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

Percentage Accurate: 99.3% → 99.3%

Time: 29.5s

Alternatives: 38

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in N/A

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

   2. +-commutative N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. div-sub N/A

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

   6. --lowering--.f64 N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cos-lowering-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

   16. distribute-lft-in N/A

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

   17. metadata-eval N/A

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

4. Applied egg-rr 99.3%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

6. Applied egg-rr 99.3%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 2: 61.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)\\ \mathbf{if}\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{t\_1} \leq 0.572:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ t_0 2.0))))))
   (if (<=
        (/
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (*
            (- (sin y) (/ (sin x) 16.0))
            (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))))
         t_1)
        0.572)
     (/
      (fma (pow (sin y) 2.0) (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) 2.0)
      (fma 1.5 (fma (cos y) t_0 (sqrt 5.0)) 1.5))
     (/ 2.0 t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0)));
	double tmp;
	if (((2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))))) / t_1) <= 0.572) {
		tmp = fma(pow(sin(y), 2.0), ((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), 2.0) / fma(1.5, fma(cos(y), t_0, sqrt(5.0)), 1.5);
	} else {
		tmp = 2.0 / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0))))
	tmp = 0.0
	if (Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))))) / t_1) <= 0.572)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), 2.0) / fma(1.5, fma(cos(y), t_0, sqrt(5.0)), 1.5));
	else
		tmp = Float64(2.0 / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 0.572], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(2.0 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y))))) < 0.571999999999999953

   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 x around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

   5. Simplified 67.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

   8. Simplified 73.9%

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

   if 0.571999999999999953 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y)))))

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. metadata-eval 57.8

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

   5. Simplified 57.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 25.5%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \leq 0.572:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in N/A

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

   2. +-commutative N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. div-sub N/A

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

   6. --lowering--.f64 N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cos-lowering-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

   16. distribute-lft-in N/A

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

   17. metadata-eval N/A

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

4. Applied egg-rr 99.3%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

6. Applied egg-rr 99.3%

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

8. Applied egg-rr 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in N/A

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

   2. +-commutative N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. div-sub N/A

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

   6. --lowering--.f64 N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cos-lowering-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

   16. distribute-lft-in N/A

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

   17. metadata-eval N/A

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

4. Applied egg-rr 99.3%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in x around inf

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (fma
   (- (cos x) (cos y))
   (*
    (sqrt 2.0)
    (* (fma (sin x) -0.0625 (sin y)) (fma -0.0625 (sin y) (sin x))))
   2.0)
  (/
   0.3333333333333333
   (fma
    (fma (sqrt 5.0) -0.5 1.5)
    (cos y)
    (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))))

C

double code(double x, double y) {
	return fma((cos(x) - cos(y)), (sqrt(2.0) * (fma(sin(x), -0.0625, sin(y)) * fma(-0.0625, sin(y), sin(x)))), 2.0) * (0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
}

Julia

function code(x, y)
	return Float64(fma(Float64(cos(x) - cos(y)), Float64(sqrt(2.0) * Float64(fma(sin(x), -0.0625, sin(y)) * fma(-0.0625, sin(y), sin(x)))), 2.0) * Float64(0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in N/A

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

   2. +-commutative N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. div-sub N/A

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

   6. --lowering--.f64 N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cos-lowering-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

   16. distribute-lft-in N/A

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

   17. metadata-eval N/A

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

4. Applied egg-rr 99.3%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

6. Applied egg-rr 99.3%

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

8. Applied egg-rr 99.4%

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

10. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]

11. Final simplification 99.2%

    \[\leadsto \mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \]
12. Add Preprocessing

Alternative 6: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot t\_0\right), 3\right)}\\ t_2 := \cos x - \cos y\\ t_3 := \sin x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), t\_2 \cdot t\_3, 2\right)\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right), -0.0625, \sin y\right), t\_2 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_3 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1
         (/
          1.0
          (fma
           3.0
           (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) (* (cos x) t_0))
           3.0)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (* (sin x) (sqrt 2.0))))
   (if (<= x -0.96)
     (* t_1 (fma (fma (sin x) -0.0625 (sin y)) (* t_2 t_3) 2.0))
     (if (<= x 0.92)
       (*
        t_1
        (fma
         (fma
          (*
           x
           (fma
            (* x x)
            (fma
             (* x x)
             (fma (* x x) -0.0001984126984126984 0.008333333333333333)
             -0.16666666666666666)
            1.0))
          -0.0625
          (sin y))
         (* t_2 (* (sqrt 2.0) (fma -0.0625 (sin y) (sin x))))
         2.0))
       (/
        (+ 2.0 (* t_2 (* t_3 (- (sin y) (/ (sin x) 16.0)))))
        (*
         3.0
         (fma t_0 (cos x) (+ 1.0 (* (- 3.0 (sqrt 5.0)) (* (cos y) 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = 1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), (cos(x) * t_0)), 3.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = sin(x) * sqrt(2.0);
	double tmp;
	if (x <= -0.96) {
		tmp = t_1 * fma(fma(sin(x), -0.0625, sin(y)), (t_2 * t_3), 2.0);
	} else if (x <= 0.92) {
		tmp = t_1 * fma(fma((x * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0)), -0.0625, sin(y)), (t_2 * (sqrt(2.0) * fma(-0.0625, sin(y), sin(x)))), 2.0);
	} else {
		tmp = (2.0 + (t_2 * (t_3 * (sin(y) - (sin(x) / 16.0))))) / (3.0 * fma(t_0, cos(x), (1.0 + ((3.0 - sqrt(5.0)) * (cos(y) * 0.5)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), Float64(cos(x) * t_0)), 3.0))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(sin(x) * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.96)
		tmp = Float64(t_1 * fma(fma(sin(x), -0.0625, sin(y)), Float64(t_2 * t_3), 2.0));
	elseif (x <= 0.92)
		tmp = Float64(t_1 * fma(fma(Float64(x * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0)), -0.0625, sin(y)), Float64(t_2 * Float64(sqrt(2.0) * fma(-0.0625, sin(y), sin(x)))), 2.0));
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_3 * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * fma(t_0, cos(x), Float64(1.0 + Float64(Float64(3.0 - sqrt(5.0)) * Float64(cos(y) * 0.5))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.96], N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$3), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.92], N[(t$95$1 * N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(t$95$3 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(1.0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.95999999999999996

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.2%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 N/A

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

       4. sin-lowering-sin.f64 68.2

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

   11. Simplified 68.2%

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

   if -0.95999999999999996 < x < 0.92000000000000004

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}, \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}, \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       2. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}, \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}, 1\right)}, \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}, 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}, 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       6. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}, \frac{-1}{6}\right)}, 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}, \frac{-1}{6}\right), 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}, \frac{-1}{6}\right), 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       11. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{5040} \cdot {x}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right), \frac{-1}{16}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       15. *-lowering-*.f64 99.6

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right), -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   11. Simplified 99.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   if 0.92000000000000004 < x

   1. Initial program 98.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. div-sub N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. cos-lowering-cos.f64 N/A

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

      15. +-lowering-+.f64 N/A

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

      16. div-inv N/A

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

      17. metadata-eval N/A

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

      18. associate-*l* N/A

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

      19. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sin-lowering-sin.f64 66.0

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

   7. Simplified 66.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right), -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot t\_0\right), 3\right)}\\ t_2 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_3 := \cos x - \cos y\\ t_4 := \sin x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.38:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(t\_2, t\_3 \cdot t\_4, 2\right)\\ \mathbf{elif}\;x \leq 0.41:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(t\_2, \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_3 \cdot \left(t\_4 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1
         (/
          1.0
          (fma
           3.0
           (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) (* (cos x) t_0))
           3.0)))
        (t_2 (fma (sin x) -0.0625 (sin y)))
        (t_3 (- (cos x) (cos y)))
        (t_4 (* (sin x) (sqrt 2.0))))
   (if (<= x -0.38)
     (* t_1 (fma t_2 (* t_3 t_4) 2.0))
     (if (<= x 0.41)
       (*
        t_1
        (fma
         t_2
         (*
          (* (sqrt 2.0) (fma -0.0625 (sin y) (sin x)))
          (fma
           (* x x)
           (fma
            (* x x)
            (fma (* x x) -0.001388888888888889 0.041666666666666664)
            -0.5)
           (- 1.0 (cos y))))
         2.0))
       (/
        (+ 2.0 (* t_3 (* t_4 (- (sin y) (/ (sin x) 16.0)))))
        (*
         3.0
         (fma t_0 (cos x) (+ 1.0 (* (- 3.0 (sqrt 5.0)) (* (cos y) 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = 1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), (cos(x) * t_0)), 3.0);
	double t_2 = fma(sin(x), -0.0625, sin(y));
	double t_3 = cos(x) - cos(y);
	double t_4 = sin(x) * sqrt(2.0);
	double tmp;
	if (x <= -0.38) {
		tmp = t_1 * fma(t_2, (t_3 * t_4), 2.0);
	} else if (x <= 0.41) {
		tmp = t_1 * fma(t_2, ((sqrt(2.0) * fma(-0.0625, sin(y), sin(x))) * fma((x * x), fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5), (1.0 - cos(y)))), 2.0);
	} else {
		tmp = (2.0 + (t_3 * (t_4 * (sin(y) - (sin(x) / 16.0))))) / (3.0 * fma(t_0, cos(x), (1.0 + ((3.0 - sqrt(5.0)) * (cos(y) * 0.5)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), Float64(cos(x) * t_0)), 3.0))
	t_2 = fma(sin(x), -0.0625, sin(y))
	t_3 = Float64(cos(x) - cos(y))
	t_4 = Float64(sin(x) * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.38)
		tmp = Float64(t_1 * fma(t_2, Float64(t_3 * t_4), 2.0));
	elseif (x <= 0.41)
		tmp = Float64(t_1 * fma(t_2, Float64(Float64(sqrt(2.0) * fma(-0.0625, sin(y), sin(x))) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5), Float64(1.0 - cos(y)))), 2.0));
	else
		tmp = Float64(Float64(2.0 + Float64(t_3 * Float64(t_4 * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * fma(t_0, cos(x), Float64(1.0 + Float64(Float64(3.0 - sqrt(5.0)) * Float64(cos(y) * 0.5))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.38], N[(t$95$1 * N[(t$95$2 * N[(t$95$3 * t$95$4), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.41], N[(t$95$1 * N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$3 * N[(t$95$4 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(1.0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.38

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.2%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 N/A

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

       4. sin-lowering-sin.f64 68.2

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

   11. Simplified 68.2%

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

   if -0.38 < x < 0.409999999999999976

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. sub-neg N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. +-commutative N/A

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

       14. *-commutative N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 N/A

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

       18. --lowering--.f64 N/A

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

       19. cos-lowering-cos.f64 99.5

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

   11. Simplified 99.5%

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

   if 0.409999999999999976 < x

   1. Initial program 98.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. div-sub N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. cos-lowering-cos.f64 N/A

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

      15. +-lowering-+.f64 N/A

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

      16. div-inv N/A

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

      17. metadata-eval N/A

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

      18. associate-*l* N/A

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

      19. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sin-lowering-sin.f64 66.0

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

   7. Simplified 66.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.38:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)\\ \mathbf{elif}\;x \leq 0.41:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot t\_0\right), 3\right)}\\ t_2 := \cos x - \cos y\\ t_3 := \sin x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.26:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), t\_2 \cdot t\_3, 2\right)\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0005208333333333333, 0.010416666666666666\right), -0.0625\right), \sin y\right), t\_2 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_3 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1
         (/
          1.0
          (fma
           3.0
           (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) (* (cos x) t_0))
           3.0)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (* (sin x) (sqrt 2.0))))
   (if (<= x -0.26)
     (* t_1 (fma (fma (sin x) -0.0625 (sin y)) (* t_2 t_3) 2.0))
     (if (<= x 0.4)
       (*
        t_1
        (fma
         (fma
          x
          (fma
           (* x x)
           (fma (* x x) -0.0005208333333333333 0.010416666666666666)
           -0.0625)
          (sin y))
         (* t_2 (* (sqrt 2.0) (fma -0.0625 (sin y) (sin x))))
         2.0))
       (/
        (+ 2.0 (* t_2 (* t_3 (- (sin y) (/ (sin x) 16.0)))))
        (*
         3.0
         (fma t_0 (cos x) (+ 1.0 (* (- 3.0 (sqrt 5.0)) (* (cos y) 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = 1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), (cos(x) * t_0)), 3.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = sin(x) * sqrt(2.0);
	double tmp;
	if (x <= -0.26) {
		tmp = t_1 * fma(fma(sin(x), -0.0625, sin(y)), (t_2 * t_3), 2.0);
	} else if (x <= 0.4) {
		tmp = t_1 * fma(fma(x, fma((x * x), fma((x * x), -0.0005208333333333333, 0.010416666666666666), -0.0625), sin(y)), (t_2 * (sqrt(2.0) * fma(-0.0625, sin(y), sin(x)))), 2.0);
	} else {
		tmp = (2.0 + (t_2 * (t_3 * (sin(y) - (sin(x) / 16.0))))) / (3.0 * fma(t_0, cos(x), (1.0 + ((3.0 - sqrt(5.0)) * (cos(y) * 0.5)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), Float64(cos(x) * t_0)), 3.0))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(sin(x) * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.26)
		tmp = Float64(t_1 * fma(fma(sin(x), -0.0625, sin(y)), Float64(t_2 * t_3), 2.0));
	elseif (x <= 0.4)
		tmp = Float64(t_1 * fma(fma(x, fma(Float64(x * x), fma(Float64(x * x), -0.0005208333333333333, 0.010416666666666666), -0.0625), sin(y)), Float64(t_2 * Float64(sqrt(2.0) * fma(-0.0625, sin(y), sin(x)))), 2.0));
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_3 * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * fma(t_0, cos(x), Float64(1.0 + Float64(Float64(3.0 - sqrt(5.0)) * Float64(cos(y) * 0.5))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.26], N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$3), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.4], N[(t$95$1 * N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0005208333333333333 + 0.010416666666666666), $MachinePrecision] + -0.0625), $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(t$95$3 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(1.0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.26000000000000001

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.2%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 N/A

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

       4. sin-lowering-sin.f64 68.2

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

   11. Simplified 68.2%

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

   if -0.26000000000000001 < x < 0.40000000000000002

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)}, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y}, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, \sin y\right)}, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       3. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, {x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{16}}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}, \frac{-1}{16}\right)}, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}, \frac{-1}{16}\right), \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}, \frac{-1}{16}\right), \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       8. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}}, \frac{-1}{16}\right), \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{1920}} + \frac{1}{96}, \frac{-1}{16}\right), \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{1920}, \frac{1}{96}\right)}, \frac{-1}{16}\right), \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{1920}, \frac{1}{96}\right), \frac{-1}{16}\right), \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{1920}, \frac{1}{96}\right), \frac{-1}{16}\right), \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       13. sin-lowering-sin.f64 99.5

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0005208333333333333, 0.010416666666666666\right), -0.0625\right), \color{blue}{\sin y}\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   11. Simplified 99.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0005208333333333333, 0.010416666666666666\right), -0.0625\right), \sin y\right)}, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   if 0.40000000000000002 < x

   1. Initial program 98.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. div-sub N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. cos-lowering-cos.f64 N/A

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

      15. +-lowering-+.f64 N/A

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

      16. div-inv N/A

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

      17. metadata-eval N/A

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

      18. associate-*l* N/A

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

      19. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sin-lowering-sin.f64 66.0

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

   7. Simplified 66.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.26:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0005208333333333333, 0.010416666666666666\right), -0.0625\right), \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot t\_0\right), 3\right)}\\ t_2 := \cos x - \cos y\\ t_3 := \sin x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), t\_2 \cdot t\_3, 2\right)\\ \mathbf{elif}\;x \leq 0.235:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 0.010416666666666666, -0.0625\right), \sin y\right), t\_2 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_3 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1
         (/
          1.0
          (fma
           3.0
           (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) (* (cos x) t_0))
           3.0)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (* (sin x) (sqrt 2.0))))
   (if (<= x -0.2)
     (* t_1 (fma (fma (sin x) -0.0625 (sin y)) (* t_2 t_3) 2.0))
     (if (<= x 0.235)
       (*
        t_1
        (fma
         (fma x (fma (* x x) 0.010416666666666666 -0.0625) (sin y))
         (* t_2 (* (sqrt 2.0) (fma -0.0625 (sin y) (sin x))))
         2.0))
       (/
        (+ 2.0 (* t_2 (* t_3 (- (sin y) (/ (sin x) 16.0)))))
        (*
         3.0
         (fma t_0 (cos x) (+ 1.0 (* (- 3.0 (sqrt 5.0)) (* (cos y) 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = 1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), (cos(x) * t_0)), 3.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = sin(x) * sqrt(2.0);
	double tmp;
	if (x <= -0.2) {
		tmp = t_1 * fma(fma(sin(x), -0.0625, sin(y)), (t_2 * t_3), 2.0);
	} else if (x <= 0.235) {
		tmp = t_1 * fma(fma(x, fma((x * x), 0.010416666666666666, -0.0625), sin(y)), (t_2 * (sqrt(2.0) * fma(-0.0625, sin(y), sin(x)))), 2.0);
	} else {
		tmp = (2.0 + (t_2 * (t_3 * (sin(y) - (sin(x) / 16.0))))) / (3.0 * fma(t_0, cos(x), (1.0 + ((3.0 - sqrt(5.0)) * (cos(y) * 0.5)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), Float64(cos(x) * t_0)), 3.0))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(sin(x) * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.2)
		tmp = Float64(t_1 * fma(fma(sin(x), -0.0625, sin(y)), Float64(t_2 * t_3), 2.0));
	elseif (x <= 0.235)
		tmp = Float64(t_1 * fma(fma(x, fma(Float64(x * x), 0.010416666666666666, -0.0625), sin(y)), Float64(t_2 * Float64(sqrt(2.0) * fma(-0.0625, sin(y), sin(x)))), 2.0));
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_3 * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * fma(t_0, cos(x), Float64(1.0 + Float64(Float64(3.0 - sqrt(5.0)) * Float64(cos(y) * 0.5))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.2], N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$3), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.235], N[(t$95$1 * N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.010416666666666666 + -0.0625), $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(t$95$3 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(1.0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.20000000000000001

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.2%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 N/A

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

       4. sin-lowering-sin.f64 68.2

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

   11. Simplified 68.2%

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

   if -0.20000000000000001 < x < 0.23499999999999999

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. sin-lowering-sin.f64 99.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 0.010416666666666666, -0.0625\right), \color{blue}{\sin y}\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   11. Simplified 99.3%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 0.010416666666666666, -0.0625\right), \sin y\right)}, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   if 0.23499999999999999 < x

   1. Initial program 98.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. div-sub N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. cos-lowering-cos.f64 N/A

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

      15. +-lowering-+.f64 N/A

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

      16. div-inv N/A

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

      17. metadata-eval N/A

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

      18. associate-*l* N/A

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

      19. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sin-lowering-sin.f64 66.0

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

   7. Simplified 66.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)\\ \mathbf{elif}\;x \leq 0.235:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 0.010416666666666666, -0.0625\right), \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \cos x - \cos y\\ t_2 := \sin x \cdot \sqrt{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot t\_0\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), t\_1 \cdot t\_2, 2\right)\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.0625, \sqrt{2}, \left(x \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.16731770833333334\right)\right), \sqrt{2} \cdot \left(\sin y \cdot 1.00390625\right)\right), \sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_3, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(t\_2 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, 1 + t\_3 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (- (cos x) (cos y)))
        (t_2 (* (sin x) (sqrt 2.0)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.185)
     (*
      (/
       1.0
       (fma 3.0 (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) (* (cos x) t_0)) 3.0))
      (fma (fma (sin x) -0.0625 (sin y)) (* t_1 t_2) 2.0))
     (if (<= x 0.02)
       (/
        (+
         2.0
         (*
          (fma
           x
           (fma
            x
            (fma
             -0.0625
             (sqrt 2.0)
             (* (* x (sqrt 2.0)) (* (sin y) -0.16731770833333334)))
            (* (sqrt 2.0) (* (sin y) 1.00390625)))
           (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0))))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) t_3 (sqrt 5.0)) -1.5)))
       (/
        (+ 2.0 (* t_1 (* t_2 (- (sin y) (/ (sin x) 16.0)))))
        (* 3.0 (fma t_0 (cos x) (+ 1.0 (* t_3 (* (cos y) 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = cos(x) - cos(y);
	double t_2 = sin(x) * sqrt(2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.185) {
		tmp = (1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), (cos(x) * t_0)), 3.0)) * fma(fma(sin(x), -0.0625, sin(y)), (t_1 * t_2), 2.0);
	} else if (x <= 0.02) {
		tmp = (2.0 + (fma(x, fma(x, fma(-0.0625, sqrt(2.0), ((x * sqrt(2.0)) * (sin(y) * -0.16731770833333334))), (sqrt(2.0) * (sin(y) * 1.00390625))), (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0)))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_3, sqrt(5.0)), -1.5));
	} else {
		tmp = (2.0 + (t_1 * (t_2 * (sin(y) - (sin(x) / 16.0))))) / (3.0 * fma(t_0, cos(x), (1.0 + (t_3 * (cos(y) * 0.5)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sin(x) * sqrt(2.0))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(Float64(1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), Float64(cos(x) * t_0)), 3.0)) * fma(fma(sin(x), -0.0625, sin(y)), Float64(t_1 * t_2), 2.0));
	elseif (x <= 0.02)
		tmp = Float64(Float64(2.0 + Float64(fma(x, fma(x, fma(-0.0625, sqrt(2.0), Float64(Float64(x * sqrt(2.0)) * Float64(sin(y) * -0.16731770833333334))), Float64(sqrt(2.0) * Float64(sin(y) * 1.00390625))), Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0)))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_3, sqrt(5.0)), -1.5)));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(t_2 * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * fma(t_0, cos(x), Float64(1.0 + Float64(t_3 * Float64(cos(y) * 0.5))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(1.0 / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.02], N[(N[(2.0 + N[(N[(x * N[(x * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision] + N[(N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.16731770833333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$3 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 * N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(1.0 + N[(t$95$3 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.2%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 N/A

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

       4. sin-lowering-sin.f64 68.2

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

   11. Simplified 68.2%

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

   if -0.185 < x < 0.0200000000000000004

   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

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(x \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + x \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right)\right)\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2 + \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + x \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right)\right)\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + x \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right)\right)\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right), \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

   11. Simplified 99.2%

       \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.0625, \sqrt{2}, \left(x \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.16731770833333334\right)\right), \sqrt{2} \cdot \left(1.00390625 \cdot \sin y\right)\right), \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 0.0200000000000000004 < x

   1. Initial program 98.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. div-sub N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. cos-lowering-cos.f64 N/A

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

      15. +-lowering-+.f64 N/A

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

      16. div-inv N/A

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

      17. metadata-eval N/A

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

      18. associate-*l* N/A

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

      19. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sin-lowering-sin.f64 66.0

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

   7. Simplified 66.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.0625, \sqrt{2}, \left(x \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.16731770833333334\right)\right), \sqrt{2} \cdot \left(\sin y \cdot 1.00390625\right)\right), \sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.0625, \sqrt{2}, \left(x \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.16731770833333334\right)\right), \sqrt{2} \cdot \left(\sin y \cdot 1.00390625\right)\right), \sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (/
           1.0
           (fma
            3.0
            (fma
             (fma (sqrt 5.0) -0.5 1.5)
             (cos y)
             (* (cos x) (fma (sqrt 5.0) 0.5 -0.5)))
            3.0))
          (fma
           (fma (sin x) -0.0625 (sin y))
           (* (- (cos x) (cos y)) (* (sin x) (sqrt 2.0)))
           2.0))))
   (if (<= x -0.185)
     t_0
     (if (<= x 0.0048)
       (/
        (+
         2.0
         (*
          (fma
           x
           (fma
            x
            (fma
             -0.0625
             (sqrt 2.0)
             (* (* x (sqrt 2.0)) (* (sin y) -0.16731770833333334)))
            (* (sqrt 2.0) (* (sin y) 1.00390625)))
           (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0))))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -1.5)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), (cos(x) * fma(sqrt(5.0), 0.5, -0.5))), 3.0)) * fma(fma(sin(x), -0.0625, sin(y)), ((cos(x) - cos(y)) * (sin(x) * sqrt(2.0))), 2.0);
	double tmp;
	if (x <= -0.185) {
		tmp = t_0;
	} else if (x <= 0.0048) {
		tmp = (2.0 + (fma(x, fma(x, fma(-0.0625, sqrt(2.0), ((x * sqrt(2.0)) * (sin(y) * -0.16731770833333334))), (sqrt(2.0) * (sin(y) * 1.00390625))), (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0)))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -1.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), Float64(cos(x) * fma(sqrt(5.0), 0.5, -0.5))), 3.0)) * fma(fma(sin(x), -0.0625, sin(y)), Float64(Float64(cos(x) - cos(y)) * Float64(sin(x) * sqrt(2.0))), 2.0))
	tmp = 0.0
	if (x <= -0.185)
		tmp = t_0;
	elseif (x <= 0.0048)
		tmp = Float64(Float64(2.0 + Float64(fma(x, fma(x, fma(-0.0625, sqrt(2.0), Float64(Float64(x * sqrt(2.0)) * Float64(sin(y) * -0.16731770833333334))), Float64(sqrt(2.0) * Float64(sin(y) * 1.00390625))), Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0)))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -1.5)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.185], t$95$0, If[LessEqual[x, 0.0048], N[(N[(2.0 + N[(N[(x * N[(x * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision] + N[(N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.16731770833333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.185 or 0.00479999999999999958 < x

   1. Initial program 98.9%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.0%

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

   8. Applied egg-rr 99.1%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sqrt-lowering-sqrt.f64 N/A

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

       4. sin-lowering-sin.f64 67.2

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

   11. Simplified 67.2%

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

   if -0.185 < x < 0.00479999999999999958

   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

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(x \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + x \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right)\right)\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2 + \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + x \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right)\right)\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + x \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right)\right)\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right), \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

   11. Simplified 99.2%

       \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.0625, \sqrt{2}, \left(x \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.16731770833333334\right)\right), \sqrt{2} \cdot \left(1.00390625 \cdot \sin y\right)\right), \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.0625, \sqrt{2}, \left(x \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.16731770833333334\right)\right), \sqrt{2} \cdot \left(\sin y \cdot 1.00390625\right)\right), \sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right)\\ t_1 := \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\sin y, t\_0, 2\right)\\ \mathbf{if}\;y \leq -0.00011:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), t\_0, 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (* (- (cos x) (cos y)) (* (sqrt 2.0) (fma -0.0625 (sin y) (sin x)))))
        (t_1
         (*
          (/
           1.0
           (fma
            3.0
            (fma
             (fma (sqrt 5.0) -0.5 1.5)
             (cos y)
             (* (cos x) (fma (sqrt 5.0) 0.5 -0.5)))
            3.0))
          (fma (sin y) t_0 2.0))))
   (if (<= y -0.00011)
     t_1
     (if (<= y 4.2e-15)
       (*
        (fma (fma (sin x) -0.0625 (sin y)) t_0 2.0)
        (/
         1.0
         (fma 1.5 (- (fma (cos x) (+ (sqrt 5.0) -1.0) 3.0) (sqrt 5.0)) 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = (cos(x) - cos(y)) * (sqrt(2.0) * fma(-0.0625, sin(y), sin(x)));
	double t_1 = (1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), (cos(x) * fma(sqrt(5.0), 0.5, -0.5))), 3.0)) * fma(sin(y), t_0, 2.0);
	double tmp;
	if (y <= -0.00011) {
		tmp = t_1;
	} else if (y <= 4.2e-15) {
		tmp = fma(fma(sin(x), -0.0625, sin(y)), t_0, 2.0) * (1.0 / fma(1.5, (fma(cos(x), (sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * fma(-0.0625, sin(y), sin(x))))
	t_1 = Float64(Float64(1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), Float64(cos(x) * fma(sqrt(5.0), 0.5, -0.5))), 3.0)) * fma(sin(y), t_0, 2.0))
	tmp = 0.0
	if (y <= -0.00011)
		tmp = t_1;
	elseif (y <= 4.2e-15)
		tmp = Float64(fma(fma(sin(x), -0.0625, sin(y)), t_0, 2.0) * Float64(1.0 / fma(1.5, Float64(fma(cos(x), Float64(sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00011], t$95$1, If[LessEqual[y, 4.2e-15], N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * N[(1.0 / N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.10000000000000004e-4 or 4.19999999999999962e-15 < y

   1. Initial program 99.0%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. div-sub N/A

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

      6. --lowering--.f64 N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

   4. Applied egg-rr 99.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.1%

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in x around 0

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

       1. sin-lowering-sin.f64 63.4

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

   11. Simplified 63.4%

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

   if -1.10000000000000004e-4 < y < 4.19999999999999962e-15

   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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.5%

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00011:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;x \leq 0.015:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.0625, \sqrt{2}, \left(x \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.16731770833333334\right)\right), \sqrt{2} \cdot \left(\sin y \cdot 1.00390625\right)\right), \sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.185)
     (/
      (+
       2.0
       (*
        (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
        (+ (cos x) -1.0)))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ t_0 2.0)))))
     (if (<= x 0.015)
       (/
        (+
         2.0
         (*
          (fma
           x
           (fma
            x
            (fma
             -0.0625
             (sqrt 2.0)
             (* (* x (sqrt 2.0)) (* (sin y) -0.16731770833333334)))
            (* (sqrt 2.0) (* (sin y) 1.00390625)))
           (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0))))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) t_0 (sqrt 5.0)) -1.5)))
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma
         (cos x)
         (fma (sqrt 5.0) 0.5 -0.5)
         (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.185) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else if (x <= 0.015) {
		tmp = (2.0 + (fma(x, fma(x, fma(-0.0625, sqrt(2.0), ((x * sqrt(2.0)) * (sin(y) * -0.16731770833333334))), (sqrt(2.0) * (sin(y) * 1.00390625))), (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0)))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_0, sqrt(5.0)), -1.5));
	} else {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (x <= 0.015)
		tmp = Float64(Float64(2.0 + Float64(fma(x, fma(x, fma(-0.0625, sqrt(2.0), Float64(Float64(x * sqrt(2.0)) * Float64(sin(y) * -0.16731770833333334))), Float64(sqrt(2.0) * Float64(sin(y) * 1.00390625))), Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0)))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_0, sqrt(5.0)), -1.5)));
	else
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.015], N[(N[(2.0 + N[(N[(x * N[(x * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision] + N[(N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.16731770833333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. metadata-eval 65.2

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

   5. Simplified 65.2%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sin-lowering-sin.f64 64.8

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

   8. Simplified 64.8%

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

   if -0.185 < x < 0.014999999999999999

   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

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(x \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + x \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right)\right)\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2 + \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + x \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right)\right)\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{16} \cdot \sqrt{2} + x \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right)\right)\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right), \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

   11. Simplified 99.2%

       \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.0625, \sqrt{2}, \left(x \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.16731770833333334\right)\right), \sqrt{2} \cdot \left(1.00390625 \cdot \sin y\right)\right), \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 0.014999999999999999 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 62.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 62.8%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.015:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.0625, \sqrt{2}, \left(x \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.16731770833333334\right)\right), \sqrt{2} \cdot \left(\sin y \cdot 1.00390625\right)\right), \sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{2 + \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.185)
     (/
      (+
       2.0
       (*
        (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
        (+ (cos x) -1.0)))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ t_0 2.0)))))
     (if (<= x 0.0135)
       (/
        (+
         2.0
         (*
          (- (fma (* x x) -0.5 1.0) (cos y))
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (/ x 16.0)))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) t_0 (sqrt 5.0)) -1.5)))
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma
         (cos x)
         (fma (sqrt 5.0) 0.5 -0.5)
         (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.185) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else if (x <= 0.0135) {
		tmp = (2.0 + ((fma((x * x), -0.5, 1.0) - cos(y)) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_0, sqrt(5.0)), -1.5));
	} else {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (x <= 0.0135)
		tmp = Float64(Float64(2.0 + Float64(Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)) * Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_0, sqrt(5.0)), -1.5)));
	else
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(2.0 + N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * 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[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. metadata-eval 65.2

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

   5. Simplified 65.2%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sin-lowering-sin.f64 64.8

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

   8. Simplified 64.8%

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

   if -0.185 < x < 0.0134999999999999998

   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

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 99.1%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 0.0134999999999999998 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 62.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 62.8%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{2 + \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_1, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos x \cdot t\_2\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{2 + \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_0 \cdot t\_1, 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= x -0.185)
     (/
      (fma t_0 t_1 2.0)
      (fma 3.0 (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (* (cos x) t_2)) 3.0))
     (if (<= x 0.0135)
       (/
        (+
         2.0
         (*
          (- (fma (* x x) -0.5 1.0) (cos y))
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (/ x 16.0)))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -1.5)))
       (/
        (fma 0.3333333333333333 (* t_0 t_1) 0.6666666666666666)
        (fma (cos x) t_2 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(2.0) * fma(cos(x), -0.0625, 0.0625);
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (x <= -0.185) {
		tmp = fma(t_0, t_1, 2.0) / fma(3.0, fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), (cos(x) * t_2)), 3.0);
	} else if (x <= 0.0135) {
		tmp = (2.0 + ((fma((x * x), -0.5, 1.0) - cos(y)) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -1.5));
	} else {
		tmp = fma(0.3333333333333333, (t_0 * t_1), 0.6666666666666666) / fma(cos(x), t_2, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(fma(t_0, t_1, 2.0) / fma(3.0, fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), Float64(cos(x) * t_2)), 3.0));
	elseif (x <= 0.0135)
		tmp = Float64(Float64(2.0 + Float64(Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)) * Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -1.5)));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(t_0 * t_1), 0.6666666666666666) / fma(cos(x), t_2, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(t$95$0 * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(2.0 + N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * 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[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$0 * t$95$1), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 64.5%

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

   7. Applied egg-rr 64.5%

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

   if -0.185 < x < 0.0134999999999999998

   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

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 99.1%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 0.0134999999999999998 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 62.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 62.8%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{2 + \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_1, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos x \cdot t\_2\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0058:\\ \;\;\;\;\frac{2 + \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_0 \cdot t\_1, 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= x -0.185)
     (/
      (fma t_0 t_1 2.0)
      (fma 3.0 (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (* (cos x) t_2)) 3.0))
     (if (<= x 0.0058)
       (/
        (+
         2.0
         (*
          (- (fma (* x x) -0.5 1.0) (cos y))
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (fma -0.0625 (sin y) x)))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -1.5)))
       (/
        (fma 0.3333333333333333 (* t_0 t_1) 0.6666666666666666)
        (fma (cos x) t_2 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(2.0) * fma(cos(x), -0.0625, 0.0625);
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (x <= -0.185) {
		tmp = fma(t_0, t_1, 2.0) / fma(3.0, fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), (cos(x) * t_2)), 3.0);
	} else if (x <= 0.0058) {
		tmp = (2.0 + ((fma((x * x), -0.5, 1.0) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * fma(-0.0625, sin(y), x))))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -1.5));
	} else {
		tmp = fma(0.3333333333333333, (t_0 * t_1), 0.6666666666666666) / fma(cos(x), t_2, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(fma(t_0, t_1, 2.0) / fma(3.0, fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), Float64(cos(x) * t_2)), 3.0));
	elseif (x <= 0.0058)
		tmp = Float64(Float64(2.0 + Float64(Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * fma(-0.0625, sin(y), x))))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -1.5)));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(t_0 * t_1), 0.6666666666666666) / fma(cos(x), t_2, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(t$95$0 * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0058], N[(N[(2.0 + N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$0 * t$95$1), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 64.5%

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

   7. Applied egg-rr 64.5%

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

   if -0.185 < x < 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 x around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. *-lowering-*.f64 N/A

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

       4. sqrt-lowering-sqrt.f64 N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. sin-lowering-sin.f64 99.0

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin y}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   11. Simplified 99.0%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 0.0058 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 62.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0058:\\ \;\;\;\;\frac{2 + \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := {\sin y}^{2}\\ t_2 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.00086:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot t\_0\right), 3\right)} \cdot \mathsf{fma}\left(-0.0625 \cdot t\_1, \sqrt{2} \cdot t\_2, 2\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_2 \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right), 3, t\_0 \cdot \left(\cos x \cdot 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (pow (sin y) 2.0))
        (t_2 (- 1.0 (cos y))))
   (if (<= y -0.00086)
     (*
      (/
       1.0
       (fma 3.0 (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) (* (cos x) t_0)) 3.0))
      (fma (* -0.0625 t_1) (* (sqrt 2.0) t_2) 2.0))
     (if (<= y 4.2e-15)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma (cos x) t_0 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))
       (/
        (fma t_1 (* t_2 (* -0.0625 (sqrt 2.0))) 2.0)
        (fma
         (fma (- 3.0 (sqrt 5.0)) (* (cos y) 0.5) 1.0)
         3.0
         (* t_0 (* (cos x) 3.0))))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = pow(sin(y), 2.0);
	double t_2 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.00086) {
		tmp = (1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), (cos(x) * t_0)), 3.0)) * fma((-0.0625 * t_1), (sqrt(2.0) * t_2), 2.0);
	} else if (y <= 4.2e-15) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), t_0, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	} else {
		tmp = fma(t_1, (t_2 * (-0.0625 * sqrt(2.0))), 2.0) / fma(fma((3.0 - sqrt(5.0)), (cos(y) * 0.5), 1.0), 3.0, (t_0 * (cos(x) * 3.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.00086)
		tmp = Float64(Float64(1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), Float64(cos(x) * t_0)), 3.0)) * fma(Float64(-0.0625 * t_1), Float64(sqrt(2.0) * t_2), 2.0));
	elseif (y <= 4.2e-15)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), t_0, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	else
		tmp = Float64(fma(t_1, Float64(t_2 * Float64(-0.0625 * sqrt(2.0))), 2.0) / fma(fma(Float64(3.0 - sqrt(5.0)), Float64(cos(y) * 0.5), 1.0), 3.0, Float64(t_0 * Float64(cos(x) * 3.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00086], N[(N[(1.0 / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-15], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(t$95$2 * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.59999999999999979e-4

   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. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

      5. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5}} \cdot \frac{1}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y \cdot 3}, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y} \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)}\right)} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 3 \cdot 1}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{3}\right)} \]

   4. Applied egg-rr 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

   6. Applied egg-rr 99.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)} \]

   8. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2\right)} \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)} \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       5. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       6. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       10. cos-lowering-cos.f64 57.8

           \[\leadsto \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   11. Simplified 57.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)} \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   if -8.59999999999999979e-4 < y < 4.19999999999999962e-15

   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-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 99.4%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]

   if 4.19999999999999962e-15 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      4. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} - \frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5}}{2} - \color{blue}{\frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{1}{-2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{1}{-2}\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{1}{-2}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\frac{-1}{2}}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x}, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      16. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

   4. Applied egg-rr 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\sqrt{2} \cdot \frac{-1}{16}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. cos-lowering-cos.f64 62.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) + \left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{\left(1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3 + \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\left(1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3 + \color{blue}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right)}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right), 3, 3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right) + 1}, 3, 3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right)\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \frac{1}{2} \cdot \cos y, 1\right)}, 3, 3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right)\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \frac{1}{2} \cdot \cos y, 1\right), 3, 3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right)\right)} \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \frac{1}{2} \cdot \cos y, 1\right), 3, 3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\frac{1}{2} \cdot \cos y}, 1\right), 3, 3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right)\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \frac{1}{2} \cdot \color{blue}{\cos y}, 1\right), 3, 3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \frac{1}{2} \cdot \cos y, 1\right), 3, \color{blue}{\left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \frac{1}{2} \cdot \cos y, 1\right), 3, \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \frac{1}{2} \cdot \cos y, 1\right), 3, \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)}\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \frac{1}{2} \cdot \cos y, 1\right), 3, \color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \left(\cos x \cdot 3\right)\right)} \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \frac{1}{2} \cdot \cos y, 1\right), 3, \mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \frac{1}{2} \cdot \cos y, 1\right), 3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \color{blue}{\left(\cos x \cdot 3\right)}\right)} \]

   9. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, 1\right), 3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \left(\cos x \cdot 3\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00086:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right), 3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \left(\cos x \cdot 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := {\sin y}^{2}\\ t_2 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.001:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot t\_0\right), 3\right)} \cdot \mathsf{fma}\left(-0.0625 \cdot t\_1, \sqrt{2} \cdot t\_2, 2\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_2 \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, \mathsf{fma}\left(\cos y \cdot \left(3 - \sqrt{5}\right), 0.5, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (pow (sin y) 2.0))
        (t_2 (- 1.0 (cos y))))
   (if (<= y -0.001)
     (*
      (/
       1.0
       (fma 3.0 (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) (* (cos x) t_0)) 3.0))
      (fma (* -0.0625 t_1) (* (sqrt 2.0) t_2) 2.0))
     (if (<= y 4.2e-15)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma (cos x) t_0 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))
       (/
        (fma t_1 (* t_2 (* -0.0625 (sqrt 2.0))) 2.0)
        (*
         3.0
         (fma t_0 (cos x) (fma (* (cos y) (- 3.0 (sqrt 5.0))) 0.5 1.0))))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = pow(sin(y), 2.0);
	double t_2 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.001) {
		tmp = (1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), (cos(x) * t_0)), 3.0)) * fma((-0.0625 * t_1), (sqrt(2.0) * t_2), 2.0);
	} else if (y <= 4.2e-15) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), t_0, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	} else {
		tmp = fma(t_1, (t_2 * (-0.0625 * sqrt(2.0))), 2.0) / (3.0 * fma(t_0, cos(x), fma((cos(y) * (3.0 - sqrt(5.0))), 0.5, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.001)
		tmp = Float64(Float64(1.0 / fma(3.0, fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), Float64(cos(x) * t_0)), 3.0)) * fma(Float64(-0.0625 * t_1), Float64(sqrt(2.0) * t_2), 2.0));
	elseif (y <= 4.2e-15)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), t_0, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	else
		tmp = Float64(fma(t_1, Float64(t_2 * Float64(-0.0625 * sqrt(2.0))), 2.0) / Float64(3.0 * fma(t_0, cos(x), fma(Float64(cos(y) * Float64(3.0 - sqrt(5.0))), 0.5, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.001], N[(N[(1.0 / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-15], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(t$95$2 * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e-3

   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. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

      5. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5}} \cdot \frac{1}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y \cdot 3}, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y} \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)}\right)} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 3 \cdot 1}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{3}\right)} \]

   4. Applied egg-rr 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

   6. Applied egg-rr 99.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)} \]

   8. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2\right)} \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)} \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       5. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       6. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right), 3\right)} \]

       10. cos-lowering-cos.f64 57.8

           \[\leadsto \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   11. Simplified 57.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)} \cdot \frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)} \]

   if -1e-3 < y < 4.19999999999999962e-15

   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-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 99.4%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]

   if 4.19999999999999962e-15 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      4. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} - \frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5}}{2} - \color{blue}{\frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{1}{-2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{1}{-2}\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{1}{-2}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\frac{-1}{2}}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x}, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      16. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

   4. Applied egg-rr 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\sqrt{2} \cdot \frac{-1}{16}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. cos-lowering-cos.f64 62.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right) + 1}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\left(\cos y \cdot \frac{1}{2}\right)} + 1\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{1}{2}, 1\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}, \frac{1}{2}, 1\right)\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y, \frac{1}{2}, 1\right)\right)} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y, \frac{1}{2}, 1\right)\right)} \]

      8. cos-lowering-cos.f64 62.8

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}, 0.5, 1\right)\right)} \]

   9. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, 1\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.001:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)} \cdot \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\cos y \cdot \left(3 - \sqrt{5}\right), 0.5, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos y \cdot \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, t\_1, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 \cdot \mathsf{fma}\left(t\_1, \cos x, \mathsf{fma}\left(\cos y \cdot \left(3 - \sqrt{5}\right), 0.5, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (pow (sin y) 2.0)
          (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0)))
          2.0))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= y -0.0009)
     (/
      t_0
      (fma
       3.0
       (fma
        (cos x)
        (fma 0.5 (sqrt 5.0) -0.5)
        (* (cos y) (fma (sqrt 5.0) -0.5 1.5)))
       3.0))
     (if (<= y 4.2e-15)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma (cos x) t_1 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))
       (/
        t_0
        (*
         3.0
         (fma t_1 (cos x) (fma (* (cos y) (- 3.0 (sqrt 5.0))) 0.5 1.0))))))))

C

double code(double x, double y) {
	double t_0 = fma(pow(sin(y), 2.0), ((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), 2.0);
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (y <= -0.0009) {
		tmp = t_0 / fma(3.0, fma(cos(x), fma(0.5, sqrt(5.0), -0.5), (cos(y) * fma(sqrt(5.0), -0.5, 1.5))), 3.0);
	} else if (y <= 4.2e-15) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), t_1, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	} else {
		tmp = t_0 / (3.0 * fma(t_1, cos(x), fma((cos(y) * (3.0 - sqrt(5.0))), 0.5, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma((sin(y) ^ 2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), 2.0)
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (y <= -0.0009)
		tmp = Float64(t_0 / fma(3.0, fma(cos(x), fma(0.5, sqrt(5.0), -0.5), Float64(cos(y) * fma(sqrt(5.0), -0.5, 1.5))), 3.0));
	elseif (y <= 4.2e-15)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), t_1, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	else
		tmp = Float64(t_0 / Float64(3.0 * fma(t_1, cos(x), fma(Float64(cos(y) * Float64(3.0 - sqrt(5.0))), 0.5, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[y, -0.0009], N[(t$95$0 / N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-15], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999998e-4

   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. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      4. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} - \frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5}}{2} - \color{blue}{\frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{1}{-2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{1}{-2}\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{1}{-2}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\frac{-1}{2}}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x}, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      16. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

   4. Applied egg-rr 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\sqrt{2} \cdot \frac{-1}{16}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. cos-lowering-cos.f64 57.7

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Simplified 57.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3 \cdot 1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + \color{blue}{3}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right), 3\right)}} \]

   10. Simplified 57.8%

       \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right) \cdot \cos y\right), 3\right)}} \]

   if -8.9999999999999998e-4 < y < 4.19999999999999962e-15

   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-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 99.4%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]

   if 4.19999999999999962e-15 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      4. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} - \frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5}}{2} - \color{blue}{\frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{1}{-2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{1}{-2}\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{1}{-2}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\frac{-1}{2}}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x}, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      16. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

   4. Applied egg-rr 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\sqrt{2} \cdot \frac{-1}{16}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. cos-lowering-cos.f64 62.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right) + 1}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\left(\cos y \cdot \frac{1}{2}\right)} + 1\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{1}{2}, 1\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}, \frac{1}{2}, 1\right)\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y, \frac{1}{2}, 1\right)\right)} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y, \frac{1}{2}, 1\right)\right)} \]

      8. cos-lowering-cos.f64 62.8

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}, 0.5, 1\right)\right)} \]

   9. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, 1\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos y \cdot \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\cos y \cdot \left(3 - \sqrt{5}\right), 0.5, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos y \cdot \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right)\right), 3\right)}\\ \mathbf{if}\;y \leq -0.00075:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (pow (sin y) 2.0)
           (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0)))
           2.0)
          (fma
           3.0
           (fma
            (cos x)
            (fma 0.5 (sqrt 5.0) -0.5)
            (* (cos y) (fma (sqrt 5.0) -0.5 1.5)))
           3.0))))
   (if (<= y -0.00075)
     t_0
     (if (<= y 4.2e-15)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma
         (cos x)
         (fma (sqrt 5.0) 0.5 -0.5)
         (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma(pow(sin(y), 2.0), ((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), 2.0) / fma(3.0, fma(cos(x), fma(0.5, sqrt(5.0), -0.5), (cos(y) * fma(sqrt(5.0), -0.5, 1.5))), 3.0);
	double tmp;
	if (y <= -0.00075) {
		tmp = t_0;
	} else if (y <= 4.2e-15) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma((sin(y) ^ 2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), 2.0) / fma(3.0, fma(cos(x), fma(0.5, sqrt(5.0), -0.5), Float64(cos(y) * fma(sqrt(5.0), -0.5, 1.5))), 3.0))
	tmp = 0.0
	if (y <= -0.00075)
		tmp = t_0;
	elseif (y <= 4.2e-15)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00075], t$95$0, If[LessEqual[y, 4.2e-15], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5000000000000002e-4 or 4.19999999999999962e-15 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      4. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} - \frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5}}{2} - \color{blue}{\frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{1}{-2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{1}{-2}\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{1}{-2}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\frac{-1}{2}}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x}, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      16. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

   4. Applied egg-rr 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\sqrt{2} \cdot \frac{-1}{16}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. cos-lowering-cos.f64 60.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Simplified 60.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3 \cdot 1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + \color{blue}{3}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right), 3\right)}} \]

   10. Simplified 60.0%

       \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right) \cdot \cos y\right), 3\right)}} \]

   if -7.5000000000000002e-4 < y < 4.19999999999999962e-15

   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-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 99.4%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00075:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos y \cdot \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos y \cdot \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{if}\;y \leq -0.0006:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1
         (/
          (fma
           (pow (sin y) 2.0)
           (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0)))
           2.0)
          (*
           3.0
           (fma t_0 (cos x) (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) 1.0))))))
   (if (<= y -0.0006)
     t_1
     (if (<= y 4.2e-15)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma (cos x) t_0 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = fma(pow(sin(y), 2.0), ((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), 2.0) / (3.0 * fma(t_0, cos(x), fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), 1.0)));
	double tmp;
	if (y <= -0.0006) {
		tmp = t_1;
	} else if (y <= 4.2e-15) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), t_0, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(fma((sin(y) ^ 2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), 2.0) / Float64(3.0 * fma(t_0, cos(x), fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), 1.0))))
	tmp = 0.0
	if (y <= -0.0006)
		tmp = t_1;
	elseif (y <= 4.2e-15)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(cos(x), t_0, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0006], t$95$1, If[LessEqual[y, 4.2e-15], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.99999999999999947e-4 or 4.19999999999999962e-15 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      4. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} - \frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5}}{2} - \color{blue}{\frac{1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{1}{-2}}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{1}{-2}\right)}, \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{1}{-2}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\frac{-1}{2}}\right), \cos x, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x}, 1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      16. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\right)} \]

   4. Applied egg-rr 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\sqrt{2} \cdot \frac{-1}{16}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. cos-lowering-cos.f64 60.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Simplified 60.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \frac{1}{2} \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)} + 1\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y} + 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(\frac{1}{2} \cdot \color{blue}{\left(3 + \left(\mathsf{neg}\left(\sqrt{5}\right)\right)\right)}\right) \cdot \cos y + 1\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(\frac{1}{2} \cdot 3 + \frac{1}{2} \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)\right)} \cdot \cos y + 1\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(\color{blue}{\frac{3}{2}} + \frac{1}{2} \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)\right) \cdot \cos y + 1\right)} \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(\frac{3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \sqrt{5}\right)\right)}\right) \cdot \cos y + 1\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(\frac{3}{2} - \frac{1}{2} \cdot \sqrt{5}\right)} \cdot \cos y + 1\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\mathsf{fma}\left(\frac{3}{2} - \frac{1}{2} \cdot \sqrt{5}, \cos y, 1\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\color{blue}{\frac{3}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \sqrt{5}\right)\right)}, \cos y, 1\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \sqrt{5}\right)\right) + \frac{3}{2}}, \cos y, 1\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right) + \frac{3}{2}, \cos y, 1\right)\right)} \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\color{blue}{\sqrt{5} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{3}{2}, \cos y, 1\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\sqrt{5} \cdot \color{blue}{\frac{-1}{2}} + \frac{3}{2}, \cos y, 1\right)\right)} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right)}, \cos y, 1\right)\right)} \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{-1}{2}, \frac{3}{2}\right), \cos y, 1\right)\right)} \]

      17. cos-lowering-cos.f64 59.9

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \color{blue}{\cos y}, 1\right)\right)} \]

   10. Simplified 59.9%

       \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, 1\right)}\right)} \]

   if -5.99999999999999947e-4 < y < 4.19999999999999962e-15

   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-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 99.4%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0006:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_1, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos x \cdot t\_2\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0013:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_0 \cdot t\_1, 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= x -0.185)
     (/
      (fma t_0 t_1 2.0)
      (fma 3.0 (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (* (cos x) t_2)) 3.0))
     (if (<= x 0.0013)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -1.5)))
       (/
        (fma 0.3333333333333333 (* t_0 t_1) 0.6666666666666666)
        (fma (cos x) t_2 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(2.0) * fma(cos(x), -0.0625, 0.0625);
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (x <= -0.185) {
		tmp = fma(t_0, t_1, 2.0) / fma(3.0, fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), (cos(x) * t_2)), 3.0);
	} else if (x <= 0.0013) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -1.5));
	} else {
		tmp = fma(0.3333333333333333, (t_0 * t_1), 0.6666666666666666) / fma(cos(x), t_2, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(fma(t_0, t_1, 2.0) / fma(3.0, fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), Float64(cos(x) * t_2)), 3.0));
	elseif (x <= 0.0013)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -1.5)));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(t_0 * t_1), 0.6666666666666666) / fma(cos(x), t_2, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(t$95$0 * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0013], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$0 * t$95$1), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 64.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied egg-rr 64.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right), 3\right)}} \]

   if -0.185 < x < 0.0012999999999999999

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       5. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       6. sqrt-lowering-sqrt.f64 98.3

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   11. Simplified 98.3%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 0.0012999999999999999 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0013:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_1, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(t\_2, \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_0 \cdot t\_1, 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= x -0.185)
     (/
      (fma t_0 t_1 2.0)
      (* 3.0 (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (fma t_2 (cos x) 1.0))))
     (if (<= x 0.001)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -1.5)))
       (/
        (fma 0.3333333333333333 (* t_0 t_1) 0.6666666666666666)
        (fma (cos x) t_2 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(2.0) * fma(cos(x), -0.0625, 0.0625);
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (x <= -0.185) {
		tmp = fma(t_0, t_1, 2.0) / (3.0 * fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(t_2, cos(x), 1.0)));
	} else if (x <= 0.001) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -1.5));
	} else {
		tmp = fma(0.3333333333333333, (t_0 * t_1), 0.6666666666666666) / fma(cos(x), t_2, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(fma(t_0, t_1, 2.0) / Float64(3.0 * fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(t_2, cos(x), 1.0))));
	elseif (x <= 0.001)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -1.5)));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(t_0 * t_1), 0.6666666666666666) / fma(cos(x), t_2, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(t$95$0 * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.001], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$0 * t$95$1), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 64.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 64.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   if -0.185 < x < 1e-3

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       5. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       6. sqrt-lowering-sqrt.f64 98.3

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   11. Simplified 98.3%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 1e-3 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := {\sin x}^{2}\\ t_2 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_2, 2\right)}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0009:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_1 \cdot t\_2, 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (pow (sin x) 2.0))
        (t_2 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))))
   (if (<= x -0.185)
     (/
      (fma t_1 t_2 2.0)
      (* 3.0 (fma t_0 (cos x) (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) 1.0))))
     (if (<= x 0.0009)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -1.5)))
       (/
        (fma 0.3333333333333333 (* t_1 t_2) 0.6666666666666666)
        (fma (cos x) t_0 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = sqrt(2.0) * fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -0.185) {
		tmp = fma(t_1, t_2, 2.0) / (3.0 * fma(t_0, cos(x), fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), 1.0)));
	} else if (x <= 0.0009) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -1.5));
	} else {
		tmp = fma(0.3333333333333333, (t_1 * t_2), 0.6666666666666666) / fma(cos(x), t_0, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(fma(t_1, t_2, 2.0) / Float64(3.0 * fma(t_0, cos(x), fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), 1.0))));
	elseif (x <= 0.0009)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -1.5)));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(t_1 * t_2), 0.6666666666666666) / fma(cos(x), t_0, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(t$95$1 * t$95$2 + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0009], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$1 * t$95$2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 64.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied egg-rr 64.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), 1\right)\right)}} \]

   if -0.185 < x < 8.9999999999999998e-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 x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       5. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       6. sqrt-lowering-sqrt.f64 98.3

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   11. Simplified 98.3%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 8.9999999999999998e-4 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. 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-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0009:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_1, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot t\_2\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.00096:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_2, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_0 \cdot t\_1, 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.185)
     (/
      (fma t_0 t_1 2.0)
      (* 3.0 (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (* (cos y) t_2)) 1.0)))
     (if (<= x 0.00096)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) t_2 (sqrt 5.0)) -1.5)))
       (/
        (fma 0.3333333333333333 (* t_0 t_1) 0.6666666666666666)
        (fma
         (cos x)
         (fma (sqrt 5.0) 0.5 -0.5)
         (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(2.0) * fma(cos(x), -0.0625, 0.0625);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.185) {
		tmp = fma(t_0, t_1, 2.0) / (3.0 * fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), (cos(y) * t_2)), 1.0));
	} else if (x <= 0.00096) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_2, sqrt(5.0)), -1.5));
	} else {
		tmp = fma(0.3333333333333333, (t_0 * t_1), 0.6666666666666666) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(fma(t_0, t_1, 2.0) / Float64(3.0 * fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(cos(y) * t_2)), 1.0)));
	elseif (x <= 0.00096)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_2, sqrt(5.0)), -1.5)));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(t_0 * t_1), 0.6666666666666666) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(t$95$0 * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00096], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$0 * t$95$1), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 64.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 1\right)}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, 1\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + -1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right)}\right), 1\right)} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{\cos y} \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right), 1\right)} \]

      13. sqrt-lowering-sqrt.f64 64.5

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right), 1\right)} \]

   8. Simplified 64.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]

   if -0.185 < x < 9.60000000000000024e-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 x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       5. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       6. sqrt-lowering-sqrt.f64 98.3

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   11. Simplified 98.3%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 9.60000000000000024e-4 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. 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-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   10. Simplified 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.00096:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, t\_0, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot t\_1\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.0006:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), t\_0, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.185)
     (/
      (fma (pow (sin x) 2.0) t_0 2.0)
      (* 3.0 (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (* (cos y) t_1)) 1.0)))
     (if (<= x 0.0006)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) t_1 (sqrt 5.0)) -1.5)))
       (*
        (/
         0.3333333333333333
         (fma
          (fma (sqrt 5.0) -0.5 1.5)
          (cos y)
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
        (fma (- 0.5 (* 0.5 (cos (+ x x)))) t_0 2.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * fma(cos(x), -0.0625, 0.0625);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.185) {
		tmp = fma(pow(sin(x), 2.0), t_0, 2.0) / (3.0 * fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), (cos(y) * t_1)), 1.0));
	} else if (x <= 0.0006) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_1, sqrt(5.0)), -1.5));
	} else {
		tmp = (0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma((0.5 - (0.5 * cos((x + x)))), t_0, 2.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(fma((sin(x) ^ 2.0), t_0, 2.0) / Float64(3.0 * fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(cos(y) * t_1)), 1.0)));
	elseif (x <= 0.0006)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_1, sqrt(5.0)), -1.5)));
	else
		tmp = Float64(Float64(0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), t_0, 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0006], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 64.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 1\right)}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, 1\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + -1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right)}\right), 1\right)} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{\cos y} \cdot \left(3 - \sqrt{5}\right)\right), 1\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right), 1\right)} \]

      13. sqrt-lowering-sqrt.f64 64.5

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right), 1\right)} \]

   8. Simplified 64.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]

   if -0.185 < x < 5.99999999999999947e-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 x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       5. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       6. sqrt-lowering-sqrt.f64 98.3

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   11. Simplified 98.3%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 5.99999999999999947e-4 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. 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-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   9. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.0006:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, t\_0, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot t\_1\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0009:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), t\_0, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.185)
     (/
      (fma (pow (sin x) 2.0) t_0 2.0)
      (fma 1.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (* (cos y) t_1)) 3.0))
     (if (<= x 0.0009)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) t_1 (sqrt 5.0)) -1.5)))
       (*
        (/
         0.3333333333333333
         (fma
          (fma (sqrt 5.0) -0.5 1.5)
          (cos y)
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
        (fma (- 0.5 (* 0.5 (cos (+ x x)))) t_0 2.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * fma(cos(x), -0.0625, 0.0625);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.185) {
		tmp = fma(pow(sin(x), 2.0), t_0, 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) + -1.0), (cos(y) * t_1)), 3.0);
	} else if (x <= 0.0009) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_1, sqrt(5.0)), -1.5));
	} else {
		tmp = (0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma((0.5 - (0.5 * cos((x + x)))), t_0, 2.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.185)
		tmp = Float64(fma((sin(x) ^ 2.0), t_0, 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(cos(y) * t_1)), 3.0));
	elseif (x <= 0.0009)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_1, sqrt(5.0)), -1.5)));
	else
		tmp = Float64(Float64(0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), t_0, 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.185], N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0009], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.185

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 64.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Simplified 64.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]

   if -0.185 < x < 8.9999999999999998e-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 x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      7. cos-lowering-cos.f64 99.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       5. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       6. sqrt-lowering-sqrt.f64 98.3

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   11. Simplified 98.3%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 8.9999999999999998e-4 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. 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-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   9. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0009:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 78.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -0.00185:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, t\_0\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0012:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot t\_1, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -0.00185)
     (/
      (fma (sqrt 2.0) (* (pow (sin x) 2.0) t_1) 2.0)
      (fma 1.5 (fma (+ (sqrt 5.0) -1.0) (cos x) t_0) 3.0))
     (if (<= x 0.0012)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))
          (- (fma (* x x) -0.5 1.0) (cos y))))
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) t_0 (sqrt 5.0)) -1.5)))
       (*
        (/
         0.3333333333333333
         (fma
          (fma (sqrt 5.0) -0.5 1.5)
          (cos y)
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
        (fma (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_1) 2.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -0.00185) {
		tmp = fma(sqrt(2.0), (pow(sin(x), 2.0) * t_1), 2.0) / fma(1.5, fma((sqrt(5.0) + -1.0), cos(x), t_0), 3.0);
	} else if (x <= 0.0012) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))) * (fma((x * x), -0.5, 1.0) - cos(y)))) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_0, sqrt(5.0)), -1.5));
	} else {
		tmp = (0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * t_1), 2.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -0.00185)
		tmp = Float64(fma(sqrt(2.0), Float64((sin(x) ^ 2.0) * t_1), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), t_0), 3.0));
	elseif (x <= 0.0012)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y)))) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_0, sqrt(5.0)), -1.5)));
	else
		tmp = Float64(Float64(0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * t_1), 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -0.00185], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0012], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0018500000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Simplified 63.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot {\sin x}^{2}\right)} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   8. Simplified 62.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   if -0.0018500000000000001 < x < 0.00119999999999999989

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right)} - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      7. cos-lowering-cos.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       5. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

       6. sqrt-lowering-sqrt.f64 98.9

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   11. Simplified 98.9%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 0.00119999999999999989 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   9. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00185:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0012:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 78.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -0.00176:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, t\_0\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot t\_1, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -0.00176)
     (/
      (fma (sqrt 2.0) (* (pow (sin x) 2.0) t_1) 2.0)
      (fma 1.5 (fma (+ (sqrt 5.0) -1.0) (cos x) t_0) 3.0))
     (if (<= x 0.0014)
       (/
        (fma (pow (sin y) 2.0) (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) 2.0)
        (+
         (fma (* x x) (fma (sqrt 5.0) -0.75 0.75) 3.0)
         (fma 1.5 (fma (cos y) t_0 (sqrt 5.0)) -1.5)))
       (*
        (/
         0.3333333333333333
         (fma
          (fma (sqrt 5.0) -0.5 1.5)
          (cos y)
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
        (fma (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_1) 2.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -0.00176) {
		tmp = fma(sqrt(2.0), (pow(sin(x), 2.0) * t_1), 2.0) / fma(1.5, fma((sqrt(5.0) + -1.0), cos(x), t_0), 3.0);
	} else if (x <= 0.0014) {
		tmp = fma(pow(sin(y), 2.0), ((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), 2.0) / (fma((x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_0, sqrt(5.0)), -1.5));
	} else {
		tmp = (0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * t_1), 2.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -0.00176)
		tmp = Float64(fma(sqrt(2.0), Float64((sin(x) ^ 2.0) * t_1), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), t_0), 3.0));
	elseif (x <= 0.0014)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), 2.0) / Float64(fma(Float64(x * x), fma(sqrt(5.0), -0.75, 0.75), 3.0) + fma(1.5, fma(cos(y), t_0, sqrt(5.0)), -1.5)));
	else
		tmp = Float64(Float64(0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * t_1), 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -0.00176], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0014], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision] + 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00176000000000000006

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Simplified 63.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot {\sin x}^{2}\right)} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   8. Simplified 62.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   if -0.00176000000000000006 < x < 0.00139999999999999999

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\sqrt{2} \cdot \frac{-1}{16}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, \frac{-3}{4}, \frac{3}{4}\right), 3\right) + \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-3}{2}\right)} \]

      15. cos-lowering-cos.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   8. Simplified 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)} \]

   if 0.00139999999999999999 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   9. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00176:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 78.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, t\_0\right), 3\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot t\_1, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -4.6e-5)
     (/
      (fma (sqrt 2.0) (* (pow (sin x) 2.0) t_1) 2.0)
      (fma 1.5 (fma (+ (sqrt 5.0) -1.0) (cos x) t_0) 3.0))
     (if (<= x 2e-5)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (pow (sin y) 2.0))) (sqrt 2.0) 2.0)
        (fma 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) -0.5) 3.0))
       (*
        (/
         0.3333333333333333
         (fma
          (fma (sqrt 5.0) -0.5 1.5)
          (cos y)
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
        (fma (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_1) 2.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -4.6e-5) {
		tmp = fma(sqrt(2.0), (pow(sin(x), 2.0) * t_1), 2.0) / fma(1.5, fma((sqrt(5.0) + -1.0), cos(x), t_0), 3.0);
	} else if (x <= 2e-5) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * pow(sin(y), 2.0))), sqrt(2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = (0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * t_1), 2.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -4.6e-5)
		tmp = Float64(fma(sqrt(2.0), Float64((sin(x) ^ 2.0) * t_1), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), t_0), 3.0));
	elseif (x <= 2e-5)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * (sin(y) ^ 2.0))), sqrt(2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0));
	else
		tmp = Float64(Float64(0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * t_1), 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -4.6e-5], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-5], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e-5

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Simplified 63.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot {\sin x}^{2}\right)} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   8. Simplified 62.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   if -4.6e-5 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

      5. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5}} \cdot \frac{1}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y \cdot 3}, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y} \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)}\right)} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 3 \cdot 1}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{3}\right)} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(3 \cdot \left(\cos y \cdot \left(\frac{3}{2} - \frac{1}{2} \cdot \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}} \]

   if 2.00000000000000016e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   9. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 78.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, t\_0\right), 3\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot t\_1, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -8.5e-6)
     (/
      (fma (sqrt 2.0) (* (pow (sin x) 2.0) t_1) 2.0)
      (fma 1.5 (fma (+ (sqrt 5.0) -1.0) (cos x) t_0) 3.0))
     (if (<= x 2e-5)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (pow (sin y) 2.0))) (sqrt 2.0) 2.0)
        (fma 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) -0.5) 3.0))
       (*
        0.3333333333333333
        (/
         (fma (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_1) 2.0)
         (fma
          (fma (sqrt 5.0) -0.5 1.5)
          (cos y)
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -8.5e-6) {
		tmp = fma(sqrt(2.0), (pow(sin(x), 2.0) * t_1), 2.0) / fma(1.5, fma((sqrt(5.0) + -1.0), cos(x), t_0), 3.0);
	} else if (x <= 2e-5) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * pow(sin(y), 2.0))), sqrt(2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = 0.3333333333333333 * (fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * t_1), 2.0) / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -8.5e-6)
		tmp = Float64(fma(sqrt(2.0), Float64((sin(x) ^ 2.0) * t_1), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), t_0), 3.0));
	elseif (x <= 2e-5)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * (sin(y) ^ 2.0))), sqrt(2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * t_1), 2.0) / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -8.5e-6], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-5], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.4999999999999999e-6

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Simplified 63.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot {\sin x}^{2}\right)} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   8. Simplified 62.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   if -8.4999999999999999e-6 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

      5. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5}} \cdot \frac{1}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y \cdot 3}, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y} \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)}\right)} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 3 \cdot 1}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{3}\right)} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(3 \cdot \left(\cos y \cdot \left(\frac{3}{2} - \frac{1}{2} \cdot \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}} \]

   if 2.00000000000000016e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x \cdot \frac{-1}{16} + \frac{1}{16}\right)\right) + 2}{\cos y \cdot \left(\sqrt{5} \cdot \frac{-1}{2} + \frac{3}{2}\right) + \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + 1\right)}}{3}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x \cdot \frac{-1}{16} + \frac{1}{16}\right)\right) + 2}{\cos y \cdot \left(\sqrt{5} \cdot \frac{-1}{2} + \frac{3}{2}\right) + \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + 1\right)} \cdot \frac{1}{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x \cdot \frac{-1}{16} + \frac{1}{16}\right)\right) + 2}{\cos y \cdot \left(\sqrt{5} \cdot \frac{-1}{2} + \frac{3}{2}\right) + \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + 1\right)} \cdot \frac{1}{3}} \]

   9. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_1 \cdot t\_2, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, t\_0\right), 3\right)}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_1 \cdot \left(\sqrt{2} \cdot t\_2\right), 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -9e-7)
     (/
      (fma (sqrt 2.0) (* t_1 t_2) 2.0)
      (fma 1.5 (fma (+ (sqrt 5.0) -1.0) (cos x) t_0) 3.0))
     (if (<= x 2.15e-5)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (pow (sin y) 2.0))) (sqrt 2.0) 2.0)
        (fma 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) -0.5) 3.0))
       (/
        (fma 0.3333333333333333 (* t_1 (* (sqrt 2.0) t_2)) 0.6666666666666666)
        (fma -0.5 (sqrt 5.0) (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 2.5)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -9e-7) {
		tmp = fma(sqrt(2.0), (t_1 * t_2), 2.0) / fma(1.5, fma((sqrt(5.0) + -1.0), cos(x), t_0), 3.0);
	} else if (x <= 2.15e-5) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * pow(sin(y), 2.0))), sqrt(2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = fma(0.3333333333333333, (t_1 * (sqrt(2.0) * t_2)), 0.6666666666666666) / fma(-0.5, sqrt(5.0), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 2.5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -9e-7)
		tmp = Float64(fma(sqrt(2.0), Float64(t_1 * t_2), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), t_0), 3.0));
	elseif (x <= 2.15e-5)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * (sin(y) ^ 2.0))), sqrt(2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(t_1 * Float64(sqrt(2.0) * t_2)), 0.6666666666666666) / fma(-0.5, sqrt(5.0), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 2.5)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -9e-7], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.15e-5], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.99999999999999959e-7

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Simplified 63.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot {\sin x}^{2}\right)} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   8. Simplified 62.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   if -8.99999999999999959e-7 < x < 2.1500000000000001e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

      5. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5}} \cdot \frac{1}{2}, \cos y \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y \cdot 3}, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y} \cdot 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)}\right)} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \color{blue}{3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 3 \cdot 1}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, 3 \cdot \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{3}\right)} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x, 3\right)\right)} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, 3\right)\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(3 \cdot \left(\cos y \cdot \left(\frac{3}{2} - \frac{1}{2} \cdot \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}} \]

   if 2.1500000000000001e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{\frac{5}{2} + \left(\frac{-1}{2} \cdot \sqrt{5} + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Simplified 61.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 33: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_1 \cdot t\_2, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, t\_0\right), 3\right)}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_1 \cdot \left(\sqrt{2} \cdot t\_2\right), 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -4.2e-6)
     (/
      (fma (sqrt 2.0) (* t_1 t_2) 2.0)
      (fma 1.5 (fma (+ (sqrt 5.0) -1.0) (cos x) t_0) 3.0))
     (if (<= x 3.2e-5)
       (/
        (fma (pow (sin y) 2.0) (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) 2.0)
        (fma 1.5 (fma (cos y) t_0 (sqrt 5.0)) 1.5))
       (/
        (fma 0.3333333333333333 (* t_1 (* (sqrt 2.0) t_2)) 0.6666666666666666)
        (fma -0.5 (sqrt 5.0) (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 2.5)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -4.2e-6) {
		tmp = fma(sqrt(2.0), (t_1 * t_2), 2.0) / fma(1.5, fma((sqrt(5.0) + -1.0), cos(x), t_0), 3.0);
	} else if (x <= 3.2e-5) {
		tmp = fma(pow(sin(y), 2.0), ((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), 2.0) / fma(1.5, fma(cos(y), t_0, sqrt(5.0)), 1.5);
	} else {
		tmp = fma(0.3333333333333333, (t_1 * (sqrt(2.0) * t_2)), 0.6666666666666666) / fma(-0.5, sqrt(5.0), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 2.5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -4.2e-6)
		tmp = Float64(fma(sqrt(2.0), Float64(t_1 * t_2), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), t_0), 3.0));
	elseif (x <= 3.2e-5)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), 2.0) / fma(1.5, fma(cos(y), t_0, sqrt(5.0)), 1.5));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(t_1 * Float64(sqrt(2.0) * t_2)), 0.6666666666666666) / fma(-0.5, sqrt(5.0), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 2.5)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -4.2e-6], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e-5], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.1999999999999996e-6

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Simplified 63.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot {\sin x}^{2}\right)} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   8. Simplified 62.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   if -4.1999999999999996e-6 < x < 3.19999999999999986e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} + \frac{3}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} + \frac{3}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 1.5\right)}} \]

   if 3.19999999999999986e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   7. Applied egg-rr 62.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{\frac{5}{2} + \left(\frac{-1}{2} \cdot \sqrt{5} + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Simplified 61.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 34: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, t\_0\right), 3\right)}\\ \mathbf{if}\;x \leq -3.35 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (sqrt 2.0)
           (* (pow (sin x) 2.0) (fma (cos x) -0.0625 0.0625))
           2.0)
          (fma 1.5 (fma (+ (sqrt 5.0) -1.0) (cos x) t_0) 3.0))))
   (if (<= x -3.35e-6)
     t_1
     (if (<= x 6.6e-5)
       (/
        (fma (pow (sin y) 2.0) (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) 2.0)
        (fma 1.5 (fma (cos y) t_0 (sqrt 5.0)) 1.5))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(sqrt(2.0), (pow(sin(x), 2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / fma(1.5, fma((sqrt(5.0) + -1.0), cos(x), t_0), 3.0);
	double tmp;
	if (x <= -3.35e-6) {
		tmp = t_1;
	} else if (x <= 6.6e-5) {
		tmp = fma(pow(sin(y), 2.0), ((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), 2.0) / fma(1.5, fma(cos(y), t_0, sqrt(5.0)), 1.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(sqrt(2.0), Float64((sin(x) ^ 2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), t_0), 3.0))
	tmp = 0.0
	if (x <= -3.35e-6)
		tmp = t_1;
	elseif (x <= 6.6e-5)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), 2.0) / fma(1.5, fma(cos(y), t_0, sqrt(5.0)), 1.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.35e-6], t$95$1, If[LessEqual[x, 6.6e-5], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.35e-6 or 6.6000000000000005e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Simplified 62.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot {\sin x}^{2}\right)} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   8. Simplified 62.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \]

   if -3.35e-6 < x < 6.6000000000000005e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} + \frac{3}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} + \frac{3}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin y}^{2}, \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 1.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 35: 45.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   3.0
   (+
    (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
    (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))

C

double code(double x, double y) {
	return 2.0 / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
end function

Java

public static double code(double x, double y) {
	return 2.0 / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
}

Python

def code(x, y):
	return 2.0 / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))

Julia

function code(x, y)
	return Float64(2.0 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
end

Wolfram

code[x_, y_] := N[(2.0 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\cos x} + \left(\mathsf{neg}\left(1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. metadata-eval 63.7

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + \color{blue}{-1}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 63.7%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x + -1\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

8. Simplified 43.9%

   \[\leadsto \frac{\color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Final simplification 43.9%

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \]
10. Add Preprocessing

Alternative 36: 45.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   3.0
   (fma
    (cos y)
    (fma (sqrt 5.0) -0.5 1.5)
    (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))))

C

double code(double x, double y) {
	return 2.0 / (3.0 * fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(3.0 * fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(3.0 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 63.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

7. Applied egg-rr 63.5%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right)\right) \cdot 3} \]
9. Step-by-step derivation

10. Simplified 43.9%

    \[\leadsto \frac{\color{blue}{2}}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3} \]

11. Final simplification 43.9%

    \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \]
12. Add Preprocessing

Alternative 37: 42.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{\mathsf{fma}\left(\sqrt{5}, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (fma (sqrt 5.0) 0.5 (fma (fma -0.5 (sqrt 5.0) 1.5) (cos y) 0.5))))

C

double code(double x, double y) {
	return 0.6666666666666666 / fma(sqrt(5.0), 0.5, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 0.5));
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / fma(sqrt(5.0), 0.5, fma(fma(-0.5, sqrt(5.0), 1.5), cos(y), 0.5)))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 63.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

7. Applied egg-rr 63.5%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right) + \frac{1}{2}}} \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \sqrt{5} + \left(\cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right) + \frac{1}{2}\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(\cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right) + \frac{1}{2}\right)} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right) + \frac{1}{2}\right)}} \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right) + \frac{1}{2}\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right) \cdot \cos y} + \frac{1}{2}\right)} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\mathsf{fma}\left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}, \cos y, \frac{1}{2}\right)}\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \sqrt{5} + \frac{3}{2}}, \cos y, \frac{1}{2}\right)\right)} \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \sqrt{5}, \frac{3}{2}\right)}, \cos y, \frac{1}{2}\right)\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\sqrt{5}}, \frac{3}{2}\right), \cos y, \frac{1}{2}\right)\right)} \]

   12. cos-lowering-cos.f64 40.4

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(\sqrt{5}, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \color{blue}{\cos y}, 0.5\right)\right)} \]

10. Simplified 40.4%

    \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(\sqrt{5}, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, 0.5\right)\right)}} \]
11. Add Preprocessing

Alternative 38: 41.0% accurate, 940.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
4. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(3 \cdot 1 + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\color{blue}{3} + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3\right) + 3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

5. Simplified 47.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 3\right) + \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{6 + {x}^{2} \cdot \left(\frac{3}{4} + \frac{-3}{4} \cdot \sqrt{5}\right)}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{6 + {x}^{2} \cdot \left(\frac{3}{4} + \frac{-3}{4} \cdot \sqrt{5}\right)}} \]

8. Simplified 29.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right), 6\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3}} \]
10. Step-by-step derivation

11. Simplified 38.7%

    \[\leadsto \color{blue}{0.3333333333333333} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))))