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

Percentage Accurate: 99.3% → 99.3%

Time: 24.1s

Alternatives: 42

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (*
   3.0
   (+
    (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
    (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / (3.0d0 * ((1.0d0 + (((sqrt(5.0d0) - 1.0d0) / 2.0d0) * cos(x))) + (((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((1.0 + (((Math.sqrt(5.0) - 1.0) / 2.0) * Math.cos(x))) + (((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y))));
}

Python

def code(x, y):
	return (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / (3.0 * ((1.0 + (((math.sqrt(5.0) - 1.0) / 2.0) * math.cos(x))) + (((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
end

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (*
   3.0
   (+
    (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
    (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / (3.0d0 * ((1.0d0 + (((sqrt(5.0d0) - 1.0d0) / 2.0d0) * cos(x))) + (((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((1.0 + (((Math.sqrt(5.0) - 1.0) / 2.0) * Math.cos(x))) + (((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y))));
}

Python

def code(x, y):
	return (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / (3.0 * ((1.0 + (((math.sqrt(5.0) - 1.0) / 2.0) * math.cos(x))) + (((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
end

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval 99.5

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $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 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_1 := 3 + \sqrt{5}\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ t_3 := \mathsf{fma}\left(t\_2, \cos x, 1\right)\\ \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\frac{\cos y}{t\_1}, 2, t\_3\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{2}{t\_1} \cdot 3, \cos y, t\_3 \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
           (- (cos x) (cos y)))
          2.0))
        (t_1 (+ 3.0 (sqrt 5.0)))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5))
        (t_3 (fma t_2 (cos x) 1.0)))
   (if (<= x -0.78)
     (/ t_0 (* (fma (/ (cos y) t_1) 2.0 t_3) 3.0))
     (if (<= x 0.026)
       (/
        (fma
         (sqrt 2.0)
         (*
          (*
           (fma
            (*
             (fma
              (fma -0.001388888888888889 (* x x) 0.041666666666666664)
              (* x x)
              -0.5)
             x)
            x
            (- 1.0 (cos y)))
           (fma (sin x) -0.0625 (sin y)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (fma (* (/ 2.0 t_1) 3.0) (cos y) (* t_3 3.0)))
       (/
        t_0
        (fma
         (fma t_2 (cos x) (* (* (- 3.0 (sqrt 5.0)) 0.5) (cos y)))
         3.0
         3.0))))))

C

double code(double x, double y) {
	double t_0 = (((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0;
	double t_1 = 3.0 + sqrt(5.0);
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double t_3 = fma(t_2, cos(x), 1.0);
	double tmp;
	if (x <= -0.78) {
		tmp = t_0 / (fma((cos(y) / t_1), 2.0, t_3) * 3.0);
	} else if (x <= 0.026) {
		tmp = fma(sqrt(2.0), ((fma((fma(fma(-0.001388888888888889, (x * x), 0.041666666666666664), (x * x), -0.5) * x), x, (1.0 - cos(y))) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(((2.0 / t_1) * 3.0), cos(y), (t_3 * 3.0));
	} else {
		tmp = t_0 / fma(fma(t_2, cos(x), (((3.0 - sqrt(5.0)) * 0.5) * cos(y))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0)
	t_1 = Float64(3.0 + sqrt(5.0))
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	t_3 = fma(t_2, cos(x), 1.0)
	tmp = 0.0
	if (x <= -0.78)
		tmp = Float64(t_0 / Float64(fma(Float64(cos(y) / t_1), 2.0, t_3) * 3.0));
	elseif (x <= 0.026)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(Float64(fma(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), -0.5) * x), x, Float64(1.0 - cos(y))) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(Float64(2.0 / t_1) * 3.0), cos(y), Float64(t_3 * 3.0)));
	else
		tmp = Float64(t_0 / fma(fma(t_2, cos(x), Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * cos(y))), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.78], N[(t$95$0 / N[(N[(N[(N[Cos[y], $MachinePrecision] / t$95$1), $MachinePrecision] * 2.0 + t$95$3), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.026], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(2.0 / t$95$1), $MachinePrecision] * 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.78000000000000003

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.8%

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

   if -0.78000000000000003 < x < 0.0259999999999999988

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 99.7%

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

   if 0.0259999999999999988 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.3

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

   7. Applied rewrites 65.3%

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

   9. Applied rewrites 65.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{2}{3 + \sqrt{5}} \cdot 3, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;x \leq -0.15:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(t\_2, \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\left(\frac{t\_0}{2} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, \left(t\_0 \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (+
          (*
           (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
           (- (cos x) (cos y)))
          2.0))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= x -0.15)
     (/
      t_1
      (* (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 2.0 (fma t_2 (cos x) 1.0)) 3.0))
     (if (<= x 0.026)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma (* x x) -0.5 (- 1.0 (cos y))) (fma (sin x) -0.0625 (sin y)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (*
         (+
          (* (/ t_0 2.0) (cos y))
          (+ (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)) 1.0))
         3.0))
       (/ t_1 (fma (fma t_2 (cos x) (* (* t_0 0.5) (cos y))) 3.0 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0;
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (x <= -0.15) {
		tmp = t_1 / (fma((cos(y) / (3.0 + sqrt(5.0))), 2.0, fma(t_2, cos(x), 1.0)) * 3.0);
	} else if (x <= 0.026) {
		tmp = fma(sqrt(2.0), ((fma((x * x), -0.5, (1.0 - cos(y))) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / ((((t_0 / 2.0) * cos(y)) + ((((sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0);
	} else {
		tmp = t_1 / fma(fma(t_2, cos(x), ((t_0 * 0.5) * cos(y))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0)
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (x <= -0.15)
		tmp = Float64(t_1 / Float64(fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 2.0, fma(t_2, cos(x), 1.0)) * 3.0));
	elseif (x <= 0.026)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(Float64(x * x), -0.5, Float64(1.0 - cos(y))) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(Float64(Float64(Float64(t_0 / 2.0) * cos(y)) + Float64(Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0));
	else
		tmp = Float64(t_1 / fma(fma(t_2, cos(x), Float64(Float64(t_0 * 0.5) * cos(y))), 3.0, 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[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.15], N[(t$95$1 / N[(N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.026], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(t$95$0 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.149999999999999994

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.8%

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

   if -0.149999999999999994 < x < 0.0259999999999999988

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 0.0259999999999999988 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.3

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

   7. Applied rewrites 65.3%

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

   9. Applied rewrites 65.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.15:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \sqrt{2}\\ t_1 := \cos x - \cos y\\ t_2 := 3 + \sqrt{5}\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.036:\\ \;\;\;\;\frac{\left(t\_3 \cdot t\_0\right) \cdot t\_1 + 2}{\mathsf{fma}\left(\frac{\cos y}{t\_2}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot t\_3\right) \cdot t\_1 + 2}{\mathsf{fma}\left(t\_4, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 0.5, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\left(\frac{2}{t\_2} \cdot \cos y + \left(\frac{t\_4}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (sqrt 2.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (+ 3.0 (sqrt 5.0)))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.036)
     (/
      (+ (* (* t_3 t_0) t_1) 2.0)
      (*
       (fma (/ (cos y) t_2) 2.0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
       3.0))
     (if (<= x 0.0011)
       (/
        (+ (* (* (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)) t_3) t_1) 2.0)
        (*
         (fma
          t_4
          (fma (* -0.25 x) x 0.5)
          (fma (cos y) (* (- 3.0 (sqrt 5.0)) 0.5) 1.0))
         3.0))
       (/
        (fma t_1 (* t_0 (fma (sin x) -0.0625 (sin y))) 2.0)
        (*
         (+ (* (/ 2.0 t_2) (cos y)) (+ (* (/ t_4 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * sqrt(2.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 + sqrt(5.0);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.036) {
		tmp = (((t_3 * t_0) * t_1) + 2.0) / (fma((cos(y) / t_2), 2.0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else if (x <= 0.0011) {
		tmp = (((((sin(x) - (sin(y) / 16.0)) * sqrt(2.0)) * t_3) * t_1) + 2.0) / (fma(t_4, fma((-0.25 * x), x, 0.5), fma(cos(y), ((3.0 - sqrt(5.0)) * 0.5), 1.0)) * 3.0);
	} else {
		tmp = fma(t_1, (t_0 * fma(sin(x), -0.0625, sin(y))), 2.0) / ((((2.0 / t_2) * cos(y)) + (((t_4 / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * sqrt(2.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 + sqrt(5.0))
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.036)
		tmp = Float64(Float64(Float64(Float64(t_3 * t_0) * t_1) + 2.0) / Float64(fma(Float64(cos(y) / t_2), 2.0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	elseif (x <= 0.0011)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0)) * t_3) * t_1) + 2.0) / Float64(fma(t_4, fma(Float64(-0.25 * x), x, 0.5), fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) * 0.5), 1.0)) * 3.0));
	else
		tmp = Float64(fma(t_1, Float64(t_0 * fma(sin(x), -0.0625, sin(y))), 2.0) / Float64(Float64(Float64(Float64(2.0 / t_2) * cos(y)) + Float64(Float64(Float64(t_4 / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.036], N[(N[(N[(N[(t$95$3 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] / t$95$2), $MachinePrecision] * 2.0 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0011], N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$4 * N[(N[(-0.25 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(2.0 / t$95$2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$4 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0359999999999999973

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.8%

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

   if -0.0359999999999999973 < x < 0.00110000000000000007

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   if 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.8

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

   7. Applied rewrites 65.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 65.8

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

   9. Applied rewrites 65.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.036:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 0.5, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \sqrt{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ t_3 := 3 + \sqrt{5}\\ t_4 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ \mathbf{if}\;x \leq -0.036:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot t\_0\right) \cdot t\_2 + 2}{\mathsf{fma}\left(\frac{\cos y}{t\_3}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_4 \cdot t\_2\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\left(\left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + \mathsf{fma}\left(t\_1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_0 \cdot t\_4, 2\right)}{\left(\frac{2}{t\_3} \cdot \cos y + \left(\frac{t\_1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (sqrt 2.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (+ 3.0 (sqrt 5.0)))
        (t_4 (fma (sin x) -0.0625 (sin y))))
   (if (<= x -0.036)
     (/
      (+ (* (* (- (sin y) (/ (sin x) 16.0)) t_0) t_2) 2.0)
      (*
       (fma (/ (cos y) t_3) 2.0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
       3.0))
     (if (<= x 0.0011)
       (/
        (fma (sqrt 2.0) (* (* t_4 t_2) (fma (sin y) -0.0625 (sin x))) 2.0)
        (*
         (+
          (* (* 0.5 (cos y)) (- 3.0 (sqrt 5.0)))
          (fma t_1 (fma -0.25 (* x x) 0.5) 1.0))
         3.0))
       (/
        (fma t_2 (* t_0 t_4) 2.0)
        (*
         (+ (* (/ 2.0 t_3) (cos y)) (+ (* (/ t_1 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * sqrt(2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 + sqrt(5.0);
	double t_4 = fma(sin(x), -0.0625, sin(y));
	double tmp;
	if (x <= -0.036) {
		tmp = ((((sin(y) - (sin(x) / 16.0)) * t_0) * t_2) + 2.0) / (fma((cos(y) / t_3), 2.0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else if (x <= 0.0011) {
		tmp = fma(sqrt(2.0), ((t_4 * t_2) * fma(sin(y), -0.0625, sin(x))), 2.0) / ((((0.5 * cos(y)) * (3.0 - sqrt(5.0))) + fma(t_1, fma(-0.25, (x * x), 0.5), 1.0)) * 3.0);
	} else {
		tmp = fma(t_2, (t_0 * t_4), 2.0) / ((((2.0 / t_3) * cos(y)) + (((t_1 / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * sqrt(2.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 + sqrt(5.0))
	t_4 = fma(sin(x), -0.0625, sin(y))
	tmp = 0.0
	if (x <= -0.036)
		tmp = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * t_0) * t_2) + 2.0) / Float64(fma(Float64(cos(y) / t_3), 2.0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	elseif (x <= 0.0011)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_4 * t_2) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(Float64(Float64(Float64(0.5 * cos(y)) * Float64(3.0 - sqrt(5.0))) + fma(t_1, fma(-0.25, Float64(x * x), 0.5), 1.0)) * 3.0));
	else
		tmp = Float64(fma(t_2, Float64(t_0 * t_4), 2.0) / Float64(Float64(Float64(Float64(2.0 / t_3) * cos(y)) + Float64(Float64(Float64(t_1 / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.036], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] / t$95$3), $MachinePrecision] * 2.0 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0011], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$4 * t$95$2), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(t$95$0 * t$95$4), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(2.0 / t$95$3), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0359999999999999973

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.8%

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

   if -0.0359999999999999973 < x < 0.00110000000000000007

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

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

   if 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.8

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

   7. Applied rewrites 65.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 65.8

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

   9. Applied rewrites 65.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.036:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\left(\left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot t\_1 + 2\\ t_3 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;x \leq -0.036:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(t\_3, \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\left(\left(0.5 \cdot \cos y\right) \cdot t\_0 + \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, \cos x, \left(t\_0 \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (+
          (* (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0))) t_1)
          2.0))
        (t_3 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= x -0.036)
     (/
      t_2
      (* (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 2.0 (fma t_3 (cos x) 1.0)) 3.0))
     (if (<= x 0.0011)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma (sin x) -0.0625 (sin y)) t_1)
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (*
         (+
          (* (* 0.5 (cos y)) t_0)
          (fma (- (sqrt 5.0) 1.0) (fma -0.25 (* x x) 0.5) 1.0))
         3.0))
       (/ t_2 (fma (fma t_3 (cos x) (* (* t_0 0.5) (cos y))) 3.0 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = (((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * t_1) + 2.0;
	double t_3 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (x <= -0.036) {
		tmp = t_2 / (fma((cos(y) / (3.0 + sqrt(5.0))), 2.0, fma(t_3, cos(x), 1.0)) * 3.0);
	} else if (x <= 0.0011) {
		tmp = fma(sqrt(2.0), ((fma(sin(x), -0.0625, sin(y)) * t_1) * fma(sin(y), -0.0625, sin(x))), 2.0) / ((((0.5 * cos(y)) * t_0) + fma((sqrt(5.0) - 1.0), fma(-0.25, (x * x), 0.5), 1.0)) * 3.0);
	} else {
		tmp = t_2 / fma(fma(t_3, cos(x), ((t_0 * 0.5) * cos(y))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * t_1) + 2.0)
	t_3 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (x <= -0.036)
		tmp = Float64(t_2 / Float64(fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 2.0, fma(t_3, cos(x), 1.0)) * 3.0));
	elseif (x <= 0.0011)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(sin(x), -0.0625, sin(y)) * t_1) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(Float64(Float64(Float64(0.5 * cos(y)) * t_0) + fma(Float64(sqrt(5.0) - 1.0), fma(-0.25, Float64(x * x), 0.5), 1.0)) * 3.0));
	else
		tmp = Float64(t_2 / fma(fma(t_3, cos(x), Float64(Float64(t_0 * 0.5) * cos(y))), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.036], N[(t$95$2 / N[(N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0011], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(N[(t$95$0 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0359999999999999973

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.8%

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

   if -0.0359999999999999973 < x < 0.00110000000000000007

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

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

   if 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 65.7%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.036:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\left(\left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_1 := 3 + \sqrt{5}\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ t_3 := \mathsf{fma}\left(t\_2, \cos x, 1\right)\\ \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\frac{\cos y}{t\_1}, 2, t\_3\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{t\_1}, \cos y, t\_3 \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
           (- (cos x) (cos y)))
          2.0))
        (t_1 (+ 3.0 (sqrt 5.0)))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5))
        (t_3 (fma t_2 (cos x) 1.0)))
   (if (<= x -0.007)
     (/ t_0 (* (fma (/ (cos y) t_1) 2.0 t_3) 3.0))
     (if (<= x 0.0011)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (fma (/ 6.0 t_1) (cos y) (* t_3 3.0)))
       (/
        t_0
        (fma
         (fma t_2 (cos x) (* (* (- 3.0 (sqrt 5.0)) 0.5) (cos y)))
         3.0
         3.0))))))

C

double code(double x, double y) {
	double t_0 = (((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0;
	double t_1 = 3.0 + sqrt(5.0);
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double t_3 = fma(t_2, cos(x), 1.0);
	double tmp;
	if (x <= -0.007) {
		tmp = t_0 / (fma((cos(y) / t_1), 2.0, t_3) * 3.0);
	} else if (x <= 0.0011) {
		tmp = fma(sqrt(2.0), ((fma(-0.0625, x, sin(y)) * (1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((6.0 / t_1), cos(y), (t_3 * 3.0));
	} else {
		tmp = t_0 / fma(fma(t_2, cos(x), (((3.0 - sqrt(5.0)) * 0.5) * cos(y))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0)
	t_1 = Float64(3.0 + sqrt(5.0))
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	t_3 = fma(t_2, cos(x), 1.0)
	tmp = 0.0
	if (x <= -0.007)
		tmp = Float64(t_0 / Float64(fma(Float64(cos(y) / t_1), 2.0, t_3) * 3.0));
	elseif (x <= 0.0011)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(6.0 / t_1), cos(y), Float64(t_3 * 3.0)));
	else
		tmp = Float64(t_0 / fma(fma(t_2, cos(x), Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * cos(y))), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.007], N[(t$95$0 / N[(N[(N[(N[Cos[y], $MachinePrecision] / t$95$1), $MachinePrecision] * 2.0 + t$95$3), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0011], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00700000000000000015

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.8%

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

   if -0.00700000000000000015 < x < 0.00110000000000000007

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-sin.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 65.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_1 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ t_2 := \left(3 - \sqrt{5}\right) \cdot 0.5\\ \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{t\_0}{\left(t\_1 \cdot \cos x + \mathsf{fma}\left(t\_2, \cos y, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(t\_1, \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2 \cdot \cos y\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
           (- (cos x) (cos y)))
          2.0))
        (t_1 (fma 0.5 (sqrt 5.0) -0.5))
        (t_2 (* (- 3.0 (sqrt 5.0)) 0.5)))
   (if (<= x -0.007)
     (/ t_0 (* (+ (* t_1 (cos x)) (fma t_2 (cos y) 1.0)) 3.0))
     (if (<= x 0.0011)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* (fma t_1 (cos x) 1.0) 3.0)))
       (/ t_0 (fma (fma t_1 (cos x) (* t_2 (cos y))) 3.0 3.0))))))

C

double code(double x, double y) {
	double t_0 = (((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0;
	double t_1 = fma(0.5, sqrt(5.0), -0.5);
	double t_2 = (3.0 - sqrt(5.0)) * 0.5;
	double tmp;
	if (x <= -0.007) {
		tmp = t_0 / (((t_1 * cos(x)) + fma(t_2, cos(y), 1.0)) * 3.0);
	} else if (x <= 0.0011) {
		tmp = fma(sqrt(2.0), ((fma(-0.0625, x, sin(y)) * (1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (fma(t_1, cos(x), 1.0) * 3.0));
	} else {
		tmp = t_0 / fma(fma(t_1, cos(x), (t_2 * cos(y))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0)
	t_1 = fma(0.5, sqrt(5.0), -0.5)
	t_2 = Float64(Float64(3.0 - sqrt(5.0)) * 0.5)
	tmp = 0.0
	if (x <= -0.007)
		tmp = Float64(t_0 / Float64(Float64(Float64(t_1 * cos(x)) + fma(t_2, cos(y), 1.0)) * 3.0));
	elseif (x <= 0.0011)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(fma(t_1, cos(x), 1.0) * 3.0)));
	else
		tmp = Float64(t_0 / fma(fma(t_1, cos(x), Float64(t_2 * cos(y))), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -0.007], N[(t$95$0 / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0011], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00700000000000000015

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.8%

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

   if -0.00700000000000000015 < x < 0.00110000000000000007

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-sin.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 65.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right) \cdot \cos x + \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ t_1 := \mathsf{fma}\left(t\_0, \cos x, 1\right) \cdot 3\\ t_2 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(t\_3, 1.5 \cdot \cos y, t\_1\right)}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, \left(t\_3 \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.5 (sqrt 5.0) -0.5))
        (t_1 (* (fma t_0 (cos x) 1.0) 3.0))
        (t_2
         (+
          (*
           (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
           (- (cos x) (cos y)))
          2.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.007)
     (/ t_2 (fma t_3 (* 1.5 (cos y)) t_1))
     (if (<= x 0.0011)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) t_1))
       (/ t_2 (fma (fma t_0 (cos x) (* (* t_3 0.5) (cos y))) 3.0 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, sqrt(5.0), -0.5);
	double t_1 = fma(t_0, cos(x), 1.0) * 3.0;
	double t_2 = (((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0;
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.007) {
		tmp = t_2 / fma(t_3, (1.5 * cos(y)), t_1);
	} else if (x <= 0.0011) {
		tmp = fma(sqrt(2.0), ((fma(-0.0625, x, sin(y)) * (1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), t_1);
	} else {
		tmp = t_2 / fma(fma(t_0, cos(x), ((t_3 * 0.5) * cos(y))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, sqrt(5.0), -0.5)
	t_1 = Float64(fma(t_0, cos(x), 1.0) * 3.0)
	t_2 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.007)
		tmp = Float64(t_2 / fma(t_3, Float64(1.5 * cos(y)), t_1));
	elseif (x <= 0.0011)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), t_1));
	else
		tmp = Float64(t_2 / fma(fma(t_0, cos(x), Float64(Float64(t_3 * 0.5) * cos(y))), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.007], N[(t$95$2 / N[(t$95$3 * N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0011], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(t$95$3 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00700000000000000015

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.7%

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

   if -0.00700000000000000015 < x < 0.00110000000000000007

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-sin.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 65.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(3 - \sqrt{5}, 1.5 \cdot \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\\ t_1 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(t\_2, 1.5 \cdot \cos y, t\_0\right)}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(t\_2 \cdot 1.5, \cos y, t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0))
        (t_1
         (+
          (*
           (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
           (- (cos x) (cos y)))
          2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.007)
     (/ t_1 (fma t_2 (* 1.5 (cos y)) t_0))
     (if (<= x 0.0011)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) t_0))
       (/ t_1 (fma (* t_2 1.5) (cos y) t_0))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0;
	double t_1 = (((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.007) {
		tmp = t_1 / fma(t_2, (1.5 * cos(y)), t_0);
	} else if (x <= 0.0011) {
		tmp = fma(sqrt(2.0), ((fma(-0.0625, x, sin(y)) * (1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), t_0);
	} else {
		tmp = t_1 / fma((t_2 * 1.5), cos(y), t_0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0)
	t_1 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.007)
		tmp = Float64(t_1 / fma(t_2, Float64(1.5 * cos(y)), t_0));
	elseif (x <= 0.0011)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), t_0));
	else
		tmp = Float64(t_1 / fma(Float64(t_2 * 1.5), cos(y), t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.007], N[(t$95$1 / N[(t$95$2 * N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0011], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(t$95$2 * 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00700000000000000015

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.7%

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

   if -0.00700000000000000015 < x < 0.00110000000000000007

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-sin.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 65.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(3 - \sqrt{5}, 1.5 \cdot \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\\ t_1 := \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, t\_0\right)}\\ \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0))
        (t_1
         (/
          (+
           (*
            (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
            (- (cos x) (cos y)))
           2.0)
          (fma (* (- 3.0 (sqrt 5.0)) 1.5) (cos y) t_0))))
   (if (<= x -0.007)
     t_1
     (if (<= x 0.0011)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) t_0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0;
	double t_1 = ((((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0) / fma(((3.0 - sqrt(5.0)) * 1.5), cos(y), t_0);
	double tmp;
	if (x <= -0.007) {
		tmp = t_1;
	} else if (x <= 0.0011) {
		tmp = fma(sqrt(2.0), ((fma(-0.0625, x, sin(y)) * (1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0) / fma(Float64(Float64(3.0 - sqrt(5.0)) * 1.5), cos(y), t_0))
	tmp = 0.0
	if (x <= -0.007)
		tmp = t_1;
	elseif (x <= 0.0011)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $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[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.007], t$95$1, If[LessEqual[x, 0.0011], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00700000000000000015 or 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 62.8

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

   7. Applied rewrites 62.8%

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

   9. Applied rewrites 62.7%

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

   if -0.00700000000000000015 < x < 0.00110000000000000007

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-sin.f64 99.5

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

   11. Applied rewrites 99.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 1.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\\ \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(t\_1 \cdot 0.5, \cos y, t\_2\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_2 \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(t\_1 \cdot \cos y, 0.5, t\_2\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
           (- (cos x) (cos y)))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
   (if (<= x -0.007)
     (/ t_0 (* (fma (* t_1 0.5) (cos y) t_2) 3.0))
     (if (<= x 0.0011)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* t_2 3.0)))
       (/ t_0 (* (fma (* t_1 (cos y)) 0.5 t_2) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0);
	double tmp;
	if (x <= -0.007) {
		tmp = t_0 / (fma((t_1 * 0.5), cos(y), t_2) * 3.0);
	} else if (x <= 0.0011) {
		tmp = fma(sqrt(2.0), ((fma(-0.0625, x, sin(y)) * (1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (t_2 * 3.0));
	} else {
		tmp = t_0 / (fma((t_1 * cos(y)), 0.5, t_2) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)
	tmp = 0.0
	if (x <= -0.007)
		tmp = Float64(t_0 / Float64(fma(Float64(t_1 * 0.5), cos(y), t_2) * 3.0));
	elseif (x <= 0.0011)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(t_2 * 3.0)));
	else
		tmp = Float64(t_0 / Float64(fma(Float64(t_1 * cos(y)), 0.5, t_2) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.007], N[(t$95$0 / N[(N[(N[(t$95$1 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0011], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$2 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00700000000000000015

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 58.8

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 58.7%

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

   if -0.00700000000000000015 < x < 0.00110000000000000007

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-sin.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 65.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\\ t_1 := \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, t\_0\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_0 \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
        (t_1
         (/
          (+
           (*
            (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
            (- (cos x) (cos y)))
           2.0)
          (* (fma (* (- 3.0 (sqrt 5.0)) (cos y)) 0.5 t_0) 3.0))))
   (if (<= x -0.007)
     t_1
     (if (<= x 0.0011)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* t_0 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0);
	double t_1 = ((((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0) / (fma(((3.0 - sqrt(5.0)) * cos(y)), 0.5, t_0) * 3.0);
	double tmp;
	if (x <= -0.007) {
		tmp = t_1;
	} else if (x <= 0.0011) {
		tmp = fma(sqrt(2.0), ((fma(-0.0625, x, sin(y)) * (1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (t_0 * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0) / Float64(fma(Float64(Float64(3.0 - sqrt(5.0)) * cos(y)), 0.5, t_0) * 3.0))
	tmp = 0.0
	if (x <= -0.007)
		tmp = t_1;
	elseif (x <= 0.0011)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(t_0 * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $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[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.007], t$95$1, If[LessEqual[x, 0.0011], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00700000000000000015 or 0.00110000000000000007 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.2

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin x} \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 62.8

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 62.8%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if -0.00700000000000000015 < x < 0.00110000000000000007

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(x \cdot \left(1 - \cos y\right)\right) + \sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(\frac{-1}{16} \cdot x\right) \cdot \left(1 - \cos y\right)} + \sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       2. distribute-rgt-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot x + \sin y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot x + \sin y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \left(\frac{-1}{16} \cdot x + \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       5. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \left(\frac{-1}{16} \cdot x + \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)}\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       7. lower-sin.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, x, \color{blue}{\sin y}\right)\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.007:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, t\_2 \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.031:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, y \cdot y, -0.75\right) \cdot t\_1, y \cdot y, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_1\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(t\_1, 1.5 \cdot \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (sqrt 2.0)
          (*
           (* (- 1.0 (cos y)) (fma (sin x) -0.0625 (sin y)))
           (fma (sin y) -0.0625 (sin x)))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.95)
     (/ t_0 (fma 1.5 (fma t_1 (cos y) (* t_2 (cos x))) 3.0))
     (if (<= y 0.031)
       (/
        (+
         (*
          (*
           (* (fma -0.0625 y (sin x)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) (cos y)))
         2.0)
        (fma
         (* (fma 0.0625 (* y y) -0.75) t_1)
         (* y y)
         (fma 1.5 (fma t_2 (cos x) t_1) 3.0)))
       (/
        t_0
        (fma
         t_1
         (* 1.5 (cos y))
         (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(2.0), (((1.0 - cos(y)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.95) {
		tmp = t_0 / fma(1.5, fma(t_1, cos(y), (t_2 * cos(x))), 3.0);
	} else if (y <= 0.031) {
		tmp = ((((fma(-0.0625, y, sin(x)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y))) + 2.0) / fma((fma(0.0625, (y * y), -0.75) * t_1), (y * y), fma(1.5, fma(t_2, cos(x), t_1), 3.0));
	} else {
		tmp = t_0 / fma(t_1, (1.5 * cos(y)), (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(2.0), Float64(Float64(Float64(1.0 - cos(y)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.95)
		tmp = Float64(t_0 / fma(1.5, fma(t_1, cos(y), Float64(t_2 * cos(x))), 3.0));
	elseif (y <= 0.031)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y))) + 2.0) / fma(Float64(fma(0.0625, Float64(y * y), -0.75) * t_1), Float64(y * y), fma(1.5, fma(t_2, cos(x), t_1), 3.0)));
	else
		tmp = Float64(t_0 / fma(t_1, Float64(1.5 * cos(y)), Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.95], N[(t$95$0 / N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.031], N[(N[(N[(N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.0625 * N[(y * y), $MachinePrecision] + -0.75), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(t$95$1 * N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.94999999999999996

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 65.2

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 65.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   10. Applied rewrites 65.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]

   if -0.94999999999999996 < y < 0.031

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(y \cdot \sqrt{2}\right) + \sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot y\right) \cdot \sqrt{2}} + \sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sin.f64 98.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + {y}^{2} \cdot \left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{{y}^{2} \cdot \left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot {y}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right), {y}^{2}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \mathsf{fma}\left(0.0625, y \cdot y, -0.75\right), y \cdot y, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)\right)}} \]

   if 0.031 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 65.3

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 65.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Applied rewrites 65.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 1.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.031:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, y \cdot y, -0.75\right) \cdot \left(3 - \sqrt{5}\right), y \cdot y, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3 - \sqrt{5}, 1.5 \cdot \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos y, t\_1 \cdot \cos x\right), 3\right)}\\ \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.031:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, y \cdot y, -0.75\right) \cdot t\_0, y \cdot y, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos x, t\_0\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (sqrt 2.0)
           (*
            (* (- 1.0 (cos y)) (fma (sin x) -0.0625 (sin y)))
            (fma (sin y) -0.0625 (sin x)))
           2.0)
          (fma 1.5 (fma t_0 (cos y) (* t_1 (cos x))) 3.0))))
   (if (<= y -0.95)
     t_2
     (if (<= y 0.031)
       (/
        (+
         (*
          (*
           (* (fma -0.0625 y (sin x)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) (cos y)))
         2.0)
        (fma
         (* (fma 0.0625 (* y y) -0.75) t_0)
         (* y y)
         (fma 1.5 (fma t_1 (cos x) t_0) 3.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(sqrt(2.0), (((1.0 - cos(y)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(1.5, fma(t_0, cos(y), (t_1 * cos(x))), 3.0);
	double tmp;
	if (y <= -0.95) {
		tmp = t_2;
	} else if (y <= 0.031) {
		tmp = ((((fma(-0.0625, y, sin(x)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y))) + 2.0) / fma((fma(0.0625, (y * y), -0.75) * t_0), (y * y), fma(1.5, fma(t_1, cos(x), t_0), 3.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(fma(sqrt(2.0), Float64(Float64(Float64(1.0 - cos(y)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(1.5, fma(t_0, cos(y), Float64(t_1 * cos(x))), 3.0))
	tmp = 0.0
	if (y <= -0.95)
		tmp = t_2;
	elseif (y <= 0.031)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y))) + 2.0) / fma(Float64(fma(0.0625, Float64(y * y), -0.75) * t_0), Float64(y * y), fma(1.5, fma(t_1, cos(x), t_0), 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.95], t$95$2, If[LessEqual[y, 0.031], N[(N[(N[(N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.0625 * N[(y * y), $MachinePrecision] + -0.75), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(1.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.94999999999999996 or 0.031 < y

   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. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 65.3

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 65.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   10. Applied rewrites 65.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]

   if -0.94999999999999996 < y < 0.031

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(y \cdot \sqrt{2}\right) + \sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot y\right) \cdot \sqrt{2}} + \sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sin.f64 98.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + {y}^{2} \cdot \left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{{y}^{2} \cdot \left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot {y}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right), {y}^{2}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \mathsf{fma}\left(0.0625, y \cdot y, -0.75\right), y \cdot y, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.031:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, y \cdot y, -0.75\right) \cdot \left(3 - \sqrt{5}\right), y \cdot y, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 - \cos y\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_2 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, t\_3\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.031:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, y \cdot y, -0.75\right) \cdot t\_1, y \cdot y, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_3, \cos x, t\_1\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_2 \cdot \sin y\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(t\_1 \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- 1.0 (cos y)))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.95)
     (/
      (fma (sqrt 2.0) (* (* t_2 (fma (sin x) -0.0625 (sin y))) t_0) 2.0)
      (fma 1.5 (fma t_1 (cos y) t_3) 3.0))
     (if (<= y 0.031)
       (/
        (+
         (*
          (*
           (* (fma -0.0625 y (sin x)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) (cos y)))
         2.0)
        (fma
         (* (fma 0.0625 (* y y) -0.75) t_1)
         (* y y)
         (fma 1.5 (fma t_3 (cos x) t_1) 3.0)))
       (/
        (fma (sqrt 2.0) (* (* t_2 (sin y)) t_0) 2.0)
        (*
         (fma (* t_1 0.5) (cos y) (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 1.0 - cos(y);
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.95) {
		tmp = fma(sqrt(2.0), ((t_2 * fma(sin(x), -0.0625, sin(y))) * t_0), 2.0) / fma(1.5, fma(t_1, cos(y), t_3), 3.0);
	} else if (y <= 0.031) {
		tmp = ((((fma(-0.0625, y, sin(x)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y))) + 2.0) / fma((fma(0.0625, (y * y), -0.75) * t_1), (y * y), fma(1.5, fma(t_3, cos(x), t_1), 3.0));
	} else {
		tmp = fma(sqrt(2.0), ((t_2 * sin(y)) * t_0), 2.0) / (fma((t_1 * 0.5), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(1.0 - cos(y))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.95)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_2 * fma(sin(x), -0.0625, sin(y))) * t_0), 2.0) / fma(1.5, fma(t_1, cos(y), t_3), 3.0));
	elseif (y <= 0.031)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y))) + 2.0) / fma(Float64(fma(0.0625, Float64(y * y), -0.75) * t_1), Float64(y * y), fma(1.5, fma(t_3, cos(x), t_1), 3.0)));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_2 * sin(y)) * t_0), 2.0) / Float64(fma(Float64(t_1 * 0.5), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.95], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.031], N[(N[(N[(N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.0625 * N[(y * y), $MachinePrecision] + -0.75), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(1.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$2 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * 0.5), $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] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.94999999999999996

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 65.2

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 65.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   10. Applied rewrites 65.0%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

   if -0.94999999999999996 < y < 0.031

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(y \cdot \sqrt{2}\right) + \sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot y\right) \cdot \sqrt{2}} + \sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sin.f64 98.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + {y}^{2} \cdot \left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{{y}^{2} \cdot \left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot {y}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{16} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right), {y}^{2}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \mathsf{fma}\left(0.0625, y \cdot y, -0.75\right), y \cdot y, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)\right)}} \]

   if 0.031 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 65.2

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 65.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.031:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, y \cdot y, -0.75\right) \cdot \left(3 - \sqrt{5}\right), y \cdot y, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 - \cos y\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_2 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, t\_3\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0058:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(-0.75 \cdot \left(y \cdot y\right), t\_1, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_3, \cos x, t\_1\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_2 \cdot \sin y\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(t\_1 \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- 1.0 (cos y)))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.95)
     (/
      (fma (sqrt 2.0) (* (* t_2 (fma (sin x) -0.0625 (sin y))) t_0) 2.0)
      (fma 1.5 (fma t_1 (cos y) t_3) 3.0))
     (if (<= y 0.0058)
       (/
        (+
         (*
          (*
           (* (fma -0.0625 y (sin x)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) (cos y)))
         2.0)
        (fma (* -0.75 (* y y)) t_1 (fma 1.5 (fma t_3 (cos x) t_1) 3.0)))
       (/
        (fma (sqrt 2.0) (* (* t_2 (sin y)) t_0) 2.0)
        (*
         (fma (* t_1 0.5) (cos y) (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 1.0 - cos(y);
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.95) {
		tmp = fma(sqrt(2.0), ((t_2 * fma(sin(x), -0.0625, sin(y))) * t_0), 2.0) / fma(1.5, fma(t_1, cos(y), t_3), 3.0);
	} else if (y <= 0.0058) {
		tmp = ((((fma(-0.0625, y, sin(x)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y))) + 2.0) / fma((-0.75 * (y * y)), t_1, fma(1.5, fma(t_3, cos(x), t_1), 3.0));
	} else {
		tmp = fma(sqrt(2.0), ((t_2 * sin(y)) * t_0), 2.0) / (fma((t_1 * 0.5), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(1.0 - cos(y))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.95)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_2 * fma(sin(x), -0.0625, sin(y))) * t_0), 2.0) / fma(1.5, fma(t_1, cos(y), t_3), 3.0));
	elseif (y <= 0.0058)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y))) + 2.0) / fma(Float64(-0.75 * Float64(y * y)), t_1, fma(1.5, fma(t_3, cos(x), t_1), 3.0)));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_2 * sin(y)) * t_0), 2.0) / Float64(fma(Float64(t_1 * 0.5), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.95], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0058], N[(N[(N[(N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(-0.75 * N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(1.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$2 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * 0.5), $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] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.94999999999999996

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 65.2

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 65.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   10. Applied rewrites 65.0%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

   if -0.94999999999999996 < y < 0.0058

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(y \cdot \sqrt{2}\right) + \sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot y\right) \cdot \sqrt{2}} + \sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sin.f64 98.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{-3}{4}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{{y}^{2} \cdot \left(\left(3 - \sqrt{5}\right) \cdot \frac{-3}{4}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{{y}^{2} \cdot \color{blue}{\left(\frac{-3}{4} \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left({y}^{2} \cdot \frac{-3}{4}\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left({y}^{2} \cdot \frac{-3}{4}, 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \frac{-3}{4}}, 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{-3}{4}, 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{-3}{4}, 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{-3}{4}, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{-3}{4}, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{-3}{4}, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}\right)} \]

      12. distribute-lft-in N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{-3}{4}, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}\right)} \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot -0.75, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)\right)}} \]

   if 0.0058 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 65.2

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 65.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0058:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(-0.75 \cdot \left(y \cdot y\right), 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 - \cos y\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_2 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0026:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, \sin x, y\right) \cdot \left(\cos x - 1\right)\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_3 \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_2 \cdot \sin y\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(t\_1 \cdot 0.5, \cos y, t\_3\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- 1.0 (cos y)))
        (t_3 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
   (if (<= y -0.095)
     (/
      (fma (sqrt 2.0) (* (* t_2 (fma (sin x) -0.0625 (sin y))) t_0) 2.0)
      (fma 1.5 (fma t_1 (cos y) (- (sqrt 5.0) 1.0)) 3.0))
     (if (<= y 0.0026)
       (/
        (fma
         (sqrt 2.0)
         (* (* (fma -0.0625 (sin x) y) (- (cos x) 1.0)) t_0)
         2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* t_3 3.0)))
       (/
        (fma (sqrt 2.0) (* (* t_2 (sin y)) t_0) 2.0)
        (* (fma (* t_1 0.5) (cos y) t_3) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 1.0 - cos(y);
	double t_3 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0);
	double tmp;
	if (y <= -0.095) {
		tmp = fma(sqrt(2.0), ((t_2 * fma(sin(x), -0.0625, sin(y))) * t_0), 2.0) / fma(1.5, fma(t_1, cos(y), (sqrt(5.0) - 1.0)), 3.0);
	} else if (y <= 0.0026) {
		tmp = fma(sqrt(2.0), ((fma(-0.0625, sin(x), y) * (cos(x) - 1.0)) * t_0), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (t_3 * 3.0));
	} else {
		tmp = fma(sqrt(2.0), ((t_2 * sin(y)) * t_0), 2.0) / (fma((t_1 * 0.5), cos(y), t_3) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(1.0 - cos(y))
	t_3 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)
	tmp = 0.0
	if (y <= -0.095)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_2 * fma(sin(x), -0.0625, sin(y))) * t_0), 2.0) / fma(1.5, fma(t_1, cos(y), Float64(sqrt(5.0) - 1.0)), 3.0));
	elseif (y <= 0.0026)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(-0.0625, sin(x), y) * Float64(cos(x) - 1.0)) * t_0), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(t_3 * 3.0)));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_2 * sin(y)) * t_0), 2.0) / Float64(fma(Float64(t_1 * 0.5), cos(y), t_3) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.095], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0026], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$2 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.095000000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 64.5

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 64.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   10. Applied rewrites 64.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

   if -0.095000000000000001 < y < 0.0025999999999999999

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + y \cdot \left(\cos x - 1\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(\frac{-1}{16} \cdot \sin x\right) \cdot \left(\cos x - 1\right)} + y \cdot \left(\cos x - 1\right)\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       2. distribute-rgt-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \left(\frac{-1}{16} \cdot \sin x + y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \left(\frac{-1}{16} \cdot \sin x + y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(\cos x - 1\right)} \cdot \left(\frac{-1}{16} \cdot \sin x + y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       5. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\color{blue}{\cos x} - 1\right) \cdot \left(\frac{-1}{16} \cdot \sin x + y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \sin x, y\right)}\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       7. lower-sin.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - 1\right) \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin x}, y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if 0.0025999999999999999 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 65.2

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 65.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0026:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(-0.0625, \sin x, y\right) \cdot \left(\cos x - 1\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 - \cos y\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_2 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, t\_3\right), 3\right)}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_3, \cos x, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_2 \cdot \sin y\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(t\_1 \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- 1.0 (cos y)))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.095)
     (/
      (fma (sqrt 2.0) (* (* t_2 (fma (sin x) -0.0625 (sin y))) t_0) 2.0)
      (fma 1.5 (fma t_1 (cos y) t_3) 3.0))
     (if (<= y 3.5e-5)
       (/
        (+
         (*
          (*
           (* (fma -0.0625 y (sin x)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) (cos y)))
         2.0)
        (fma 1.5 (fma t_3 (cos x) t_1) 3.0))
       (/
        (fma (sqrt 2.0) (* (* t_2 (sin y)) t_0) 2.0)
        (*
         (fma (* t_1 0.5) (cos y) (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 1.0 - cos(y);
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.095) {
		tmp = fma(sqrt(2.0), ((t_2 * fma(sin(x), -0.0625, sin(y))) * t_0), 2.0) / fma(1.5, fma(t_1, cos(y), t_3), 3.0);
	} else if (y <= 3.5e-5) {
		tmp = ((((fma(-0.0625, y, sin(x)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(t_3, cos(x), t_1), 3.0);
	} else {
		tmp = fma(sqrt(2.0), ((t_2 * sin(y)) * t_0), 2.0) / (fma((t_1 * 0.5), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(1.0 - cos(y))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.095)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_2 * fma(sin(x), -0.0625, sin(y))) * t_0), 2.0) / fma(1.5, fma(t_1, cos(y), t_3), 3.0));
	elseif (y <= 3.5e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(t_3, cos(x), t_1), 3.0));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_2 * sin(y)) * t_0), 2.0) / Float64(fma(Float64(t_1 * 0.5), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.095], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-5], N[(N[(N[(N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$2 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * 0.5), $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] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.095000000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 64.5

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 64.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   10. Applied rewrites 64.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

   if -0.095000000000000001 < y < 3.4999999999999997e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(y \cdot \sqrt{2}\right) + \sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot y\right) \cdot \sqrt{2}} + \sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{16} \cdot y + \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sin.f64 99.2

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if 3.4999999999999997e-5 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 65.2

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 65.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\\ t_1 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_3 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00055:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, t\_0\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_3 \cdot \sin y\right) \cdot t\_1, 2\right)}{\mathsf{fma}\left(t\_2 \cdot 0.5, \cos y, t\_0\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
        (t_1 (fma (sin y) -0.0625 (sin x)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (- 1.0 (cos y))))
   (if (<= y -0.095)
     (/
      (fma (sqrt 2.0) (* (* t_3 (fma (sin x) -0.0625 (sin y))) t_1) 2.0)
      (fma 1.5 (fma t_2 (cos y) (- (sqrt 5.0) 1.0)) 3.0))
     (if (<= y 0.00055)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (*
         (fma
          (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
          (- 14.0 (* 3.0 (sqrt 5.0)))
          t_0)
         3.0))
       (/
        (fma (sqrt 2.0) (* (* t_3 (sin y)) t_1) 2.0)
        (* (fma (* t_2 0.5) (cos y) t_0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0);
	double t_1 = fma(sin(y), -0.0625, sin(x));
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.095) {
		tmp = fma(sqrt(2.0), ((t_3 * fma(sin(x), -0.0625, sin(y))) * t_1), 2.0) / fma(1.5, fma(t_2, cos(y), (sqrt(5.0) - 1.0)), 3.0);
	} else if (y <= 0.00055) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), t_0) * 3.0);
	} else {
		tmp = fma(sqrt(2.0), ((t_3 * sin(y)) * t_1), 2.0) / (fma((t_2 * 0.5), cos(y), t_0) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)
	t_1 = fma(sin(y), -0.0625, sin(x))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.095)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_3 * fma(sin(x), -0.0625, sin(y))) * t_1), 2.0) / fma(1.5, fma(t_2, cos(y), Float64(sqrt(5.0) - 1.0)), 3.0));
	elseif (y <= 0.00055)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), t_0) * 3.0));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_3 * sin(y)) * t_1), 2.0) / Float64(fma(Float64(t_2 * 0.5), cos(y), t_0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.095], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$3 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00055], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$3 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$2 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.095000000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 64.5

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 64.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   10. Applied rewrites 64.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

   if -0.095000000000000001 < y < 5.50000000000000033e-4

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if 5.50000000000000033e-4 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 65.2

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 65.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00055:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_2 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00055:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, t\_0\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(t\_1 \cdot 0.5, \cos y, t\_0\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, t\_2 \cdot \sin y, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- 1.0 (cos y))))
   (if (<= y -0.095)
     (/
      (fma
       (sqrt 2.0)
       (* (* t_2 (fma (sin x) -0.0625 (sin y))) (fma (sin y) -0.0625 (sin x)))
       2.0)
      (fma 1.5 (fma t_1 (cos y) (- (sqrt 5.0) 1.0)) 3.0))
     (if (<= y 0.00055)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (*
         (fma
          (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
          (- 14.0 (* 3.0 (sqrt 5.0)))
          t_0)
         3.0))
       (*
        (/ 0.3333333333333333 (fma (* t_1 0.5) (cos y) t_0))
        (fma
         (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
         (* t_2 (sin y))
         2.0))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.095) {
		tmp = fma(sqrt(2.0), ((t_2 * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(1.5, fma(t_1, cos(y), (sqrt(5.0) - 1.0)), 3.0);
	} else if (y <= 0.00055) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), t_0) * 3.0);
	} else {
		tmp = (0.3333333333333333 / fma((t_1 * 0.5), cos(y), t_0)) * fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), (t_2 * sin(y)), 2.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.095)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_2 * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(1.5, fma(t_1, cos(y), Float64(sqrt(5.0) - 1.0)), 3.0));
	elseif (y <= 0.00055)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), t_0) * 3.0));
	else
		tmp = Float64(Float64(0.3333333333333333 / fma(Float64(t_1 * 0.5), cos(y), t_0)) * fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(t_2 * sin(y)), 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.095], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00055], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(t$95$1 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.095000000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 64.5

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 64.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   10. Applied rewrites 64.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

   if -0.095000000000000001 < y < 5.50000000000000033e-4

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if 5.50000000000000033e-4 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 65.2

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 65.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 65.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \left(1 - \cos y\right) \cdot \sin y, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00055:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \left(1 - \cos y\right) \cdot \sin y, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_0 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, t\_1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00066:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, t\_0 \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{t\_1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y))) (t_1 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.095)
     (/
      (fma
       (sqrt 2.0)
       (* (* t_0 (fma (sin x) -0.0625 (sin y))) (fma (sin y) -0.0625 (sin x)))
       2.0)
      (fma 1.5 (fma (- 3.0 (sqrt 5.0)) (cos y) t_1) 3.0))
     (if (<= y 0.00066)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (*
         (fma
          (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
          (- 14.0 (* 3.0 (sqrt 5.0)))
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
         3.0))
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* t_0 (sqrt 2.0)) 2.0)
        (*
         (+
          (* (/ 2.0 (+ 3.0 (sqrt 5.0))) (cos y))
          (+ (* (/ t_1 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.095) {
		tmp = fma(sqrt(2.0), ((t_0 * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(1.5, fma((3.0 - sqrt(5.0)), cos(y), t_1), 3.0);
	} else if (y <= 0.00066) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), (t_0 * sqrt(2.0)), 2.0) / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + (((t_1 / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.095)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_0 * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(1.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), t_1), 3.0));
	elseif (y <= 0.00066)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(t_0 * sqrt(2.0)), 2.0) / Float64(Float64(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * cos(y)) + Float64(Float64(Float64(t_1 / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.095], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00066], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.095000000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-cos.f64 64.5

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 64.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   10. Applied rewrites 64.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

   if -0.095000000000000001 < y < 6.6e-4

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if 6.6e-4 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 64.9

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 64.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00066:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_0 \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00066:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, t\_0 \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y))))
   (if (<= y -0.095)
     (/
      (fma (sqrt 2.0) (* (* t_0 (sin y)) (fma (sin y) -0.0625 (sin x))) 2.0)
      (fma
       (* 1.5 (cos y))
       (- 3.0 (sqrt 5.0))
       (* (fma 0.5 (sqrt 5.0) 0.5) 3.0)))
     (if (<= y 0.00066)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (*
         (fma
          (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
          (- 14.0 (* 3.0 (sqrt 5.0)))
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
         3.0))
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* t_0 (sqrt 2.0)) 2.0)
        (*
         (+
          (* (/ 2.0 (+ 3.0 (sqrt 5.0))) (cos y))
          (+ (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.095) {
		tmp = fma(sqrt(2.0), ((t_0 * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((1.5 * cos(y)), (3.0 - sqrt(5.0)), (fma(0.5, sqrt(5.0), 0.5) * 3.0));
	} else if (y <= 0.00066) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), (t_0 * sqrt(2.0)), 2.0) / ((((2.0 / (3.0 + sqrt(5.0))) * cos(y)) + ((((sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.095)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_0 * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(1.5 * cos(y)), Float64(3.0 - sqrt(5.0)), Float64(fma(0.5, sqrt(5.0), 0.5) * 3.0)));
	elseif (y <= 0.00066)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(t_0 * sqrt(2.0)), 2.0) / Float64(Float64(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * cos(y)) + Float64(Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.095], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$0 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00066], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.095000000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 64.1

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 64.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       4. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       9. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)} \cdot 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{1}{2}\right)} \cdot 3\right)} \]

       11. lower-sqrt.f64 64.2

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, 0.5\right) \cdot 3\right)} \]

   12. Applied rewrites 64.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}} \]

   if -0.095000000000000001 < y < 6.6e-4

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if 6.6e-4 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 64.9

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 64.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00066:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3 + \sqrt{5}}\\ t_1 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_1 \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00066:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(t\_0 \cdot 3, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, t\_1 \cdot \sqrt{2}, 2\right)}{\left(t\_0 \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 3.0 (sqrt 5.0)))) (t_1 (- 1.0 (cos y))))
   (if (<= y -0.095)
     (/
      (fma (sqrt 2.0) (* (* t_1 (sin y)) (fma (sin y) -0.0625 (sin x))) 2.0)
      (fma
       (* 1.5 (cos y))
       (- 3.0 (sqrt 5.0))
       (* (fma 0.5 (sqrt 5.0) 0.5) 3.0)))
     (if (<= y 0.00066)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (fma
         (* t_0 3.0)
         (cos y)
         (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0)))
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* t_1 (sqrt 2.0)) 2.0)
        (*
         (+ (* t_0 (cos y)) (+ (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = 2.0 / (3.0 + sqrt(5.0));
	double t_1 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.095) {
		tmp = fma(sqrt(2.0), ((t_1 * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((1.5 * cos(y)), (3.0 - sqrt(5.0)), (fma(0.5, sqrt(5.0), 0.5) * 3.0));
	} else if (y <= 0.00066) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma((t_0 * 3.0), cos(y), (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0));
	} else {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), (t_1 * sqrt(2.0)), 2.0) / (((t_0 * cos(y)) + ((((sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 / Float64(3.0 + sqrt(5.0)))
	t_1 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.095)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_1 * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(1.5 * cos(y)), Float64(3.0 - sqrt(5.0)), Float64(fma(0.5, sqrt(5.0), 0.5) * 3.0)));
	elseif (y <= 0.00066)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(Float64(t_0 * 3.0), cos(y), Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0)));
	else
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(t_1 * sqrt(2.0)), 2.0) / Float64(Float64(Float64(t_0 * cos(y)) + Float64(Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.095], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$1 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00066], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.095000000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 64.1

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 64.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       4. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       9. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)} \cdot 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{1}{2}\right)} \cdot 3\right)} \]

       11. lower-sqrt.f64 64.2

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, 0.5\right) \cdot 3\right)} \]

   12. Applied rewrites 64.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}} \]

   if -0.095000000000000001 < y < 6.6e-4

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   if 6.6e-4 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 64.9

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 64.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00066:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{3 + \sqrt{5}} \cdot 3, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 1 - \cos y\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(t\_1 \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00065:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{3 + \sqrt{5}} \cdot 3, \cos y, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left({\sin y}^{2} \cdot -0.0625\right) \cdot t\_1, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{2} \cdot t\_0, 3, t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- 1.0 (cos y)))
        (t_2 (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0)))
   (if (<= y -0.095)
     (/
      (fma (sqrt 2.0) (* (* t_1 (sin y)) (fma (sin y) -0.0625 (sin x))) 2.0)
      (fma (* 1.5 (cos y)) t_0 (* (fma 0.5 (sqrt 5.0) 0.5) 3.0)))
     (if (<= y 0.00065)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (fma (* (/ 2.0 (+ 3.0 (sqrt 5.0))) 3.0) (cos y) t_2))
       (/
        (fma (sqrt 2.0) (* (* (pow (sin y) 2.0) -0.0625) t_1) 2.0)
        (fma (* (/ (cos y) 2.0) t_0) 3.0 t_2))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 1.0 - cos(y);
	double t_2 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0;
	double tmp;
	if (y <= -0.095) {
		tmp = fma(sqrt(2.0), ((t_1 * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((1.5 * cos(y)), t_0, (fma(0.5, sqrt(5.0), 0.5) * 3.0));
	} else if (y <= 0.00065) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(((2.0 / (3.0 + sqrt(5.0))) * 3.0), cos(y), t_2);
	} else {
		tmp = fma(sqrt(2.0), ((pow(sin(y), 2.0) * -0.0625) * t_1), 2.0) / fma(((cos(y) / 2.0) * t_0), 3.0, t_2);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(1.0 - cos(y))
	t_2 = Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0)
	tmp = 0.0
	if (y <= -0.095)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(t_1 * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(1.5 * cos(y)), t_0, Float64(fma(0.5, sqrt(5.0), 0.5) * 3.0)));
	elseif (y <= 0.00065)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * 3.0), cos(y), t_2));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64((sin(y) ^ 2.0) * -0.0625) * t_1), 2.0) / fma(Float64(Float64(cos(y) / 2.0) * t_0), 3.0, t_2));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]}, If[LessEqual[y, -0.095], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$1 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00065], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] / 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.095000000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 64.1

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 64.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       4. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       9. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)} \cdot 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{1}{2}\right)} \cdot 3\right)} \]

       11. lower-sqrt.f64 64.2

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, 0.5\right) \cdot 3\right)} \]

   12. Applied rewrites 64.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}} \]

   if -0.095000000000000001 < y < 6.4999999999999997e-4

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   if 6.4999999999999997e-4 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      7. lower-cos.f64 64.8

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 64.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00065:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{3 + \sqrt{5}} \cdot 3, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{\cos y}{2} \cdot \left(3 - \sqrt{5}\right), 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0016:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{3 + \sqrt{5}} \cdot 3, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (sqrt 2.0)
           (* (* (- 1.0 (cos y)) (sin y)) (fma (sin y) -0.0625 (sin x)))
           2.0)
          (fma
           (* 1.5 (cos y))
           (- 3.0 (sqrt 5.0))
           (* (fma 0.5 (sqrt 5.0) 0.5) 3.0)))))
   (if (<= y -0.095)
     t_0
     (if (<= y 0.0016)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (fma
         (* (/ 2.0 (+ 3.0 (sqrt 5.0))) 3.0)
         (cos y)
         (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(2.0), (((1.0 - cos(y)) * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((1.5 * cos(y)), (3.0 - sqrt(5.0)), (fma(0.5, sqrt(5.0), 0.5) * 3.0));
	double tmp;
	if (y <= -0.095) {
		tmp = t_0;
	} else if (y <= 0.0016) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(((2.0 / (3.0 + sqrt(5.0))) * 3.0), cos(y), (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(sqrt(2.0), Float64(Float64(Float64(1.0 - cos(y)) * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(1.5 * cos(y)), Float64(3.0 - sqrt(5.0)), Float64(fma(0.5, sqrt(5.0), 0.5) * 3.0)))
	tmp = 0.0
	if (y <= -0.095)
		tmp = t_0;
	elseif (y <= 0.0016)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) * 3.0), cos(y), Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.095], t$95$0, If[LessEqual[y, 0.0016], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.095000000000000001 or 0.00160000000000000008 < y

   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. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 64.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 64.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       4. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       9. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)} \cdot 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{1}{2}\right)} \cdot 3\right)} \]

       11. lower-sqrt.f64 64.5

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, 0.5\right) \cdot 3\right)} \]

   12. Applied rewrites 64.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}} \]

   if -0.095000000000000001 < y < 0.00160000000000000008

   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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0016:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{3 + \sqrt{5}} \cdot 3, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0016:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (sqrt 2.0)
           (* (* (- 1.0 (cos y)) (sin y)) (fma (sin y) -0.0625 (sin x)))
           2.0)
          (fma
           (* 1.5 (cos y))
           (- 3.0 (sqrt 5.0))
           (* (fma 0.5 (sqrt 5.0) 0.5) 3.0)))))
   (if (<= y -0.095)
     t_0
     (if (<= y 0.0016)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (*
         (fma
          (/ 2.0 (+ 3.0 (sqrt 5.0)))
          (cos y)
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
         3.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(2.0), (((1.0 - cos(y)) * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((1.5 * cos(y)), (3.0 - sqrt(5.0)), (fma(0.5, sqrt(5.0), 0.5) * 3.0));
	double tmp;
	if (y <= -0.095) {
		tmp = t_0;
	} else if (y <= 0.0016) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (fma((2.0 / (3.0 + sqrt(5.0))), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(sqrt(2.0), Float64(Float64(Float64(1.0 - cos(y)) * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(1.5 * cos(y)), Float64(3.0 - sqrt(5.0)), Float64(fma(0.5, sqrt(5.0), 0.5) * 3.0)))
	tmp = 0.0
	if (y <= -0.095)
		tmp = t_0;
	elseif (y <= 0.0016)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(fma(Float64(2.0 / Float64(3.0 + sqrt(5.0))), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.095], t$95$0, If[LessEqual[y, 0.0016], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $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] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.095000000000000001 or 0.00160000000000000008 < y

   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. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 64.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 64.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       4. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       9. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)} \cdot 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{1}{2}\right)} \cdot 3\right)} \]

       11. lower-sqrt.f64 64.5

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, 0.5\right) \cdot 3\right)} \]

   12. Applied rewrites 64.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}} \]

   if -0.095000000000000001 < y < 0.00160000000000000008

   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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0016:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (sqrt 2.0)
           (* (* (- 1.0 (cos y)) (sin y)) (fma (sin y) -0.0625 (sin x)))
           2.0)
          (fma
           (* 1.5 (cos y))
           (- 3.0 (sqrt 5.0))
           (* (fma 0.5 (sqrt 5.0) 0.5) 3.0)))))
   (if (<= y -0.095)
     t_0
     (if (<= y 1.1e-5)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (*
         (fma
          (/ 2.0 (fma (sqrt 5.0) 5.0 27.0))
          (- 14.0 (* 3.0 (sqrt 5.0)))
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
         3.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(2.0), (((1.0 - cos(y)) * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma((1.5 * cos(y)), (3.0 - sqrt(5.0)), (fma(0.5, sqrt(5.0), 0.5) * 3.0));
	double tmp;
	if (y <= -0.095) {
		tmp = t_0;
	} else if (y <= 1.1e-5) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (fma((2.0 / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(sqrt(2.0), Float64(Float64(Float64(1.0 - cos(y)) * sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(1.5 * cos(y)), Float64(3.0 - sqrt(5.0)), Float64(fma(0.5, sqrt(5.0), 0.5) * 3.0)))
	tmp = 0.0
	if (y <= -0.095)
		tmp = t_0;
	elseif (y <= 1.1e-5)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(fma(Float64(2.0 / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.095], t$95$0, If[LessEqual[y, 1.1e-5], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.095000000000000001 or 1.1e-5 < y

   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. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 64.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 64.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       4. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3}\right)} \]

       9. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)} \cdot 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{1}{2}\right)} \cdot 3\right)} \]

       11. lower-sqrt.f64 64.5

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, 0.5\right) \cdot 3\right)} \]

   12. Applied rewrites 64.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}} \]

   if -0.095000000000000001 < y < 1.1e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{2}{27 + 5 \cdot \sqrt{5}}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{2}{27 + 5 \cdot \sqrt{5}}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{2}{\color{blue}{5 \cdot \sqrt{5} + 27}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{2}{\color{blue}{\sqrt{5} \cdot 5} + 27}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{2}{\color{blue}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-sqrt.f64 98.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{2}{\mathsf{fma}\left(\color{blue}{\sqrt{5}}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   10. Applied rewrites 98.7%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right)\\ t_1 := \mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(0.5, t\_0, 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(1.5, t\_0, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- 3.0 (sqrt 5.0)) (cos y) (* (- (sqrt 5.0) 1.0) (cos x))))
        (t_1
         (fma
          (* (pow (sin x) 2.0) (sqrt 2.0))
          (fma -0.0625 (cos x) 0.0625)
          2.0)))
   (if (<= x -5.5e-5)
     (/ t_1 (* (fma 0.5 t_0 1.0) 3.0))
     (if (<= x 1.35e-6)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma
         (fma 0.5 (sqrt 5.0) 0.5)
         3.0
         (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))
       (/ t_1 (fma 1.5 t_0 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((3.0 - sqrt(5.0)), cos(y), ((sqrt(5.0) - 1.0) * cos(x)));
	double t_1 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0);
	double tmp;
	if (x <= -5.5e-5) {
		tmp = t_1 / (fma(0.5, t_0, 1.0) * 3.0);
	} else if (x <= 1.35e-6) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	} else {
		tmp = t_1 / fma(1.5, t_0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x)))
	t_1 = fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0)
	tmp = 0.0
	if (x <= -5.5e-5)
		tmp = Float64(t_1 / Float64(fma(0.5, t_0, 1.0) * 3.0));
	elseif (x <= 1.35e-6)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))));
	else
		tmp = Float64(t_1 / fma(1.5, t_0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[x, -5.5e-5], N[(t$95$1 / N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-6], 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[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(1.5 * t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5000000000000002e-5

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 55.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 55.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right)}, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \cos x \cdot \left(\sqrt{5} - 1\right), 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right)}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right)} \cdot \cos x\right), 1\right)} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\color{blue}{\sqrt{5}} - 1\right) \cdot \cos x\right), 1\right)} \]

      14. lower-cos.f64 55.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \color{blue}{\cos x}\right), 1\right)} \]

   8. Applied rewrites 55.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}} \]

   if -5.5000000000000002e-5 < x < 1.34999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       6. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       7. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       11. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3} + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{\sqrt{5} + 3}\right)}} \]

   if 1.34999999999999999e-6 < x

   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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 62.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma -0.0625 (cos x) 0.0625)
           2.0)
          (fma
           1.5
           (fma (- 3.0 (sqrt 5.0)) (cos y) (* (- (sqrt 5.0) 1.0) (cos x)))
           3.0))))
   (if (<= x -5.5e-5)
     t_0
     (if (<= x 1.35e-6)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma
         (fma 0.5 (sqrt 5.0) 0.5)
         3.0
         (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma((3.0 - sqrt(5.0)), cos(y), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	double tmp;
	if (x <= -5.5e-5) {
		tmp = t_0;
	} else if (x <= 1.35e-6) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0))
	tmp = 0.0
	if (x <= -5.5e-5)
		tmp = t_0;
	elseif (x <= 1.35e-6)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e-5], t$95$0, If[LessEqual[x, 1.35e-6], 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[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000002e-5 or 1.34999999999999999e-6 < x

   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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 59.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 59.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 59.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]

   if -5.5000000000000002e-5 < x < 1.34999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       6. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       7. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       11. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3} + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{\sqrt{5} + 3}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 33: 78.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma -0.0625 (cos x) 0.0625)
           2.0)
          (*
           (fma
            (/ 2.0 (fma (sqrt 5.0) 5.0 27.0))
            (- 14.0 (* 3.0 (sqrt 5.0)))
            (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
           3.0))))
   (if (<= x -6.5e-5)
     t_0
     (if (<= x 6.8e-6)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma
         (fma 0.5 (sqrt 5.0) 0.5)
         3.0
         (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (fma((2.0 / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	double tmp;
	if (x <= -6.5e-5) {
		tmp = t_0;
	} else if (x <= 6.8e-6) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(fma(Float64(2.0 / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0))
	tmp = 0.0
	if (x <= -6.5e-5)
		tmp = t_0;
	elseif (x <= 6.8e-6)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e-5], t$95$0, If[LessEqual[x, 6.8e-6], 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[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999943e-5 or 6.80000000000000012e-6 < x

   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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 59.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 59.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{2}{27 + 5 \cdot \sqrt{5}}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{2}{27 + 5 \cdot \sqrt{5}}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{2}{\color{blue}{5 \cdot \sqrt{5} + 27}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{2}{\color{blue}{\sqrt{5} \cdot 5} + 27}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{2}{\color{blue}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-sqrt.f64 58.8

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{2}{\mathsf{fma}\left(\color{blue}{\sqrt{5}}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   10. Applied rewrites 58.8%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -6.49999999999999943e-5 < x < 6.80000000000000012e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       6. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       7. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       11. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3} + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{\sqrt{5} + 3}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\frac{2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 34: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* (pow (sin x) 2.0) -0.0625)
           (* (- (cos x) 1.0) (sqrt 2.0))
           2.0)
          (fma
           1.5
           (- 3.0 (sqrt 5.0))
           (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0)))))
   (if (<= x -6.5e-5)
     t_0
     (if (<= x 6.8e-6)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma
         (fma 0.5 (sqrt 5.0) 0.5)
         3.0
         (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(x), 2.0) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, (3.0 - sqrt(5.0)), (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0));
	double tmp;
	if (x <= -6.5e-5) {
		tmp = t_0;
	} else if (x <= 6.8e-6) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, Float64(3.0 - sqrt(5.0)), Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0)))
	tmp = 0.0
	if (x <= -6.5e-5)
		tmp = t_0;
	elseif (x <= 6.8e-6)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e-5], t$95$0, If[LessEqual[x, 6.8e-6], 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[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999943e-5 or 6.80000000000000012e-6 < x

   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. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 26.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       6. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       7. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       11. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       13. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, 3 - \sqrt{5}, 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)}} \]

   12. Applied rewrites 58.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   if -6.49999999999999943e-5 < x < 6.80000000000000012e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\frac{2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 \cdot 2}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       6. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       7. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       11. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) \cdot 3} + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{\sqrt{5} + 3}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 35: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* (pow (sin x) 2.0) -0.0625)
           (* (- (cos x) 1.0) (sqrt 2.0))
           2.0)
          (fma
           1.5
           (- 3.0 (sqrt 5.0))
           (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0)))))
   (if (<= x -6.5e-5)
     t_0
     (if (<= x 6.8e-6)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 6.0 (fma 1.5 (sqrt 5.0) 1.5)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(x), 2.0) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, (3.0 - sqrt(5.0)), (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0));
	double tmp;
	if (x <= -6.5e-5) {
		tmp = t_0;
	} else if (x <= 6.8e-6) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((cos(y) / (3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, Float64(3.0 - sqrt(5.0)), Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) * 3.0)))
	tmp = 0.0
	if (x <= -6.5e-5)
		tmp = t_0;
	elseif (x <= 6.8e-6)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e-5], t$95$0, If[LessEqual[x, 6.8e-6], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999943e-5 or 6.80000000000000012e-6 < x

   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. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 26.7

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \frac{\cos y}{2}, 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       6. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       7. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       11. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \]

       13. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, 3 - \sqrt{5}, 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)}} \]

   12. Applied rewrites 58.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   if -6.49999999999999943e-5 < x < 6.80000000000000012e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   9. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 36: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma -0.0625 (cos x) 0.0625)
           2.0)
          (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 3.0))))
   (if (<= x -6.5e-5)
     t_0
     (if (<= x 6.8e-6)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 6.0 (fma 1.5 (sqrt 5.0) 1.5)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 3.0);
	double tmp;
	if (x <= -6.5e-5) {
		tmp = t_0;
	} else if (x <= 6.8e-6) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((cos(y) / (3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 3.0))
	tmp = 0.0
	if (x <= -6.5e-5)
		tmp = t_0;
	elseif (x <= 6.8e-6)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e-5], t$95$0, If[LessEqual[x, 6.8e-6], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999943e-5 or 6.80000000000000012e-6 < x

   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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 59.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 59.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if -6.49999999999999943e-5 < x < 6.80000000000000012e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{2}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   9. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 37: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma -0.0625 (cos x) 0.0625)
           2.0)
          (fma 1.5 (fma t_0 (cos x) t_1) 3.0))))
   (if (<= x -6.5e-5)
     t_2
     (if (<= x 6.8e-6)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma t_1 (cos y) t_0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	double tmp;
	if (x <= -6.5e-5) {
		tmp = t_2;
	} else if (x <= 6.8e-6) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(t_1, cos(y), t_0), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0))
	tmp = 0.0
	if (x <= -6.5e-5)
		tmp = t_2;
	elseif (x <= 6.8e-6)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(t_1, cos(y), t_0), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e-5], t$95$2, If[LessEqual[x, 6.8e-6], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999943e-5 or 6.80000000000000012e-6 < x

   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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 59.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 59.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if -6.49999999999999943e-5 < x < 6.80000000000000012e-6

   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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 59.7

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 59.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\frac{-1}{16}}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\color{blue}{-0.0625}, \cos x, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

       4. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

       5. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   11. Applied rewrites 59.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

       11. lower-sqrt.f64 99.0

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]

   14. Applied rewrites 99.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 38: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
  (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   17. lower-cos.f64 59.6

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 59.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 57.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

9. Final simplification 57.4%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]
10. Add Preprocessing

Alternative 39: 45.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   (fma
    (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
    (- 14.0 (* 3.0 (sqrt 5.0)))
    (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / (fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   17. lower-cos.f64 59.6

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 59.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

7. Applied rewrites 59.7%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 43.7%

    \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

11. Final simplification 43.7%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3} \]
12. Add Preprocessing

Alternative 40: 42.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\cos y \cdot 2, \frac{\mathsf{fma}\left(-3, \sqrt{5}, 14\right)}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \mathsf{fma}\left(\sqrt{5}, 0.5, 0.5\right)\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   2.0
   (fma
    (* (cos y) 2.0)
    (/ (fma -3.0 (sqrt 5.0) 14.0) (fma (sqrt 5.0) 5.0 27.0))
    (fma (sqrt 5.0) 0.5 0.5)))
  0.3333333333333333))

C

double code(double x, double y) {
	return (2.0 / fma((cos(y) * 2.0), (fma(-3.0, sqrt(5.0), 14.0) / fma(sqrt(5.0), 5.0, 27.0)), fma(sqrt(5.0), 0.5, 0.5))) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / fma(Float64(cos(y) * 2.0), Float64(fma(-3.0, sqrt(5.0), 14.0) / fma(sqrt(5.0), 5.0, 27.0)), fma(sqrt(5.0), 0.5, 0.5))) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(-3.0 * N[Sqrt[5.0], $MachinePrecision] + 14.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   17. lower-cos.f64 59.6

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 59.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

7. Applied rewrites 59.7%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + 2 \cdot \frac{\cos y \cdot \left(14 - 3 \cdot \sqrt{5}\right)}{27 + 5 \cdot \sqrt{5}}\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + 2 \cdot \frac{\cos y \cdot \left(14 - 3 \cdot \sqrt{5}\right)}{27 + 5 \cdot \sqrt{5}}\right)} \cdot \frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + 2 \cdot \frac{\cos y \cdot \left(14 - 3 \cdot \sqrt{5}\right)}{27 + 5 \cdot \sqrt{5}}\right)} \cdot \frac{1}{3}} \]

10. Applied rewrites 61.5%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(2 \cdot \cos y, \frac{\mathsf{fma}\left(-3, \sqrt{5}, 14\right)}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \mathsf{fma}\left(\sqrt{5}, 0.5, 0.5\right)\right)} \cdot 0.3333333333333333} \]

11. Taylor expanded in y around 0

    \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot \cos y, \frac{\mathsf{fma}\left(-3, \sqrt{5}, 14\right)}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{1}{2}\right)\right)} \cdot \frac{1}{3} \]
12. Step-by-step derivation

13. Applied rewrites 41.2%

    \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot \cos y, \frac{\mathsf{fma}\left(-3, \sqrt{5}, 14\right)}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \mathsf{fma}\left(\sqrt{5}, 0.5, 0.5\right)\right)} \cdot 0.3333333333333333 \]

14. Final simplification 41.2%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\cos y \cdot 2, \frac{\mathsf{fma}\left(-3, \sqrt{5}, 14\right)}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \mathsf{fma}\left(\sqrt{5}, 0.5, 0.5\right)\right)} \cdot 0.3333333333333333 \]
15. Add Preprocessing

Alternative 41: 42.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (- 3.0 (sqrt 5.0)) (cos y) (- (sqrt 5.0) 1.0)) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma((3.0 - sqrt(5.0)), cos(y), (sqrt(5.0) - 1.0)), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(sqrt(5.0) - 1.0)), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   17. lower-cos.f64 59.6

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 59.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left({x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\frac{-1}{16}}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

8. Applied rewrites 34.0%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\color{blue}{-0.0625}, \cos x, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

11. Applied rewrites 33.6%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)}} \]

12. Taylor expanded in x around 0

    \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]
13. Step-by-step derivation

14. Applied rewrites 41.2%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)} \]
15. Add Preprocessing

Alternative 42: 40.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-3, \sqrt{5}, 14\right)}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 2, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (fma
   (/ (fma -3.0 (sqrt 5.0) 14.0) (fma (sqrt 5.0) 5.0 27.0))
   2.0
   (fma 0.5 (sqrt 5.0) 0.5))))

C

double code(double x, double y) {
	return 0.6666666666666666 / fma((fma(-3.0, sqrt(5.0), 14.0) / fma(sqrt(5.0), 5.0, 27.0)), 2.0, fma(0.5, sqrt(5.0), 0.5));
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / fma(Float64(fma(-3.0, sqrt(5.0), 14.0) / fma(sqrt(5.0), 5.0, 27.0)), 2.0, fma(0.5, sqrt(5.0), 0.5)))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(N[(N[(-3.0 * N[Sqrt[5.0], $MachinePrecision] + 14.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot \color{blue}{-1}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   17. lower-cos.f64 59.6

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 59.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

7. Applied rewrites 59.7%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - \sqrt{5} \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + 2 \cdot \frac{\cos y \cdot \left(14 - 3 \cdot \sqrt{5}\right)}{27 + 5 \cdot \sqrt{5}}\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + 2 \cdot \frac{\cos y \cdot \left(14 - 3 \cdot \sqrt{5}\right)}{27 + 5 \cdot \sqrt{5}}\right)} \cdot \frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + 2 \cdot \frac{\cos y \cdot \left(14 - 3 \cdot \sqrt{5}\right)}{27 + 5 \cdot \sqrt{5}}\right)} \cdot \frac{1}{3}} \]

10. Applied rewrites 61.5%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(2 \cdot \cos y, \frac{\mathsf{fma}\left(-3, \sqrt{5}, 14\right)}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \mathsf{fma}\left(\sqrt{5}, 0.5, 0.5\right)\right)} \cdot 0.3333333333333333} \]

11. Taylor expanded in y around 0

    \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + 2 \cdot \frac{14 + -3 \cdot \sqrt{5}}{27 + 5 \cdot \sqrt{5}}\right)}} \]
12. Step-by-step derivation

13. Applied rewrites 39.1%

    \[\leadsto \frac{0.6666666666666666}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-3, \sqrt{5}, 14\right)}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 2, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)\right)}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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))))))