0), (t_3 * 3.0));
	} else if (y <= 5e-6) {
		tmp = (fma((pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_1 / (3.0 * fma(t_0, cos(x), t_3));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, sqrt(5.0), -0.5)
	t_1 = fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(Float64(0.5 * t_2), cos(y), 1.0)
	tmp = 0.0
	if (y <= -2.3e-5)
		tmp = Float64(t_1 / fma(t_0, Float64(cos(x) * 3.0), Float64(t_3 * 3.0)));
	elseif (y <= 5e-6)
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(t_1 / Float64(3.0 * fma(t_0, cos(x), t_3)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -2.3e-5], N[(t$95$1 / N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision] + N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-6], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e-5

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 55.8%

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

   if -2.3e-5 < y < 5.00000000000000041e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if 5.00000000000000041e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      9. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)\right)} \]

   7. Applied rewrites 61.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin y}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_2\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, t\_1, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(0.5 \cdot t\_2, \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin y) 2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -2.3e-5)
     (/
      (fma (* -0.0625 t_1) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma 1.5 (fma (cos x) t_0 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y))) 3.0))
     (if (<= y 5e-6)
       (*
        (/
         (fma
          (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625))
          (sqrt 2.0)
          2.0)
         (fma (fma t_0 (cos x) t_2) 0.5 1.0))
        0.3333333333333333)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) t_1 2.0)
        (*
         3.0
         (fma
          (fma 0.5 (sqrt 5.0) -0.5)
          (cos x)
          (fma (* 0.5 t_2) (cos y) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(y), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -2.3e-5) {
		tmp = fma((-0.0625 * t_1), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	} else if (y <= 5e-6) {
		tmp = (fma((pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_1, 2.0) / (3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma((0.5 * t_2), cos(y), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -2.3e-5)
		tmp = Float64(fma(Float64(-0.0625 * t_1), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	elseif (y <= 5e-6)
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_1, 2.0) / Float64(3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma(Float64(0.5 * t_2), cos(y), 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[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-5], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-6], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e-5

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      11. lower-sqrt.f64 55.8

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 55.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   if -2.3e-5 < y < 5.00000000000000041e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if 5.00000000000000041e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      9. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)\right)} \]

   7. Applied rewrites 61.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_0, \mathsf{fma}\left(t\_2, \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 \cdot \mathsf{fma}\left(t\_2, \cos x, \mathsf{fma}\left(0.5 \cdot t\_0, \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (fma
          (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0))
          (pow (sin y) 2.0)
          2.0))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= y -2.3e-5)
     (/ t_1 (* (fma (* 0.5 (cos y)) t_0 (fma t_2 (cos x) 1.0)) 3.0))
     (if (<= y 5e-6)
       (*
        (/
         (fma
          (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625))
          (sqrt 2.0)
          2.0)
         (fma (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 0.5 1.0))
        0.3333333333333333)
       (/ t_1 (* 3.0 (fma t_2 (cos x) (fma (* 0.5 t_0) (cos y) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0);
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (y <= -2.3e-5) {
		tmp = t_1 / (fma((0.5 * cos(y)), t_0, fma(t_2, cos(x), 1.0)) * 3.0);
	} else if (y <= 5e-6) {
		tmp = (fma((pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_1 / (3.0 * fma(t_2, cos(x), fma((0.5 * t_0), cos(y), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0)
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (y <= -2.3e-5)
		tmp = Float64(t_1 / Float64(fma(Float64(0.5 * cos(y)), t_0, fma(t_2, cos(x), 1.0)) * 3.0));
	elseif (y <= 5e-6)
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(t_1 / Float64(3.0 * fma(t_2, cos(x), fma(Float64(0.5 * t_0), cos(y), 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[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[y, -2.3e-5], N[(t$95$1 / N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-6], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e-5

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      3. lower-*.f64 55.7

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied rewrites 55.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   if -2.3e-5 < y < 5.00000000000000041e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if 5.00000000000000041e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      9. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)\right)} \]

   7. Applied rewrites 61.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin y}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_2 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_2\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, t\_1, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(0.5 \cdot t\_2, \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin y) 2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -2.3e-5)
     (/
      (fma (* -0.0625 t_1) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma 1.5 (fma (cos x) t_0 (* t_2 (cos y))) 3.0))
     (if (<= y 5e-6)
       (*
        (/
         (fma
          (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625))
          (sqrt 2.0)
          2.0)
         (fma (fma t_0 (cos x) t_2) 0.5 1.0))
        0.3333333333333333)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) t_1 2.0)
        (*
         3.0
         (fma
          (fma 0.5 (sqrt 5.0) -0.5)
          (cos x)
          (fma (* 0.5 t_2) (cos y) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(y), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -2.3e-5) {
		tmp = fma((-0.0625 * t_1), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, (t_2 * cos(y))), 3.0);
	} else if (y <= 5e-6) {
		tmp = (fma((pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_1, 2.0) / (3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma((0.5 * t_2), cos(y), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -2.3e-5)
		tmp = Float64(fma(Float64(-0.0625 * t_1), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(t_2 * cos(y))), 3.0));
	elseif (y <= 5e-6)
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_1, 2.0) / Float64(3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma(Float64(0.5 * t_2), cos(y), 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[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-5], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-6], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e-5

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. lower-sqrt.f64 55.7

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 55.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -2.3e-5 < y < 5.00000000000000041e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if 5.00000000000000041e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      9. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x + \left(1 + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)\right)} \]

   7. Applied rewrites 61.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin y}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_2 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_2\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, t\_1, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, \cos x \cdot t\_0\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin y) 2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -2.3e-5)
     (/
      (fma (* -0.0625 t_1) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma 1.5 (fma (cos x) t_0 (* t_2 (cos y))) 3.0))
     (if (<= y 5e-6)
       (*
        (/
         (fma
          (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625))
          (sqrt 2.0)
          2.0)
         (fma (fma t_0 (cos x) t_2) 0.5 1.0))
        0.3333333333333333)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) t_1 2.0)
        (* 3.0 (fma 0.5 (fma (cos y) t_2 (* (cos x) t_0)) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(y), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -2.3e-5) {
		tmp = fma((-0.0625 * t_1), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, (t_2 * cos(y))), 3.0);
	} else if (y <= 5e-6) {
		tmp = (fma((pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_1, 2.0) / (3.0 * fma(0.5, fma(cos(y), t_2, (cos(x) * t_0)), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -2.3e-5)
		tmp = Float64(fma(Float64(-0.0625 * t_1), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(t_2 * cos(y))), 3.0));
	elseif (y <= 5e-6)
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_1, 2.0) / Float64(3.0 * fma(0.5, fma(cos(y), t_2, Float64(cos(x) * 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[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-5], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-6], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e-5

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. lower-sqrt.f64 55.7

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 55.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -2.3e-5 < y < 5.00000000000000041e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if 5.00000000000000041e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right)}, 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right)}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}\right), 1\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\cos x} \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right), 1\right)} \]

      12. lower-sqrt.f64 61.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right)\right), 1\right)} \]

   8. Applied rewrites 61.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-5} \lor \neg \left(y \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= y -2.3e-5) (not (<= y 5e-6)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma 1.5 (fma (cos x) t_0 (* t_1 (cos y))) 3.0))
     (*
      (/
       (fma (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)) (sqrt 2.0) 2.0)
       (fma (fma t_0 (cos x) t_1) 0.5 1.0))
      0.3333333333333333))))

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 ((y <= -2.3e-5) || !(y <= 5e-6)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, (t_1 * cos(y))), 3.0);
	} else {
		tmp = (fma((pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0)) * 0.3333333333333333;
	}
	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 ((y <= -2.3e-5) || !(y <= 5e-6))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(t_1 * cos(y))), 3.0));
	else
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0)) * 0.3333333333333333);
	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[y, -2.3e-5], N[Not[LessEqual[y, 5e-6]], $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[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e-5 or 5.00000000000000041e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. lower-sqrt.f64 58.7

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 58.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -2.3e-5 < y < 5.00000000000000041e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-5} \lor \neg \left(y \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-5} \lor \neg \left(y \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_0\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -2.3e-5) (not (<= y 5e-6)))
     (/
      (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
      (fma 1.5 (fma (cos y) t_0 (* (cos x) t_1)) 3.0))
     (*
      (/
       (fma (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)) (sqrt 2.0) 2.0)
       (fma (fma t_1 (cos x) t_0) 0.5 1.0))
      0.3333333333333333))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -2.3e-5) || !(y <= 5e-6)) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(1.5, fma(cos(y), t_0, (cos(x) * t_1)), 3.0);
	} else {
		tmp = (fma((pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_1, cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -2.3e-5) || !(y <= 5e-6))
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), t_0, Float64(cos(x) * t_1)), 3.0));
	else
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625)), sqrt(2.0), 2.0) / fma(fma(t_1, cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333);
	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[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -2.3e-5], N[Not[LessEqual[y, 5e-6]], $MachinePrecision]], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e-5 or 5.00000000000000041e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 58.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3} + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), 3, \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 47.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

       4. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

       5. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   11. Applied rewrites 58.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 3\right)}} \]

   if -2.3e-5 < y < 5.00000000000000041e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-5} \lor \neg \left(y \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_0\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)))
   (if (or (<= x -0.0011) (not (<= x 3.6e-12)))
     (/
      (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
      (fma 1.5 (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))
     (/
      (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
      (fma
       (fma 0.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_0) 1.0)
       3.0
       (* (fma -0.75 (sqrt 5.0) 0.75) (* x x)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double tmp;
	if ((x <= -0.0011) || !(x <= 3.6e-12)) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(fma(0.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_0), 1.0), 3.0, (fma(-0.75, sqrt(5.0), 0.75) * (x * x)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((x <= -0.0011) || !(x <= 3.6e-12))
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(fma(0.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_0), 1.0), 3.0, Float64(fma(-0.75, sqrt(5.0), 0.75) * Float64(x * x))));
	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, -0.0011], N[Not[LessEqual[x, 3.6e-12]], $MachinePrecision]], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00110000000000000007 or 3.6e-12 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      16. lower-cos.f64 62.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -0.00110000000000000007 < x < 3.6e-12

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3} + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), 3, \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 78.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)\\ t_2 := {\sin x}^{2}\\ t_3 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ \mathbf{if}\;x \leq -0.0013:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot t\_3, \sqrt{2}, 2\right)}{t\_1} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_0\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{t\_1}{\mathsf{fma}\left(t\_3, t\_2 \cdot \sqrt{2}, 2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (fma (fma t_0 (cos x) (- 3.0 (sqrt 5.0))) 0.5 1.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (fma -0.0625 (cos x) 0.0625)))
   (if (<= x -0.0013)
     (* (/ (fma (* t_2 t_3) (sqrt 2.0) 2.0) t_1) 0.3333333333333333)
     (if (<= x 3.6e-12)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (fma
         (fma 0.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_0) 1.0)
         3.0
         (* (fma -0.75 (sqrt 5.0) 0.75) (* x x))))
       (/ 0.3333333333333333 (/ t_1 (fma t_3 (* t_2 (sqrt 2.0)) 2.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(fma(t_0, cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0);
	double t_2 = pow(sin(x), 2.0);
	double t_3 = fma(-0.0625, cos(x), 0.0625);
	double tmp;
	if (x <= -0.0013) {
		tmp = (fma((t_2 * t_3), sqrt(2.0), 2.0) / t_1) * 0.3333333333333333;
	} else if (x <= 3.6e-12) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(fma(0.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_0), 1.0), 3.0, (fma(-0.75, sqrt(5.0), 0.75) * (x * x)));
	} else {
		tmp = 0.3333333333333333 / (t_1 / fma(t_3, (t_2 * sqrt(2.0)), 2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(fma(t_0, cos(x), Float64(3.0 - sqrt(5.0))), 0.5, 1.0)
	t_2 = sin(x) ^ 2.0
	t_3 = fma(-0.0625, cos(x), 0.0625)
	tmp = 0.0
	if (x <= -0.0013)
		tmp = Float64(Float64(fma(Float64(t_2 * t_3), sqrt(2.0), 2.0) / t_1) * 0.3333333333333333);
	elseif (x <= 3.6e-12)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(fma(0.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_0), 1.0), 3.0, Float64(fma(-0.75, sqrt(5.0), 0.75) * Float64(x * x))));
	else
		tmp = Float64(0.3333333333333333 / Float64(t_1 / fma(t_3, Float64(t_2 * sqrt(2.0)), 2.0)));
	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[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, If[LessEqual[x, -0.0013], N[(N[(N[(N[(t$95$2 * t$95$3), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.6e-12], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(t$95$1 / N[(t$95$3 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0012999999999999999

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 98.9

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if -0.0012999999999999999 < x < 3.6e-12

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3} + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), 3, \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)} \]

   if 3.6e-12 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.6%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 0.5, 1\right)\\ t_2 := {\sin x}^{2}\\ t_3 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ \mathbf{if}\;x \leq -5.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot t\_3, \sqrt{2}, 2\right)}{t\_1} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{t\_1}{\mathsf{fma}\left(t\_3, t\_2 \cdot \sqrt{2}, 2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 0.5 1.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (fma -0.0625 (cos x) 0.0625)))
   (if (<= x -5.2)
     (* (/ (fma (* t_2 t_3) (sqrt 2.0) 2.0) t_1) 0.3333333333333333)
     (if (<= x 3.6e-12)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (* 3.0 (+ (+ 1.0 (fma 0.5 (sqrt 5.0) -0.5)) (* (/ t_0 2.0) (cos y)))))
       (/ 0.3333333333333333 (/ t_1 (fma t_3 (* t_2 (sqrt 2.0)) 2.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(fma((sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0);
	double t_2 = pow(sin(x), 2.0);
	double t_3 = fma(-0.0625, cos(x), 0.0625);
	double tmp;
	if (x <= -5.2) {
		tmp = (fma((t_2 * t_3), sqrt(2.0), 2.0) / t_1) * 0.3333333333333333;
	} else if (x <= 3.6e-12) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / (3.0 * ((1.0 + fma(0.5, sqrt(5.0), -0.5)) + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = 0.3333333333333333 / (t_1 / fma(t_3, (t_2 * sqrt(2.0)), 2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)
	t_2 = sin(x) ^ 2.0
	t_3 = fma(-0.0625, cos(x), 0.0625)
	tmp = 0.0
	if (x <= -5.2)
		tmp = Float64(Float64(fma(Float64(t_2 * t_3), sqrt(2.0), 2.0) / t_1) * 0.3333333333333333);
	elseif (x <= 3.6e-12)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + fma(0.5, sqrt(5.0), -0.5)) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = Float64(0.3333333333333333 / Float64(t_1 / fma(t_3, Float64(t_2 * sqrt(2.0)), 2.0)));
	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[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, If[LessEqual[x, -5.2], N[(N[(N[(N[(t$95$2 * t$95$3), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.6e-12], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(t$95$1 / N[(t$95$3 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000018

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 98.9

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if -5.20000000000000018 < x < 3.6e-12

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot -1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 98.3

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 98.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 3.6e-12 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.6%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)\\ t_3 := {\sin x}^{2}\\ t_4 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ \mathbf{if}\;x \leq -5.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3 \cdot t\_4, \sqrt{2}, 2\right)}{t\_2} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{t\_2}{\mathsf{fma}\left(t\_4, t\_3 \cdot \sqrt{2}, 2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma (fma t_0 (cos x) t_1) 0.5 1.0))
        (t_3 (pow (sin x) 2.0))
        (t_4 (fma -0.0625 (cos x) 0.0625)))
   (if (<= x -5.2)
     (* (/ (fma (* t_3 t_4) (sqrt 2.0) 2.0) t_2) 0.3333333333333333)
     (if (<= x 3.6e-12)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       (/ 0.3333333333333333 (/ t_2 (fma t_4 (* t_3 (sqrt 2.0)) 2.0)))))))

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 = fma(fma(t_0, cos(x), t_1), 0.5, 1.0);
	double t_3 = pow(sin(x), 2.0);
	double t_4 = fma(-0.0625, cos(x), 0.0625);
	double tmp;
	if (x <= -5.2) {
		tmp = (fma((t_3 * t_4), sqrt(2.0), 2.0) / t_2) * 0.3333333333333333;
	} else if (x <= 3.6e-12) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = 0.3333333333333333 / (t_2 / fma(t_4, (t_3 * sqrt(2.0)), 2.0));
	}
	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 = fma(fma(t_0, cos(x), t_1), 0.5, 1.0)
	t_3 = sin(x) ^ 2.0
	t_4 = fma(-0.0625, cos(x), 0.0625)
	tmp = 0.0
	if (x <= -5.2)
		tmp = Float64(Float64(fma(Float64(t_3 * t_4), sqrt(2.0), 2.0) / t_2) * 0.3333333333333333);
	elseif (x <= 3.6e-12)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = Float64(0.3333333333333333 / Float64(t_2 / fma(t_4, Float64(t_3 * sqrt(2.0)), 2.0)));
	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[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, If[LessEqual[x, -5.2], N[(N[(N[(N[(t$95$3 * t$95$4), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.6e-12], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(t$95$2 / N[(t$95$4 * N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000018

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 98.9

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if -5.20000000000000018 < x < 3.6e-12

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3} + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), 3, \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 98.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

       4. associate-+r- N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) - 1\right)}\right) + 3 \cdot 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} - 1\right)\right) + 3 \cdot 1} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} + 3 \cdot 1} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       9. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + \color{blue}{3}} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1, 3\right)}} \]

   11. Applied rewrites 98.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   if 3.6e-12 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.6%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -5.2) (not (<= x 3.6e-12)))
     (*
      (/
       (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
       (fma 0.5 (fma (cos x) t_0 t_1) 1.0))
      0.3333333333333333)
     (/
      (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
      (fma 1.5 (fma (cos y) t_1 t_0) 3.0)))))

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 <= -5.2) || !(x <= 3.6e-12)) {
		tmp = (fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), t_0, t_1), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	}
	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 <= -5.2) || !(x <= 3.6e-12))
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), t_0, t_1), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -5.2], N[Not[LessEqual[x, 3.6e-12]], $MachinePrecision]], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000018 or 3.6e-12 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -5.20000000000000018 < x < 3.6e-12

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3} + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), 3, \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 98.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

       4. associate-+r- N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) - 1\right)}\right) + 3 \cdot 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} - 1\right)\right) + 3 \cdot 1} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} + 3 \cdot 1} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       9. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + \color{blue}{3}} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1, 3\right)}} \]

   11. Applied rewrites 98.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)))
   (if (or (<= x -5.2) (not (<= x 3.6e-12)))
     (*
      (/
       (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
       (fma 0.5 (- (fma t_0 (cos x) 3.0) (sqrt 5.0)) 1.0))
      0.3333333333333333)
     (/
      (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
      (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_0) 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double tmp;
	if ((x <= -5.2) || !(x <= 3.6e-12)) {
		tmp = (fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, (fma(t_0, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), t_0), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((x <= -5.2) || !(x <= 3.6e-12))
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, Float64(fma(t_0, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_0), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -5.2], N[Not[LessEqual[x, 3.6e-12]], $MachinePrecision]], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000018 or 3.6e-12 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.8%

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]

   if -5.20000000000000018 < x < 3.6e-12

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3} + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), 3, \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 98.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

       4. associate-+r- N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) - 1\right)}\right) + 3 \cdot 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} - 1\right)\right) + 3 \cdot 1} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} + 3 \cdot 1} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       9. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + \color{blue}{3}} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1, 3\right)}} \]

   11. Applied rewrites 98.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ t_3 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ \mathbf{if}\;x \leq -5.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot t\_3, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, t\_3, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

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 (fma -0.0625 (cos x) 0.0625)))
   (if (<= x -5.2)
     (*
      (/ (fma (* t_2 t_3) (sqrt 2.0) 2.0) (fma (fma t_0 (cos x) t_1) 0.5 1.0))
      0.3333333333333333)
     (if (<= x 3.6e-12)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       (*
        (/
         (fma (* t_2 (sqrt 2.0)) t_3 2.0)
         (fma 0.5 (fma (cos x) t_0 t_1) 1.0))
        0.3333333333333333)))))

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 = fma(-0.0625, cos(x), 0.0625);
	double tmp;
	if (x <= -5.2) {
		tmp = (fma((t_2 * t_3), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0)) * 0.3333333333333333;
	} else if (x <= 3.6e-12) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = (fma((t_2 * sqrt(2.0)), t_3, 2.0) / fma(0.5, fma(cos(x), t_0, t_1), 1.0)) * 0.3333333333333333;
	}
	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 = fma(-0.0625, cos(x), 0.0625)
	tmp = 0.0
	if (x <= -5.2)
		tmp = Float64(Float64(fma(Float64(t_2 * t_3), sqrt(2.0), 2.0) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0)) * 0.3333333333333333);
	elseif (x <= 3.6e-12)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = Float64(Float64(fma(Float64(t_2 * sqrt(2.0)), t_3, 2.0) / fma(0.5, fma(cos(x), t_0, t_1), 1.0)) * 0.3333333333333333);
	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[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, If[LessEqual[x, -5.2], N[(N[(N[(N[(t$95$2 * t$95$3), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.6e-12], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000018

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}} + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. lower-fma.f64 98.9

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{3 - \sqrt{5}}{2}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right), 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. lower-*.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      16. lower-*.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}}\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}}\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right)} \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if -5.20000000000000018 < x < 3.6e-12

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3} + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), 3, \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 98.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

       4. associate-+r- N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) - 1\right)}\right) + 3 \cdot 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} - 1\right)\right) + 3 \cdot 1} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} + 3 \cdot 1} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       9. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + \color{blue}{3}} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1, 3\right)}} \]

   11. Applied rewrites 98.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   if 3.6e-12 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -5.2) (not (<= x 3.6e-12)))
     (/
      (fma (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0)) (pow (sin x) 2.0) 2.0)
      (fma 1.5 (fma t_0 (cos x) t_1) 3.0))
     (/
      (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
      (fma 1.5 (fma (cos y) t_1 t_0) 3.0)))))

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 <= -5.2) || !(x <= 3.6e-12)) {
		tmp = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	}
	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 <= -5.2) || !(x <= 3.6e-12))
		tmp = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0));
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -5.2], N[Not[LessEqual[x, 3.6e-12]], $MachinePrecision]], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000018 or 3.6e-12 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}\right)}\right) + 3 \cdot 1} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right) + 3 \cdot 1} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right) + \color{blue}{3}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 3\right)}} \]

   8. Applied rewrites 61.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if -5.20000000000000018 < x < 3.6e-12

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3} + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), 3, \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 98.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

       4. associate-+r- N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) - 1\right)}\right) + 3 \cdot 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} - 1\right)\right) + 3 \cdot 1} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} + 3 \cdot 1} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

       9. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + \color{blue}{3}} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1, 3\right)}} \]

   11. Applied rewrites 98.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \lor \neg \left(x \leq 3.6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 59.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
  (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (- (sqrt 5.0) 1.0)) 3.0)))

C

double code(double x, double y) {
	return fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), (sqrt(5.0) - 1.0)), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(sqrt(5.0) - 1.0)), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 60.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3} + \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), 3, \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}\right)}} \]

8. Applied rewrites 47.6%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right), 3, \mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot \left(x \cdot x\right)\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-+r- N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) - 1\right)}\right) + 3 \cdot 1} \]

    5. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} - 1\right)\right) + 3 \cdot 1} \]

    6. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} + 3 \cdot 1} \]

    7. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 3 \cdot 1} \]

    9. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + \color{blue}{3}} \]

    10. lower-fma.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1, 3\right)}} \]

11. Applied rewrites 56.9%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]
12. Add Preprocessing

Alternative 31: 43.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

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))
  0.3333333333333333))

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)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / fma(0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(2.0 / 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]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

5. Applied rewrites 99.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

8. Applied rewrites 61.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]
10. Step-by-step derivation

11. Applied rewrites 42.3%

    \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
12. Add Preprocessing

Alternative 32: 40.9% accurate, 940.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

5. Applied rewrites 99.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

8. Applied rewrites 61.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
10. Step-by-step derivation

11. Applied rewrites 39.6%

    \[\leadsto 0.3333333333333333 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(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))))))