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

Percentage Accurate: 99.3% → 99.4%

Time: 12.9s

Alternatives: 30

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

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (fma (sin y) -0.0625 (sin x))
   (* (* (- (cos x) (cos y)) (sqrt 2.0)) (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(fma(sin(y), -0.0625, sin(x)), (((cos(x) - cos(y)) * sqrt(2.0)) * 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(fma(sin(y), -0.0625, sin(x)), Float64(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)) * 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[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[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 99.3%

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

3. Taylor expanded in 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 99.4%

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

7. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (* (- (cos x) (cos y)) (fma (sin y) -0.0625 (sin x))) (sqrt 2.0))
   (fma -0.0625 (sin x) (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((((cos(x) - cos(y)) * fma(sin(y), -0.0625, sin(x))) * sqrt(2.0)), fma(-0.0625, sin(x), 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(Float64(Float64(Float64(cos(x) - cos(y)) * fma(sin(y), -0.0625, sin(x))) * sqrt(2.0)), fma(-0.0625, sin(x), 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[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $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 99.3%

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

3. Taylor expanded in 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 99.4%

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

7. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.4

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-fma.f64 99.4

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

9. Applied rewrites 99.4%

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

Alternative 3: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -0.56:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (+
           (*
            (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))
            (- (cos x) (cos y)))
           2.0)
          (fma
           1.5
           (fma (cos x) t_0 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
           3.0))))
   (if (<= x -0.56)
     t_1
     (if (<= x 0.5)
       (/
        (fma
         (*
          (*
           (fma
            (fma
             (fma -0.001388888888888889 (* x x) 0.041666666666666664)
             (* x x)
             -0.5)
            (* x x)
            (- 1.0 (cos y)))
           (sqrt 2.0))
          (fma -0.0625 (sin y) (sin x)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	double tmp;
	if (x <= -0.56) {
		tmp = t_1;
	} else if (x <= 0.5) {
		tmp = fma(((fma(fma(fma(-0.001388888888888889, (x * x), 0.041666666666666664), (x * x), -0.5), (x * x), (1.0 - cos(y))) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -0.56)
		tmp = t_1;
	elseif (x <= 0.5)
		tmp = Float64(fma(Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), -0.5), Float64(x * x), Float64(1.0 - cos(y))) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $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[x, -0.56], t$95$1, If[LessEqual[x, 0.5], N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $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], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.56000000000000005 or 0.5 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in 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 58.3

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

   8. Applied rewrites 58.3%

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

   10. Applied rewrites 58.4%

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

   if -0.56000000000000005 < x < 0.5

   1. Initial program 99.6%

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

   3. Taylor expanded in 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-cos.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1 - \cos y\right)} \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.56:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -0.355:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_0 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right), x, -0.0625 \cdot \sin y\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (+
           (* (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x))) t_0)
           2.0)
          (fma
           1.5
           (fma (cos x) t_1 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
           3.0))))
   (if (<= x -0.355)
     t_2
     (if (<= x 0.42)
       (/
        (fma
         (*
          (* t_0 (sqrt 2.0))
          (fma
           (fma
            (fma 0.008333333333333333 (* x x) -0.16666666666666666)
            (* x x)
            1.0)
           x
           (* -0.0625 (sin y))))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_1 (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))
       t_2))))

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 = ((((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * t_0) + 2.0) / fma(1.5, fma(cos(x), t_1, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	double tmp;
	if (x <= -0.355) {
		tmp = t_2;
	} else if (x <= 0.42) {
		tmp = fma(((t_0 * sqrt(2.0)) * fma(fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0), x, (-0.0625 * sin(y)))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_1, ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	} else {
		tmp = t_2;
	}
	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(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * t_0) + 2.0) / fma(1.5, fma(cos(x), t_1, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -0.355)
		tmp = t_2;
	elseif (x <= 0.42)
		tmp = Float64(fma(Float64(Float64(t_0 * sqrt(2.0)) * fma(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0), x, Float64(-0.0625 * sin(y)))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_1, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	else
		tmp = t_2;
	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[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $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[x, -0.355], t$95$2, If[LessEqual[x, 0.42], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.35499999999999998 or 0.419999999999999984 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in 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 58.3

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

   8. Applied rewrites 58.3%

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

   10. Applied rewrites 58.4%

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

   if -0.35499999999999998 < x < 0.419999999999999984

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \sin y + x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) + \frac{-1}{16} \cdot \sin y\right)} \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x} + \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right), x, \frac{-1}{16} \cdot \sin y\right)} \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1}, x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2}} + 1, x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right)}, x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {x}^{2}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}, {x}^{2}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right)}, {x}^{2}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{-1}{6}\right), {x}^{2}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{-1}{6}\right), {x}^{2}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), \color{blue}{x \cdot x}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), \color{blue}{x \cdot x}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right), x, \color{blue}{\frac{-1}{16} \cdot \sin y}\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      15. lower-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right), x, -0.0625 \cdot \color{blue}{\sin y}\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right), x, -0.0625 \cdot \sin y\right)} \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.355:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right), x, -0.0625 \cdot \sin y\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -0.22:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (+
           (*
            (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))
            (- (cos x) (cos y)))
           2.0)
          (fma
           1.5
           (fma (cos x) t_0 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
           3.0))))
   (if (<= x -0.22)
     t_1
     (if (<= x 0.215)
       (/
        (fma
         (*
          (*
           (fma
            (fma 0.041666666666666664 (* x x) -0.5)
            (* x x)
            (- 1.0 (cos y)))
           (sqrt 2.0))
          (fma -0.0625 (sin y) (sin x)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	double tmp;
	if (x <= -0.22) {
		tmp = t_1;
	} else if (x <= 0.215) {
		tmp = fma(((fma(fma(0.041666666666666664, (x * x), -0.5), (x * x), (1.0 - cos(y))) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -0.22)
		tmp = t_1;
	elseif (x <= 0.215)
		tmp = Float64(fma(Float64(Float64(fma(fma(0.041666666666666664, Float64(x * x), -0.5), Float64(x * x), Float64(1.0 - cos(y))) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $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[x, -0.22], t$95$1, If[LessEqual[x, 0.215], N[(N[(N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $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], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.220000000000000001 or 0.214999999999999997 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in 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 58.3

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

   8. Applied rewrites 58.3%

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

   10. Applied rewrites 58.4%

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

   if -0.220000000000000001 < x < 0.214999999999999997

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1 - \cos y\right)} \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.22:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -0.155:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.098:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \left(t\_0 \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (+
           (* (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x))) t_0)
           2.0)
          (fma
           1.5
           (fma (cos x) t_1 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
           3.0))))
   (if (<= x -0.155)
     t_2
     (if (<= x 0.098)
       (/
        (fma
         (*
          (fma (fma (* x x) -0.16666666666666666 1.0) x (* -0.0625 (sin y)))
          (* t_0 (sqrt 2.0)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_1 (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))
       t_2))))

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 = ((((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * t_0) + 2.0) / fma(1.5, fma(cos(x), t_1, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	double tmp;
	if (x <= -0.155) {
		tmp = t_2;
	} else if (x <= 0.098) {
		tmp = fma((fma(fma((x * x), -0.16666666666666666, 1.0), x, (-0.0625 * sin(y))) * (t_0 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_1, ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	} else {
		tmp = t_2;
	}
	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(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * t_0) + 2.0) / fma(1.5, fma(cos(x), t_1, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -0.155)
		tmp = t_2;
	elseif (x <= 0.098)
		tmp = Float64(fma(Float64(fma(fma(Float64(x * x), -0.16666666666666666, 1.0), x, Float64(-0.0625 * sin(y))) * Float64(t_0 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_1, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	else
		tmp = t_2;
	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[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $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[x, -0.155], t$95$2, If[LessEqual[x, 0.098], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.154999999999999999 or 0.098000000000000004 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in 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 58.3

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

   8. Applied rewrites 58.3%

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

   10. Applied rewrites 58.4%

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

   if -0.154999999999999999 < x < 0.098000000000000004

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \color{blue}{\sin y}\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)} \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.155:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.098:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.084:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (+
           (*
            (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))
            (- (cos x) (cos y)))
           2.0)
          (fma
           1.5
           (fma (cos x) t_0 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
           3.0))))
   (if (<= x -0.078)
     t_1
     (if (<= x 0.084)
       (/
        (fma
         (*
          (* (fma (* x x) -0.5 (- 1.0 (cos y))) (sqrt 2.0))
          (fma -0.0625 (sin y) (sin x)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	double tmp;
	if (x <= -0.078) {
		tmp = t_1;
	} else if (x <= 0.084) {
		tmp = fma(((fma((x * x), -0.5, (1.0 - cos(y))) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -0.078)
		tmp = t_1;
	elseif (x <= 0.084)
		tmp = Float64(fma(Float64(Float64(fma(Float64(x * x), -0.5, Float64(1.0 - cos(y))) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $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[x, -0.078], t$95$1, If[LessEqual[x, 0.084], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $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], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0779999999999999999 or 0.0840000000000000052 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in 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 58.3

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

   8. Applied rewrites 58.3%

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

   10. Applied rewrites 58.4%

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

   if -0.0779999999999999999 < x < 0.0840000000000000052

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 99.3

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

   10. Applied rewrites 99.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.084:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_3 := \mathsf{fma}\left(t\_1 \cdot 1.5, \cos y, 3\right)\\ \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;\frac{t\_2}{t\_3 + \left(t\_0 \cdot 1.5\right) \cdot \cos x}\\ \mathbf{elif}\;x \leq 0.084:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(t\_0 \cdot \cos x, 1.5, t\_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (+
          (*
           (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))
           (- (cos x) (cos y)))
          2.0))
        (t_3 (fma (* t_1 1.5) (cos y) 3.0)))
   (if (<= x -0.078)
     (/ t_2 (+ t_3 (* (* t_0 1.5) (cos x))))
     (if (<= x 0.084)
       (/
        (fma
         (*
          (* (fma (* x x) -0.5 (- 1.0 (cos y))) (sqrt 2.0))
          (fma -0.0625 (sin y) (sin x)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_0 (* t_1 (cos y))) 3.0))
       (/ t_2 (fma (* t_0 (cos x)) 1.5 t_3))))))

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 = (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * (cos(x) - cos(y))) + 2.0;
	double t_3 = fma((t_1 * 1.5), cos(y), 3.0);
	double tmp;
	if (x <= -0.078) {
		tmp = t_2 / (t_3 + ((t_0 * 1.5) * cos(x)));
	} else if (x <= 0.084) {
		tmp = fma(((fma((x * x), -0.5, (1.0 - cos(y))) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, (t_1 * cos(y))), 3.0);
	} else {
		tmp = t_2 / fma((t_0 * cos(x)), 1.5, t_3);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * Float64(cos(x) - cos(y))) + 2.0)
	t_3 = fma(Float64(t_1 * 1.5), cos(y), 3.0)
	tmp = 0.0
	if (x <= -0.078)
		tmp = Float64(t_2 / Float64(t_3 + Float64(Float64(t_0 * 1.5) * cos(x))));
	elseif (x <= 0.084)
		tmp = Float64(fma(Float64(Float64(fma(Float64(x * x), -0.5, Float64(1.0 - cos(y))) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(t_1 * cos(y))), 3.0));
	else
		tmp = Float64(t_2 / fma(Float64(t_0 * cos(x)), 1.5, t_3));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[x, -0.078], N[(t$95$2 / N[(t$95$3 + N[(N[(t$95$0 * 1.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.084], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $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[(t$95$2 / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 1.5 + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0779999999999999999

   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

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   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 52.7

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

   8. Applied rewrites 52.7%

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

   10. Applied rewrites 52.6%

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

   12. Applied rewrites 52.7%

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

   if -0.0779999999999999999 < x < 0.0840000000000000052

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 99.3

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

   10. Applied rewrites 99.3%

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

   if 0.0840000000000000052 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 99.0%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

   10. Applied rewrites 64.8%

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

   12. Applied rewrites 64.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, 3\right) + \left(\left(\sqrt{5} - 1\right) \cdot 1.5\right) \cdot \cos x}\\ \mathbf{elif}\;x \leq 0.084:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \cos x, 1.5, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot t\_1 + 2\\ t_4 := \mathsf{fma}\left(t\_2 \cdot 1.5, \cos y, 3\right)\\ \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{t\_3}{t\_4 + \left(t\_0 \cdot 1.5\right) \cdot \cos x}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(t\_1 \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_2 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(t\_0 \cdot \cos x, 1.5, t\_4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (+
          (* (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x))) t_1)
          2.0))
        (t_4 (fma (* t_2 1.5) (cos y) 3.0)))
   (if (<= x -0.037)
     (/ t_3 (+ t_4 (* (* t_0 1.5) (cos x))))
     (if (<= x 0.036)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (* t_1 (sqrt 2.0)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_0 (* t_2 (cos y))) 3.0))
       (/ t_3 (fma (* t_0 (cos x)) 1.5 t_4))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * t_1) + 2.0;
	double t_4 = fma((t_2 * 1.5), cos(y), 3.0);
	double tmp;
	if (x <= -0.037) {
		tmp = t_3 / (t_4 + ((t_0 * 1.5) * cos(x)));
	} else if (x <= 0.036) {
		tmp = fma((fma(-0.0625, sin(y), x) * (t_1 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, (t_2 * cos(y))), 3.0);
	} else {
		tmp = t_3 / fma((t_0 * cos(x)), 1.5, t_4);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * t_1) + 2.0)
	t_4 = fma(Float64(t_2 * 1.5), cos(y), 3.0)
	tmp = 0.0
	if (x <= -0.037)
		tmp = Float64(t_3 / Float64(t_4 + Float64(Float64(t_0 * 1.5) * cos(x))));
	elseif (x <= 0.036)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * Float64(t_1 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(t_2 * cos(y))), 3.0));
	else
		tmp = Float64(t_3 / fma(Float64(t_0 * cos(x)), 1.5, t_4));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 * 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[x, -0.037], N[(t$95$3 / N[(t$95$4 + N[(N[(t$95$0 * 1.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.036], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $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], N[(t$95$3 / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 1.5 + t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0369999999999999982

   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

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   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 52.7

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

   8. Applied rewrites 52.7%

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

   10. Applied rewrites 52.6%

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

   12. Applied rewrites 52.7%

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

   if -0.0369999999999999982 < x < 0.0359999999999999973

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if 0.0359999999999999973 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 99.0%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

   10. Applied rewrites 64.8%

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

   12. Applied rewrites 64.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, 3\right) + \left(\left(\sqrt{5} - 1\right) \cdot 1.5\right) \cdot \cos x}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \cos x, 1.5, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(t\_1 \cdot 1.5, \cos y, 3\right)\\ t_3 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot t\_0 + 2\\ t_4 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(t\_4, 1.5 \cdot \cos x, t\_2\right)}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(t\_0 \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_4, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(t\_4 \cdot \cos x, 1.5, t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma (* t_1 1.5) (cos y) 3.0))
        (t_3
         (+
          (* (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x))) t_0)
          2.0))
        (t_4 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.037)
     (/ t_3 (fma t_4 (* 1.5 (cos x)) t_2))
     (if (<= x 0.036)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (* t_0 (sqrt 2.0)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_4 (* t_1 (cos y))) 3.0))
       (/ t_3 (fma (* t_4 (cos x)) 1.5 t_2))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((t_1 * 1.5), cos(y), 3.0);
	double t_3 = (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * t_0) + 2.0;
	double t_4 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.037) {
		tmp = t_3 / fma(t_4, (1.5 * cos(x)), t_2);
	} else if (x <= 0.036) {
		tmp = fma((fma(-0.0625, sin(y), x) * (t_0 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_4, (t_1 * cos(y))), 3.0);
	} else {
		tmp = t_3 / fma((t_4 * cos(x)), 1.5, t_2);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(Float64(t_1 * 1.5), cos(y), 3.0)
	t_3 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * t_0) + 2.0)
	t_4 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.037)
		tmp = Float64(t_3 / fma(t_4, Float64(1.5 * cos(x)), t_2));
	elseif (x <= 0.036)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * Float64(t_0 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_4, Float64(t_1 * cos(y))), 3.0));
	else
		tmp = Float64(t_3 / fma(Float64(t_4 * cos(x)), 1.5, t_2));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.037], N[(t$95$3 / N[(t$95$4 * N[(1.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.036], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$4 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[(t$95$4 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 1.5 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0369999999999999982

   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

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   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 52.7

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

   8. Applied rewrites 52.7%

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

   10. Applied rewrites 52.6%

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

   12. Applied rewrites 52.7%

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

   if -0.0369999999999999982 < x < 0.0359999999999999973

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if 0.0359999999999999973 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 99.0%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

   10. Applied rewrites 64.8%

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

   12. Applied rewrites 64.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\sqrt{5} - 1, 1.5 \cdot \cos x, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, 3\right)\right)}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \cos x, 1.5, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot t\_0 + 2\\ t_2 := \sqrt{5} - 1\\ t_3 := 3 - \sqrt{5}\\ t_4 := t\_3 \cdot \cos y\\ \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(t\_2, \cos x, t\_4\right) \cdot 1.5 + 3}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(t\_0 \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_4\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(t\_2 \cdot \cos x, 1.5, \mathsf{fma}\left(t\_3 \cdot 1.5, \cos y, 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (+
          (* (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x))) t_0)
          2.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (* t_3 (cos y))))
   (if (<= x -0.037)
     (/ t_1 (+ (* (fma t_2 (cos x) t_4) 1.5) 3.0))
     (if (<= x 0.036)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (* t_0 (sqrt 2.0)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_2 t_4) 3.0))
       (/ t_1 (fma (* t_2 (cos x)) 1.5 (fma (* t_3 1.5) (cos y) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * t_0) + 2.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 * cos(y);
	double tmp;
	if (x <= -0.037) {
		tmp = t_1 / ((fma(t_2, cos(x), t_4) * 1.5) + 3.0);
	} else if (x <= 0.036) {
		tmp = fma((fma(-0.0625, sin(y), x) * (t_0 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_2, t_4), 3.0);
	} else {
		tmp = t_1 / fma((t_2 * cos(x)), 1.5, fma((t_3 * 1.5), cos(y), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * t_0) + 2.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 * cos(y))
	tmp = 0.0
	if (x <= -0.037)
		tmp = Float64(t_1 / Float64(Float64(fma(t_2, cos(x), t_4) * 1.5) + 3.0));
	elseif (x <= 0.036)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * Float64(t_0 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_2, t_4), 3.0));
	else
		tmp = Float64(t_1 / fma(Float64(t_2 * cos(x)), 1.5, fma(Float64(t_3 * 1.5), cos(y), 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.037], N[(t$95$1 / N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$4), $MachinePrecision] * 1.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.036], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$4), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(t$95$3 * 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0369999999999999982

   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

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   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 52.7

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

   8. Applied rewrites 52.7%

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

   10. Applied rewrites 52.6%

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

   12. Applied rewrites 52.7%

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

   if -0.0369999999999999982 < x < 0.0359999999999999973

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if 0.0359999999999999973 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 99.0%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

   10. Applied rewrites 64.8%

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

   12. Applied rewrites 64.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot 1.5 + 3}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \cos x, 1.5, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot t\_0 + 2\\ t_2 := \sqrt{5} - 1\\ t_3 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(t\_2, \cos x, t\_3\right) \cdot 1.5 + 3}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(t\_0 \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_3\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(1.5, t\_2 \cdot \cos x + t\_3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (+
          (* (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x))) t_0)
          2.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (* (- 3.0 (sqrt 5.0)) (cos y))))
   (if (<= x -0.037)
     (/ t_1 (+ (* (fma t_2 (cos x) t_3) 1.5) 3.0))
     (if (<= x 0.036)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (* t_0 (sqrt 2.0)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        (fma 1.5 (fma (cos x) t_2 t_3) 3.0))
       (/ t_1 (fma 1.5 (+ (* t_2 (cos x)) t_3) 3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * t_0) + 2.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = (3.0 - sqrt(5.0)) * cos(y);
	double tmp;
	if (x <= -0.037) {
		tmp = t_1 / ((fma(t_2, cos(x), t_3) * 1.5) + 3.0);
	} else if (x <= 0.036) {
		tmp = fma((fma(-0.0625, sin(y), x) * (t_0 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_2, t_3), 3.0);
	} else {
		tmp = t_1 / fma(1.5, ((t_2 * cos(x)) + t_3), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * t_0) + 2.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	tmp = 0.0
	if (x <= -0.037)
		tmp = Float64(t_1 / Float64(Float64(fma(t_2, cos(x), t_3) * 1.5) + 3.0));
	elseif (x <= 0.036)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * Float64(t_0 * sqrt(2.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(1.5, fma(cos(x), t_2, t_3), 3.0));
	else
		tmp = Float64(t_1 / fma(1.5, Float64(Float64(t_2 * cos(x)) + t_3), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.037], N[(t$95$1 / N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] * 1.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.036], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(1.5 * N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0369999999999999982

   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

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   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 52.7

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

   8. Applied rewrites 52.7%

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

   10. Applied rewrites 52.6%

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

   12. Applied rewrites 52.7%

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

   if -0.0369999999999999982 < x < 0.0359999999999999973

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if 0.0359999999999999973 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 99.0%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

   10. Applied rewrites 64.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot 1.5 + 3}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right) \cdot \cos y, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_1\right), 3\right)\\ t_4 := t\_0 \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(t\_2, \cos x, t\_1\right) \cdot 1.5 + 3}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot t\_4, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, t\_4 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma 1.5 (fma (cos x) t_2 t_1) 3.0))
        (t_4 (* t_0 (sqrt 2.0))))
   (if (<= x -0.037)
     (/
      (+ (* (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x))) t_0) 2.0)
      (+ (* (fma t_2 (cos x) t_1) 1.5) 3.0))
     (if (<= x 0.036)
       (/
        (fma (* (fma -0.0625 (sin y) x) t_4) (fma -0.0625 (sin x) (sin y)) 2.0)
        t_3)
       (/ (fma (sin x) (* t_4 (fma (sin x) -0.0625 (sin y))) 2.0) t_3)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = (3.0 - sqrt(5.0)) * cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(1.5, fma(cos(x), t_2, t_1), 3.0);
	double t_4 = t_0 * sqrt(2.0);
	double tmp;
	if (x <= -0.037) {
		tmp = ((((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * t_0) + 2.0) / ((fma(t_2, cos(x), t_1) * 1.5) + 3.0);
	} else if (x <= 0.036) {
		tmp = fma((fma(-0.0625, sin(y), x) * t_4), fma(-0.0625, sin(x), sin(y)), 2.0) / t_3;
	} else {
		tmp = fma(sin(x), (t_4 * fma(sin(x), -0.0625, sin(y))), 2.0) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(1.5, fma(cos(x), t_2, t_1), 3.0)
	t_4 = Float64(t_0 * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.037)
		tmp = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * t_0) + 2.0) / Float64(Float64(fma(t_2, cos(x), t_1) * 1.5) + 3.0));
	elseif (x <= 0.036)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * t_4), fma(-0.0625, sin(x), sin(y)), 2.0) / t_3);
	else
		tmp = Float64(fma(sin(x), Float64(t_4 * fma(sin(x), -0.0625, sin(y))), 2.0) / t_3);
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.037], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 1.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.036], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$4 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0369999999999999982

   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

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   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 52.7

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

   8. Applied rewrites 52.7%

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

   10. Applied rewrites 52.6%

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

   12. Applied rewrites 52.7%

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

   if -0.0369999999999999982 < x < 0.0359999999999999973

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if 0.0359999999999999973 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 99.0%

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

   7. Applied rewrites 99.1%

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

      1. lift-fma.f64 N/A

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in y around 0

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

       1. lower-sin.f64 64.9

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

   12. Applied rewrites 64.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot 1.5 + 3}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)\\ t_1 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ t_2 := \frac{\mathsf{fma}\left(\sin x, t\_1 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot t\_1, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          1.5
          (fma (cos x) (- (sqrt 5.0) 1.0) (* (- 3.0 (sqrt 5.0)) (cos y)))
          3.0))
        (t_1 (* (- (cos x) (cos y)) (sqrt 2.0)))
        (t_2 (/ (fma (sin x) (* t_1 (fma (sin x) -0.0625 (sin y))) 2.0) t_0)))
   (if (<= x -0.037)
     t_2
     (if (<= x 0.036)
       (/
        (fma (* (fma -0.0625 (sin y) x) t_1) (fma -0.0625 (sin x) (sin y)) 2.0)
        t_0)
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	double t_1 = (cos(x) - cos(y)) * sqrt(2.0);
	double t_2 = fma(sin(x), (t_1 * fma(sin(x), -0.0625, sin(y))), 2.0) / t_0;
	double tmp;
	if (x <= -0.037) {
		tmp = t_2;
	} else if (x <= 0.036) {
		tmp = fma((fma(-0.0625, sin(y), x) * t_1), fma(-0.0625, sin(x), sin(y)), 2.0) / t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0)
	t_1 = Float64(Float64(cos(x) - cos(y)) * sqrt(2.0))
	t_2 = Float64(fma(sin(x), Float64(t_1 * fma(sin(x), -0.0625, sin(y))), 2.0) / t_0)
	tmp = 0.0
	if (x <= -0.037)
		tmp = t_2;
	elseif (x <= 0.036)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * t_1), fma(-0.0625, sin(x), sin(y)), 2.0) / t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.037], t$95$2, If[LessEqual[x, 0.036], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0369999999999999982 or 0.0359999999999999973 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in 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%

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

   7. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in y around 0

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

       1. lower-sin.f64 58.3

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

   12. Applied rewrites 58.3%

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

   if -0.0369999999999999982 < x < 0.0359999999999999973

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 99.2

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

   10. Applied rewrites 99.2%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.037:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)\\ t_1 := \frac{\mathsf{fma}\left(\sin x, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          1.5
          (fma (cos x) (- (sqrt 5.0) 1.0) (* (- 3.0 (sqrt 5.0)) (cos y)))
          3.0))
        (t_1
         (/
          (fma
           (sin x)
           (* (* (- (cos x) (cos y)) (sqrt 2.0)) (fma (sin x) -0.0625 (sin y)))
           2.0)
          t_0)))
   (if (<= x -0.034)
     t_1
     (if (<= x 0.022)
       (/
        (fma
         (* (* (- 1.0 (cos y)) (sqrt 2.0)) (fma -0.0625 (sin y) x))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	double t_1 = fma(sin(x), (((cos(x) - cos(y)) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / t_0;
	double tmp;
	if (x <= -0.034) {
		tmp = t_1;
	} else if (x <= 0.022) {
		tmp = fma((((1.0 - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(y), x)), fma(-0.0625, sin(x), sin(y)), 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0)
	t_1 = Float64(fma(sin(x), Float64(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / t_0)
	tmp = 0.0
	if (x <= -0.034)
		tmp = t_1;
	elseif (x <= 0.022)
		tmp = Float64(fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(y), x)), fma(-0.0625, sin(x), sin(y)), 2.0) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.034], t$95$1, If[LessEqual[x, 0.022], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.034000000000000002 or 0.021999999999999999 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in 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%

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

   7. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in y around 0

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

       1. lower-sin.f64 58.3

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

   12. Applied rewrites 58.3%

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

   if -0.034000000000000002 < x < 0.021999999999999999

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sin.f64 99.0

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

   10. Applied rewrites 99.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0 \cdot \cos y\right), 3\right)\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), t\_2 \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(t\_2 \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_0 (cos y))) 3.0))
        (t_2 (pow (sin x) 2.0)))
   (if (<= x -0.034)
     (*
      (fma (fma (cos x) -0.0625 0.0625) (* t_2 (sqrt 2.0)) 2.0)
      (/
       0.3333333333333333
       (fma (* 0.5 t_0) (cos y) (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
     (if (<= x 0.022)
       (/
        (fma
         (* (* (- 1.0 (cos y)) (sqrt 2.0)) (fma -0.0625 (sin y) x))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        t_1)
       (/
        (+ (* (* (* t_2 -0.0625) (sqrt 2.0)) (- (cos x) (cos y))) 2.0)
        t_1)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_0 * cos(y))), 3.0);
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -0.034) {
		tmp = fma(fma(cos(x), -0.0625, 0.0625), (t_2 * sqrt(2.0)), 2.0) * (0.3333333333333333 / fma((0.5 * t_0), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	} else if (x <= 0.022) {
		tmp = fma((((1.0 - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(y), x)), fma(-0.0625, sin(x), sin(y)), 2.0) / t_1;
	} else {
		tmp = ((((t_2 * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_0 * cos(y))), 3.0)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -0.034)
		tmp = Float64(fma(fma(cos(x), -0.0625, 0.0625), Float64(t_2 * sqrt(2.0)), 2.0) * Float64(0.3333333333333333 / fma(Float64(0.5 * t_0), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))));
	elseif (x <= 0.022)
		tmp = Float64(fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(y), x)), fma(-0.0625, sin(x), sin(y)), 2.0) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_2 * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -0.034], N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.022], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(t$95$2 * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.034000000000000002

   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 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

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-cos.f64 48.7

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

   5. Applied rewrites 48.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

   7. Applied rewrites 48.7%

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

   9. Applied rewrites 48.8%

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

   if -0.034000000000000002 < x < 0.021999999999999999

   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 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 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sin.f64 99.0

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

   10. Applied rewrites 99.0%

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

   if 0.021999999999999999 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 99.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 61.7

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

   8. Applied rewrites 61.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -19:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), t\_1 \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, t\_2\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(t\_1 \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_0 \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -19.0)
     (*
      (fma (fma (cos x) -0.0625 0.0625) (* t_1 (sqrt 2.0)) 2.0)
      (/
       0.3333333333333333
       (fma (* 0.5 t_0) (cos y) (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
     (if (<= x 3.1e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* (fma 0.5 (fma (cos y) t_0 t_2) 1.0) 3.0))
       (/
        (+ (* (* (* t_1 -0.0625) (sqrt 2.0)) (- (cos x) (cos y))) 2.0)
        (fma 1.5 (fma (cos x) t_2 (* t_0 (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -19.0) {
		tmp = fma(fma(cos(x), -0.0625, 0.0625), (t_1 * sqrt(2.0)), 2.0) * (0.3333333333333333 / fma((0.5 * t_0), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	} else if (x <= 3.1e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(0.5, fma(cos(y), t_0, t_2), 1.0) * 3.0);
	} else {
		tmp = ((((t_1 * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_2, (t_0 * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -19.0)
		tmp = Float64(fma(fma(cos(x), -0.0625, 0.0625), Float64(t_1 * sqrt(2.0)), 2.0) * Float64(0.3333333333333333 / fma(Float64(0.5 * t_0), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))));
	elseif (x <= 3.1e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(0.5, fma(cos(y), t_0, t_2), 1.0) * 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_1 * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_2, Float64(t_0 * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -19.0], N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$1 * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -19

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-cos.f64 49.0

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

   5. Applied rewrites 49.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

   7. Applied rewrites 49.0%

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

   9. Applied rewrites 49.1%

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

   if -19 < x < 3.10000000000000014e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 97.8

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

   8. Applied rewrites 97.8%

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

   if 3.10000000000000014e-5 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 99.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 61.7

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

   8. Applied rewrites 61.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_2 := {\sin x}^{2} \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;x \leq -19:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_2, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(t\_3, \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_1, 2\right)}{\mathsf{fma}\left(t\_3 \cdot \cos x, 3, \mathsf{fma}\left(0.5 \cdot \cos y, t\_0, 1\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (cos x) -0.0625 0.0625))
        (t_2 (* (pow (sin x) 2.0) (sqrt 2.0)))
        (t_3 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= x -19.0)
     (*
      (fma t_1 t_2 2.0)
      (/ 0.3333333333333333 (fma (* 0.5 t_0) (cos y) (fma t_3 (cos x) 1.0))))
     (if (<= x 3.1e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* (fma 0.5 (fma (cos y) t_0 (- (sqrt 5.0) 1.0)) 1.0) 3.0))
       (/
        (fma t_2 t_1 2.0)
        (fma (* t_3 (cos x)) 3.0 (* (fma (* 0.5 (cos y)) t_0 1.0) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(cos(x), -0.0625, 0.0625);
	double t_2 = pow(sin(x), 2.0) * sqrt(2.0);
	double t_3 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (x <= -19.0) {
		tmp = fma(t_1, t_2, 2.0) * (0.3333333333333333 / fma((0.5 * t_0), cos(y), fma(t_3, cos(x), 1.0)));
	} else if (x <= 3.1e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(0.5, fma(cos(y), t_0, (sqrt(5.0) - 1.0)), 1.0) * 3.0);
	} else {
		tmp = fma(t_2, t_1, 2.0) / fma((t_3 * cos(x)), 3.0, (fma((0.5 * cos(y)), t_0, 1.0) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(cos(x), -0.0625, 0.0625)
	t_2 = Float64((sin(x) ^ 2.0) * sqrt(2.0))
	t_3 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (x <= -19.0)
		tmp = Float64(fma(t_1, t_2, 2.0) * Float64(0.3333333333333333 / fma(Float64(0.5 * t_0), cos(y), fma(t_3, cos(x), 1.0))));
	elseif (x <= 3.1e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(0.5, fma(cos(y), t_0, Float64(sqrt(5.0) - 1.0)), 1.0) * 3.0));
	else
		tmp = Float64(fma(t_2, t_1, 2.0) / fma(Float64(t_3 * cos(x)), 3.0, Float64(fma(Float64(0.5 * cos(y)), t_0, 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]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[x, -19.0], N[(N[(t$95$1 * t$95$2 + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$3 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * t$95$1 + 2.0), $MachinePrecision] / N[(N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -19

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-cos.f64 49.0

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

   5. Applied rewrites 49.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

   7. Applied rewrites 49.0%

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

   9. Applied rewrites 49.1%

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

   if -19 < x < 3.10000000000000014e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 97.8

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

   8. Applied rewrites 97.8%

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

   if 3.10000000000000014e-5 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-cos.f64 61.6

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

   5. Applied rewrites 61.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

   7. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. div-sub N/A

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

      13. lift--.f64 N/A

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

      14. lift-/.f64 N/A

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

   9. Applied rewrites 61.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right) \cdot \cos x, 3, \mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{if}\;x \leq -19:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (fma
           (fma (cos x) -0.0625 0.0625)
           (* (pow (sin x) 2.0) (sqrt 2.0))
           2.0)
          (/
           0.3333333333333333
           (fma
            (* 0.5 t_0)
            (cos y)
            (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))))
   (if (<= x -19.0)
     t_1
     (if (<= x 3.1e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* (fma 0.5 (fma (cos y) t_0 (- (sqrt 5.0) 1.0)) 1.0) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(fma(cos(x), -0.0625, 0.0625), (pow(sin(x), 2.0) * sqrt(2.0)), 2.0) * (0.3333333333333333 / fma((0.5 * t_0), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	double tmp;
	if (x <= -19.0) {
		tmp = t_1;
	} else if (x <= 3.1e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(0.5, fma(cos(y), t_0, (sqrt(5.0) - 1.0)), 1.0) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(fma(cos(x), -0.0625, 0.0625), Float64((sin(x) ^ 2.0) * sqrt(2.0)), 2.0) * Float64(0.3333333333333333 / fma(Float64(0.5 * t_0), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))))
	tmp = 0.0
	if (x <= -19.0)
		tmp = t_1;
	elseif (x <= 3.1e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(0.5, fma(cos(y), t_0, Float64(sqrt(5.0) - 1.0)), 1.0) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -19.0], t$95$1, If[LessEqual[x, 3.1e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -19 or 3.10000000000000014e-5 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-cos.f64 54.9

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

   5. Applied rewrites 54.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

   7. Applied rewrites 54.9%

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

   9. Applied rewrites 54.9%

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

   if -19 < x < 3.10000000000000014e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 97.8

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

   8. Applied rewrites 97.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, t\_2, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0 \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (pow (sin x) 2.0)))
   (if (<= x -19.0)
     (/
      (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) t_2 2.0)
      (fma 1.5 (fma (cos x) t_0 (* t_1 (cos y))) 3.0))
     (if (<= x 3.1e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* (fma 0.5 (fma (cos y) t_1 t_0) 1.0) 3.0))
       (/
        (fma (* t_2 (sqrt 2.0)) (fma (cos x) -0.0625 0.0625) 2.0)
        (fma 1.5 (fma (cos y) t_1 (* t_0 (cos x))) 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 t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -19.0) {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_2, 2.0) / fma(1.5, fma(cos(x), t_0, (t_1 * cos(y))), 3.0);
	} else if (x <= 3.1e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(0.5, fma(cos(y), t_1, t_0), 1.0) * 3.0);
	} else {
		tmp = fma((t_2 * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(y), t_1, (t_0 * cos(x))), 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))
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -19.0)
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_2, 2.0) / fma(1.5, fma(cos(x), t_0, Float64(t_1 * cos(y))), 3.0));
	elseif (x <= 3.1e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(0.5, fma(cos(y), t_1, t_0), 1.0) * 3.0));
	else
		tmp = Float64(fma(Float64(t_2 * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(y), t_1, Float64(t_0 * cos(x))), 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]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -19.0], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2 + 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], If[LessEqual[x, 3.1e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -19

   1. Initial program 99.1%

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

   3. Taylor expanded in 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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 49.0

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

   8. Applied rewrites 49.0%

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

   if -19 < x < 3.10000000000000014e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 97.8

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

   8. Applied rewrites 97.8%

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

   if 3.10000000000000014e-5 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-cos.f64 61.6

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 61.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0 \cdot \cos x\right), 3\right)}\\ \mathbf{if}\;x \leq -19:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma (cos x) -0.0625 0.0625)
           2.0)
          (fma 1.5 (fma (cos y) t_1 (* t_0 (cos x))) 3.0))))
   (if (<= x -19.0)
     t_2
     (if (<= x 3.1e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* (fma 0.5 (fma (cos y) t_1 t_0) 1.0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(y), t_1, (t_0 * cos(x))), 3.0);
	double tmp;
	if (x <= -19.0) {
		tmp = t_2;
	} else if (x <= 3.1e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(0.5, fma(cos(y), t_1, t_0), 1.0) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(y), t_1, Float64(t_0 * cos(x))), 3.0))
	tmp = 0.0
	if (x <= -19.0)
		tmp = t_2;
	elseif (x <= 3.1e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(0.5, fma(cos(y), t_1, t_0), 1.0) * 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -19.0], t$95$2, If[LessEqual[x, 3.1e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -19 or 3.10000000000000014e-5 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-cos.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 54.9%

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

   if -19 < x < 3.10000000000000014e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 97.8

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

   8. Applied rewrites 97.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(\frac{t\_0}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -2400000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (fma
           (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (*
           (+
            (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))
            (+ (* (/ t_0 2.0) (cos x)) 1.0))
           3.0))))
   (if (<= y -2400000000.0)
     t_1
     (if (<= y 1.75e-19)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma (cos x) -0.0625 0.0625) 2.0)
        (fma 1.5 (fma (cos x) t_0 (/ 4.0 (+ (sqrt 5.0) 3.0))) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(((0.5 - (cos((y + y)) * 0.5)) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (((((3.0 - sqrt(5.0)) / 2.0) * cos(y)) + (((t_0 / 2.0) * cos(x)) + 1.0)) * 3.0);
	double tmp;
	if (y <= -2400000000.0) {
		tmp = t_1;
	} else if (y <= 1.75e-19) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), t_0, (4.0 / (sqrt(5.0) + 3.0))), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(Float64(Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)) + Float64(Float64(Float64(t_0 / 2.0) * cos(x)) + 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -2400000000.0)
		tmp = t_1;
	elseif (y <= 1.75e-19)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(4.0 / Float64(sqrt(5.0) + 3.0))), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2400000000.0], t$95$1, If[LessEqual[y, 1.75e-19], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e9 or 1.75000000000000008e-19 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 58.2%

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

   if -2.4e9 < y < 1.75000000000000008e-19

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-cos.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 98.9

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

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.0%

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3}\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2400000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3}\right), 3\right)}\\ \mathbf{if}\;x \leq -19:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1150000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_0\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma (cos x) -0.0625 0.0625)
           2.0)
          (fma 1.5 (fma (cos x) t_0 (/ 4.0 (+ (sqrt 5.0) 3.0))) 3.0))))
   (if (<= x -19.0)
     t_1
     (if (<= x 1150000.0)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_0) 1.0) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), t_0, (4.0 / (sqrt(5.0) + 3.0))), 3.0);
	double tmp;
	if (x <= -19.0) {
		tmp = t_1;
	} else if (x <= 1150000.0) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), t_0), 1.0) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(4.0 / Float64(sqrt(5.0) + 3.0))), 3.0))
	tmp = 0.0
	if (x <= -19.0)
		tmp = t_1;
	elseif (x <= 1150000.0)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_0), 1.0) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -19.0], t$95$1, If[LessEqual[x, 1150000.0], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -19 or 1.15e6 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-cos.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 54.0

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

   8. Applied rewrites 54.0%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.1%

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

   if -19 < x < 1.15e6

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. 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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 96.5

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

   8. Applied rewrites 96.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3}\right), 3\right)}\\ \mathbf{elif}\;x \leq 1150000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\cos x, t\_1, t\_2\right)\\ \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, t\_0, 2\right)}{\mathsf{fma}\left(0.5, t\_3, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1150000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, t\_1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, t\_3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (cos x) t_1 t_2)))
   (if (<= x -19.0)
     (*
      (/
       (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) t_0 2.0)
       (fma 0.5 t_3 1.0))
      0.3333333333333333)
     (if (<= x 1150000.0)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* (fma 0.5 (fma (cos y) t_2 t_1) 1.0) 3.0))
       (/
        (fma (* t_0 (sqrt 2.0)) (fma (cos x) -0.0625 0.0625) 2.0)
        (fma 1.5 t_3 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(cos(x), t_1, t_2);
	double tmp;
	if (x <= -19.0) {
		tmp = (fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_0, 2.0) / fma(0.5, t_3, 1.0)) * 0.3333333333333333;
	} else if (x <= 1150000.0) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(0.5, fma(cos(y), t_2, t_1), 1.0) * 3.0);
	} else {
		tmp = fma((t_0 * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, t_3, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(cos(x), t_1, t_2)
	tmp = 0.0
	if (x <= -19.0)
		tmp = Float64(Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_0, 2.0) / fma(0.5, t_3, 1.0)) * 0.3333333333333333);
	elseif (x <= 1150000.0)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(0.5, fma(cos(y), t_2, t_1), 1.0) * 3.0));
	else
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, t_3, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[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[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision]}, If[LessEqual[x, -19.0], N[(N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(0.5 * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1150000.0], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * t$95$3 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -19

   1. Initial program 99.1%

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

   3. Taylor expanded in 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 47.2%

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

   if -19 < x < 1.15e6

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. 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. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 1\right)} \]

      9. lower-sqrt.f64 96.5

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

   8. Applied rewrites 96.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)}} \]

   if 1.15e6 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 62.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      13. lower-sqrt.f64 62.1

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1150000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_1\right)\\ \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, t\_0, 2\right)}{\mathsf{fma}\left(0.5, t\_2, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1150000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, \sqrt{5}\right), 0.5\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, t\_2, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma (cos x) (- (sqrt 5.0) 1.0) t_1)))
   (if (<= x -19.0)
     (*
      (/
       (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) t_0 2.0)
       (fma 0.5 t_2 1.0))
      0.3333333333333333)
     (if (<= x 1150000.0)
       (*
        (/
         (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma (cos y) t_1 (sqrt 5.0)) 0.5))
        0.3333333333333333)
       (/
        (fma (* t_0 (sqrt 2.0)) (fma (cos x) -0.0625 0.0625) 2.0)
        (fma 1.5 t_2 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(cos(x), (sqrt(5.0) - 1.0), t_1);
	double tmp;
	if (x <= -19.0) {
		tmp = (fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_0, 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333;
	} else if (x <= 1150000.0) {
		tmp = (fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_1, sqrt(5.0)), 0.5)) * 0.3333333333333333;
	} else {
		tmp = fma((t_0 * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, t_2, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(cos(x), Float64(sqrt(5.0) - 1.0), t_1)
	tmp = 0.0
	if (x <= -19.0)
		tmp = Float64(Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_0, 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333);
	elseif (x <= 1150000.0)
		tmp = Float64(Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_1, sqrt(5.0)), 0.5)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, t_2, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[x, -19.0], N[(N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(0.5 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1150000.0], N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * t$95$2 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -19

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 47.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -19 < x < 1.15e6

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\color{blue}{\frac{\left(3 - \sqrt{5}\right) \cdot \cos y}{2}} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{1}{2}} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \color{blue}{\frac{1}{2}} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      9. lower-fma.f64 99.5

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{1}{2}, \color{blue}{1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{1}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1}\right)} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \cdot \frac{1}{3}} \]

   9. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)} \cdot 0.3333333333333333} \]

   if 1.15e6 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 62.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      13. lower-sqrt.f64 62.1

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1150000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0\right), 3\right)}\\ \mathbf{if}\;x \leq -19:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1150000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), 0.5\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma (cos x) -0.0625 0.0625)
           2.0)
          (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) t_0) 3.0))))
   (if (<= x -19.0)
     t_1
     (if (<= x 1150000.0)
       (*
        (/
         (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) 0.5))
        0.3333333333333333)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), t_0), 3.0);
	double tmp;
	if (x <= -19.0) {
		tmp = t_1;
	} else if (x <= 1150000.0) {
		tmp = (fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), t_0), 3.0))
	tmp = 0.0
	if (x <= -19.0)
		tmp = t_1;
	elseif (x <= 1150000.0)
		tmp = Float64(Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -19.0], t$95$1, If[LessEqual[x, 1150000.0], N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -19 or 1.15e6 < x

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 55.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 55.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      13. lower-sqrt.f64 54.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 54.0%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

   if -19 < x < 1.15e6

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\color{blue}{\frac{\left(3 - \sqrt{5}\right) \cdot \cos y}{2}} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{1}{2}} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \left(\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \color{blue}{\frac{1}{2}} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      9. lower-fma.f64 99.5

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{1}{2}, \color{blue}{1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{1}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1}\right)} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \cdot \frac{1}{3}} \]

   9. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 1150000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 59.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma (cos x) -0.0625 0.0625) 2.0)
  (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-cos.f64 58.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 58.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   13. lower-sqrt.f64 55.6

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 55.6%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
9. Add Preprocessing

Alternative 28: 45.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   1.5
   (fma (cos y) (- 3.0 (sqrt 5.0)) (* (- (sqrt 5.0) 1.0) (cos x)))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-cos.f64 58.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 58.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   13. lower-sqrt.f64 55.6

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 55.6%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 40.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

12. Taylor expanded in x around inf

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-*r* N/A

       \[\leadsto \frac{2}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

14. Applied rewrites 43.0%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 3\right)}} \]

15. Final simplification 43.0%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]
16. Add Preprocessing

Alternative 29: 42.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (- (sqrt 5.0) 1.0)) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), (sqrt(5.0) - 1.0)), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(sqrt(5.0) - 1.0)), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-cos.f64 58.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 58.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   13. lower-sqrt.f64 55.6

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 55.6%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 40.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

12. Taylor expanded in x around 0

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-*r* N/A

       \[\leadsto \frac{2}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

14. Applied rewrites 40.7%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]
15. Add Preprocessing

Alternative 30: 43.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-cos.f64 58.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 58.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   13. lower-sqrt.f64 55.6

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 55.6%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 40.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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))))))